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=
AiVp eospV
We now discuss two cases. (A) Let Mp = 0, then it can be seen from (2.34) that 77+ is the creation operator of elementary excitation and its energy is given by
g'p = yje% + 4e,4  2a.
(2.37)
Using this, we can obtain the following from (2.35) and (2.36)
1+
d
1+
<"'H( t) ™ « H (  f ) 
(238)

From (2.34), we know that £+ is not a creation operator of the elementary excitation. Thus, another transformation must be made BP = XPCP + »PCP+,
XPI 2

IMPI2
= I
(239)
We can then prove that [BP,H] = EPBP,
(2.40)
44
Quantum Mechanics in Nonlinear Systems
where
Now, inserting (2.32), (2.39), (2.40) and Mp = 0 into (2.31), and after some reorganizations, we have n = U + E0 + ^2 [EP(B+BP + JB+pB_p) + gp(r,+r,p + r,+pr,_p)]
(2.41)
p>0
where
Eo =  2 ] [ » P  2 = J2(9'pEP). p>0
(2.42)
p>0
Both U and £ 0 are now independent of the creation and annihilation operators of the Bosons. U + Eo gives the energy of the ground state. iV0 can be determined from the condition 8(U + EQ) _ ~U SN0 which gives
f  3  l^J.
(,43,
This is the condensed density of the ground state <£0 Prom (2.36), (2.37) and (2.40), we can get
Ep = y ^  a.
g'p = ^Jela,
(2.44)
These correspond to the energy spectra of 77+ and B+, respectively, and they are similar to the energy spectra of the Cooper pair and phonon in the BCS theory. Substituting (2.44) into (2.38), we have (2.45)
(B) In the case of Mp — 0, a similar approach can be used to get the energy spectrum corresponding to £/ as
EP = yje\ + a while that corresponding to A+ = XpVp + PPVP is £
9P = \l
l + a>
45
Macroscopic Quantum Effects and Motions of QuasiParticles
where oil
\
2e% + a
1 /
2el + a
\
(2.46)
From quantum statistical physics we know that the occupation number of the level with the energy ep for a system in thermal equilibrium at temperature T (^ 0) is given by N
'
= {b
>»} = e^IKBT
_
l
(247)
where (• • •) denotes Gibbs average, defined as SP[e«/** T ] Here SP denotes the trace in Gibbs statistics. At low temperature, or T > 0 K, the majority of the Bosons or Cooper pairs in a superconductor condense to the ground state with p = 0. Therefore, (b+b0) « No, where iVo is the total number of Bosons or Cooper pairs in the system and iV0 ^> 1, i.e. {b+b) = 1 « (b+b0). As can be seen from (2.29) and (2.30), the number of particles is extremely large when they lie in the condensed state, that is (bo + b+). ^0 =
(2.48)
Because 7o<£o) = 0 and A>o) = 0, 60 and 6j can be taken to be S/NQ. The average value of 4>*(j> in the ground state then becomes <<»o^ = fa»o = r  ^ = • 4iV0 = ^ .
(2.49)
Substituting (2.43) into (2.49), we can get
or
which is the ground state of the condensed phase or the superconducting phase that we have known. Thus, the density of states, NQ/V, of the condensed phase or
46
Quantum Mechanics in Nonlinear Systems
the superconducting phase formed after the Bose condensation coincides with the average value of the Boson's (or Copper pair's) field in the ground state. We can then conclude that the macroscopic quantum state or the superconducting ground state formed after the spontaneous symmetry breakdown is indeed a BoseEinstein condensed state. In the last few decades, BoseEinstein condensation was observed in a series of remarkable experiments for weakly interacting atomic gases, such as vapors of rabidium, sodium lithium, or hydrogen. Its formation and properties have been extensively studied. These studies show that the BoseEinstein condensation is a nonlinear phenomenon, analogous to nonlinear optics, the state is coherent, and can be described by the following nonlinear Schrodinger equation or the GrossPitaerskii equation (2.50)
where t' = t/h, x' = xy/2m/h. This equation was used to discuss realization of BoseEinstein condensation in the d+l dimensions (d = 1,2,3) by Bullough et al. The corresponding Hamillonian of a condensate system was given by Elyutin et al. as follows
n=^
+ v(x'M2lx\
(2.5i)
where the nonlinearity parameter A is defined as A — 2Naai/al, with N being the number of particles trapped in the condensed state, a the ground state scattering length, ao and o; the transverse (y,z) and the longitudinal (a;) condensate sizes (without selfinteraction), respectively. (Integrations over y and z have been carried out in obtaining the above equation). A is positive for condensation with selfattraction (negative scattering length). The coherent regime was observed in BoseEinstein condensation in lithium. The specific form of the trapping potential V(x') trapping potential depends on the details of the experimental setup. Work on BoseEinstein condensation based on the above model Hamiltonian were carried out and are reported by Barenghi et al. (2001). It is not surprising to see that (2.50) is exactly the same as (2.17), and (2.51) is the same as (2.1). This further confirms the correctness of the above theory for BoseEinstein condensation. As a matter of fact, immediately after the first experimental observation of this condensation phenomenon it was realized that coherent dynamics of the condensated macroscopic wave function can lead to the formation of nonlinear solitary waves. For example, selflocalized bright, dark and vortex solitons, formed by increased (bright) or decreased (dark or vortex) probability density, respectively, were experimentally observed, particularly for the vortex soliton which has the same form as the vortex lines found in type IIsuperconductors and superfluids. These experimental results were in concordance with the results of the above theory. In the
Macroscopic Quantum Effects and Motions of QuasiParticles
47
following sections, we will study the soliton motions of quasiparticles in macroscopic quantum systems. We will see that the dynamic equations in MQS have such soliton solutions. From the above discussion we clearly understand the nature and characteristics of an MQS such as a superconducting system. It would be interesting if a comparison is made between the macroscopic quantum effect and the microscopic quantum effect which is well known. Here we give a summary of the main differences between them. (1) Concerning the origins of these quantum effects, the microscopic quantum effect is produced when microscopic particles which have the particlewave duality are confined in a finite space, while the macroscopic quantum effect is due to the collective motion of the microscopic particles in systems with nonlinear interaction. It occurs through secondorder phase transition following the spontaneous breakdown of symmetry of the systems. (2) From the point of view of their characteristics, the microscopic quantum effect is characterized by quantization of physical quantities such as energy, momentum, angular momentum, etc. of the microscopic particles. On the other hand, the macroscopic quantum effect is represented by discontinuities in macroscopic quantities such as resistance, magnetic flux, vortex lines, voltage, etc.. The macroscopic quantum effects can be directly observed in experiments, while the microscopic quantum effects can only be inferred from other effects related to them. (3) The macroscopic quantum state is a condensed and coherent state, but the microscopic quantum effect typically occurs in bound states. Certain quantization condition is satisfied for a microscopic quantum effect and the particles involved can be either Bosons or Fermions. But so far only Bosons or combination of Fermions are found in macroscopic quantum effects. (4) The microscopic quantum effect is a linear effect and motions of particles are described by linear differential equations, such as the Schrodinger equation, the Dirac equation, and the KleinGordon equations. On the other hand, the macroscopic quantum effect is a nonlinear effect and the motions of the particles are described by nonlinear partial differential equations such as the nonlinear Schrodinger equation (2.17). Thus, the fundamental nature and characteristics of the macroscopic effect are different from those of the microscopic quantum effects. For these reasons, a different approach must be used to describe the macroscopic quatum effect. Also based on these discussions, it is clear that a nonlinear quantum theory would be necessary to describe these macroscopic quantum effects. 2.3
Motion of Superconducting Electrons
In this section, we discuss the motions of quasiparticles, such as superconductive electrons or Cooper pairs in superconductors and superfluid helium, in the MQS. The properties and motion of quasiparticles are not only important for understand
48
Quantum Mechanics in Nonlinear Systems
ing the relevant quantum effects, but also provides the basis for establishing the new nonlinear quantum mechanical theory. Prom earlier discussions, we know that the superconductive state is a highly ordered coherent state or a BoseEinstein condensed state formed after a spontaneous breakdown of the symmetry of the system due to the nonlinear interaction in the system. A macroscopic quantum wavefunction, (j), can be used to describe the superconducting electrons (Cooper pairs)
¥>(£,*) =4tan" 1 [e^**)] .
99
Principles and Theories of Nonlinear Quantum Mechanics
The method of canonical quantization is to expand the soliton solution (3.77) in terms of the classical soliton solution
i(r,t) = tpi(r,zi,
°° ^ 1n(t)fi{r,zi,
,zk)+
^
,zk)
n=k+l
(3.78)
where qn{t) satisfies the canonical commutative relation [qi(x,t),qf{y,tj\ = S^x
(3.79)
 y).
The JVcomponent function tp%n satisfies the following constraint,
 ;  ^ g d r = 0.
(3.80)
Under orthogonal condition, we have N
.
5 ] \ipWn,dT = bnn,.
(3.81)
After some derivations, we can get iSkPktj,T + I
Pkdzk22[Ni
+ 2)OJn
= 27rn
'
or /•MT
J
(
1 f
1^^
^ ) + k  2 E
w
1 "1
«W
\dzk=2Tm,
(3.82)
where N[ is an occupation number and N't = 0,1,2, • • •, wn is a single vibrational frequency of static classical soliton solution tp(r, z\, • • • ,zk), under the approximation of small oscillation. If we further assume that \p) is the eigenstate of both energy and momentum, that is P\P)=P\P),
H\p) = E\p),
where E{p) = s/p2 + M2. Then the nonlinear relation of the matrix element of the canonical commutation relation for the quantum field
(3.83)
The above is the collective coordinate method of canonically quantizing the classical soliton solution proposed by Lee, et al. The key in this approach is to find the classical soliton solution. However, this method is only suitable to the case of a nonlinear scalar field
100
Quantum Mechanics in Nonlinear Systems
3.3.4
Nonlinear perturbation
theory
Most dynamic equations in nonlinear quantum mechanics cannot be solved analytically. Therefore perturbative approach is often used to solve these equations, when certain terms are small relatively to the nonlinear energy or the dispersive energy in these equations. In contrast to the linear quantum mechanics, a systematic perturbative approach and a universal formula are impossible to obtain in the nonlinear quantum mechanics due to nonlinear interactions. As a result, different perturbation methods exist in the nonlinear quantum mechanics which will be discussed in Chapter 7 in details. In this section, one of such methods is briefly described. Assume that 4>o (r, t) is the soliton solution of the nonlinear dynamic equation of a unperturbed system. The wave function of the perturbed system can be written as $ =
(3.84)
where e is a small quantity. Substituting (3.84) into the original nonlinear equation, equations of fa, fa, • • • corresponding to different orders of e can be obtained, respectively. Thus fa, fa, • • • can be determined by solving these equations, and 4> is obtained from (3.84). For example, the SineGordon equation:
(3.85)
may be solved using the perturbation method when T and Ax are small. Its solution is given by (x,t) = 4>0(x  vt) + where <j>i(x,t) is a small quantity. Equation for 4>\ (3.86) into (3.85),
fa{x,t) c a n De
fa,tt ~
fa(x,t)
(3.86)
obtained by substituting
= AX+ 2/37rsechx.
(3.87)
using the complete set
{fb(x)Jk(x)} rOO
0 i ( M ) 
dk4>x{k,t)fk(x)
(3.88)
J — OO
where fb(x)
= ^sechx, v2
1 eikx fk(x) = y=^(k
+ itajihx).
(3.89)
Principles and Theories of Nonlinear Quantum Mechanics
101
The solutions can be found and are given by
'fr(M)= U
***(*>*
,
V ^ n  sinh(7TJfc/2) (3.90)
^16(t) = 2v / 2^+^)(le^). Thus, the complete solution is fa (x,t) = 1R sin[2i?(a;  x)]e2iqx2i
(3.91)
where Rt = 2172  y sech (   ) sin x,
9t
= —sech(jSina;.
We have given a very brief account of the four basic components of the nonlinear quantum mechanics. In the following chapters, these basic theories and their applications will be discussed in more details. Using these principles and theories, nonlinear quantum mechanical problems can be studied. But applications to real systems require detailed information of the nonlinear systems which will be discussed in the next section. 3.4
Properties of Nonlinear QuantumMechanical Systems
In the last section we described the fundamental principles of the nonlinear quantum mechanics which are used to describe nonlinear quantum systems. However, what are nonlinearly quantummechanical systems? What properties do these systems have? We will look into these in this section. (1) The nonlinear quantum mechanics describes Hamiltonian systems. That is, behaviors of the systems can be determined by a set of canonical conjugate variables. Using these variables one can determine the Poisson bracket and write the equations in the form of Hamilton's equations. For equation (3.2) with V(f,t) = A((j)) = 0, the variables cj> and 4>* satisfy the Poisson bracket
{^a)(x),rW(y)} where
 iSabS(x  y)
(3.92)
102
Quantum Mechanics in Nonlinear Systems
The corresponding Lagrangian density C associated with (3.2) with A(<j>) = 0 can be written in terms of (z, t) and its conjugate <j>* viewed as independent variables C = j(4>*4>t  H*t)  ^ ( V 0 • V<£*)  V{x)P4>
+ b{
(3.93)
The action of the system can be written as
(3.94)
S{4>,<j>*) = I' [ Cdxdt. Jtb
JD
and its variation for infinitesimal 6
6S = S{cj> + 6
(3.95)
{,*}
can be written as
6s=
f f [^+^v^+i£t6Adxdt+cc
(396)
9V< > Jto JD L d
) denotes the vector with components dC/d{di(j)) (i — 1,2,3). After integrating by parts, we get 4>(t, x) =
0"(x",t") = I <j>(x,t) • e ^l 0, this means that Q
x'dx', 0 as x' » oo. For slowly varying inhomogeneities (in comparison with soliton scale), i.e., Ws 3> L, where L is the inhomogeneity scale, Ws the soliton width, expanding (4.21) into a power series in Ws/L and keeping only the leading term, we can get df ~
v
+ <M7)
»n[s (&)*(£)MK 
A necessary and sufficient condition for a function cj)(x, t) with known values 4>(x, to) and (f>(x, ti) to yield an extremum of the action 5 is that it must satisfy the EulerLagrange equation (3.98)
Equation (3.98) gives the nonlinear Schrodinger equation (3.2) if the Lagrangian (3.93) is used. Therefore, the dynamic equation, or the nonlinear Schrodinger equation, in the nonlinear quantum mechanics can be derived from the EulerLagrange equation, if the Lagrangian function of the system is known. This is different from the linear quantum mechanics, in which a dynamic equation, or the linear Schrodinger equation cannot be obtained from the EulerLagrange equation. This is a unique property of the nonlinear quantum mechanics. The above derivation for the nonlinear Schrodinger equation based on the variational principle is a foundation for other methods such as the "collective coordinates" , the "variational approach", and the "RayleighRitz optimization principle", where a solution is assumed to maintain a prescribed approximate profile (often belltype). Such methods greatly simplify the problem, reducing it to a system of ordinary differential equations for the evolution of a few characteristics of the systems.
103
Principles and Theories of Nonlinear Quantum Mechanics
The Hamiltonian density corresponding to (3.2) or (3.93) can be written as •H = l4(
(3.99)
2m
Introducing the canonical variables 1,,
,,.
dC (3.100)
1 ,,
...
dC
the Hamiltonian density takes the form (3101)
n = Y,PidtQi£ i
and the corresponding variation of the Lagrangian density can be written as
6C
= H ir5qi + ir£~^Vqi)
+
ATSTT*^*)
(3 102)

*? oqi o(Vqi) 0{Otqi) Prom (3.102), the definition of pi, and the EulerLagrange equation %
d(Vqi)
one obtains the variation of the Hamiltonian in the form SH = ^
[(dtqidpi  dtpi6qi)dx.
(3.103)
Thus the Hamilton equations can be derived
dqi_6ji dt  sPi'
dVi_jE_ dt  sqi
[6 im)

or in complex form
*a*=!p
(3105)
This is also interesting. It shows that dynamic equations, such as the nonlinear Schrodinger equation, can also be obtained from the Hamilton equation, if the Hamiltonian of the system is known. Obviously, such method of finding dynamic equations is impossible in the linear quantum mechanics. The EulerLagrange equation and the Hamilton equations are important equations in classical theoretical (analytic) mechanics, they were used to describe motions of classical particles. Now these equations are used to depict motions of microscopic particles in the nonlinear quantum mechanics. This shows clearly the classical nature of microscopic particles in the nonlinear quantum mechanics. From
104
Quantum Mechanics in Nonlinear Systems
these, we obtain new ways of finding the equation of motion of microscopic particles in the nonlinear quantum mechanics. If the Hamiltonian or the Lagrangian of a system is known in the coordinate representation, then we can obtain the equation of motion from the EulerLagrange or Hamilton equations. (2) The nonlinear dynamic equations such as the nonlinear Schrodinger equation can be written either in the Lax form, given in (3.20) and (3.21), or in the Hamilton form of a compatibility condition for overdetermined linear spectral problem. General nonlinear equations with soliton solutions are often expressed as (3.8), where K(<j>) is defined as a nonlinear operator in some suitable function space. If linear operators L and B which depend on solution
(3.106)
where the function f(L) may be chosen according to convenience. In this case we can write the overdetermined system of linear matrix equations as t/y =[/(*',*', A) V, (3.107)
rPt.=V'{x',t',\)rl>, where U and V are 2x2 matrices, t' = t/h, x' = (y/2m/h)x. The compatibility condition of this system is obtained by differentiating the first equation of (3.107) with respect to t' and the second one with respect to x' and then subtracting one from the other
Ut.Vj.,[U,V'] = 0.
(3.108)
We emphasize that the operators U and V depend not only on t' and x' but also on some parameter, A, which is called a spectral parameter. Condition (3.108)
Principles and Theories of Nonlinear Quantum Mechanics
105
must be satisfied by A. In this case the operators U and V have the form
5
+
(1
—[% VV.] U)V T1. (3.109)
+2i\
,
(l + s ) ( l  s 2 )
L(lS)(lS2)
1S21 + s J
V l  S 1S2/
The presence of a continuous timeindependent parameter is a reflection of the fact that the nonlinear Schrodinger equation (3.2) with V(f, t) = A((f>) — 0 describes a Hamiltonian system with a set of infinite number of conservation laws, which will be discussed in chapters 4. In the case of a system with a finite number (N1) of degrees of freedom one can succeed sometimes in finding 2N'firstintegrals of motion between which the Poisson brackets are zero (they are said to be in involution). Such a system is called completely integrable. In such a case, equation (3.108) with (3.109) is referred to as the compatibility condition, and its consequence is the nonlinear Schrodinger equation (3.2) with V(x,t) = A((p) = 0. This means that for each solution,
or pt = Vj,
J=pV0,
(3.110)
and 0t + ^W 2 ^=^=Av^.
(3.111)
Taking p = ^ 2 = y>2 as a density and 6 as an hydrodynamic potential, equations (3.110)  (3.111) with b < 0 can be identified with the equations for an irrotational barotropic gas. Hence, (3.110) is a continuum equation that occurs in macrofluid hydrodynamics. Thus, equation (3.110) is also called the fluiddynamical form of
106
Quantum Mechanics in Nonlinear Systems
the nonlinear Schrodinger equation. This shows the features of classical particles of the microscopic particle described by the nonlinear Schrodinger equation in the nonlinear quantum mechanics. Prom this interpretation of p — ]<$*• we know clearly that the meaning of 2 = <^2 is in truth far from the concept of probability in the linear quantum mechanics. It essentially represents the mass density of the microscopic particles in the nonlinear quantum mechanics. If (3.110) is integrated over x one finds the integral of motion N = J^ p(x, t)dx with j >• 0 when x >• ±oo. N is then the mass of the microscopic particle, and (3.110) is nothing but the mass conservation for the microscopic particle. This discussion enhances our understanding on the meaning of wave function of the microscopic particle, 4>(f, t), in nonlinear quantum mechanics. (4) In some cases, the inverse scattering problem for the operator L, in (3.20), can be solved, and a potential can be obtained from the scattering data. The role of the potential is played through the function <\>. This means that for the equations discussed one can solve the Cauchy problem and the behavior of the integrable system is strictly determinate. The localized solutions of the integrable equations which correspond to the discrete spectrum of the operator L are usually related to the solitons. In the case of the simplest integrable systems the soliton dynamics is trivial. (5) For integrable systems, an exact quantum approach can be developed that allows one to determine the ground state and excitation spectra of the system. This was studied by Faddeev et al. and Thacker et al.. It thus connects classical description and quantum objects with sufficient rigor. For integrable equations, one may construct a perturbation theory and investigate the structural and "initial" stability problems. In the first case the equation itself is weakly perturbed and the perturbation can be nonHamiltonian. In the second case the initial state studied (in particular it may be a onesoliton solution) is weakly perturbed. For Hamiltonian systems with a finite number of degrees of freedom there is a rigorous theory of structural stability. These problems were studied in details in the literature.
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Calogero, F. and Degasperis, A. (1976). Nuovo Cimento 32B 201. Chen, X. R., Gou, Q. Q. and Pang, X. F. (1996). Chin. Phys. Lett. 13 660. Chen, X. R., Gou, Q. Q. and Pang, X. F. (1997). Chin. J. Atom. Mol. Phys. 14 393; Chin. J. Chem. Phys. 10 145. Chen, X. R., Gou, Q. Q. and Pang, X. F. (1998). Acta Phys. Sin. 7 329; J. Sichuan University (nature) Sin. 35 362. Chen, X. R., Gou, Q. Q. and Pang, X. F. (1999). Acta Phys. Sin. 8 1313; Chin. J. Chem. Phys. 11 240; Chin. J. Comput. Phys. 16 346; Commun. Theor. Phys. 31 169. Christ, N. H. (1976). Phys. Rep. C 23 294. Doebner, H. D., et al. (1995). Nonlinear deformed and irreversible quantum systems, World Scientific, Singapore. Doebner, H. D., Manko, V. I. and Scherer, W. (2000). Phys. Lett. A 268 17. Faddeev, L. (1979). JINR D212462, Dubna. Faddeev, L. D. and Takhtajan, L. A. (1987). Hamiltonian methods in the theory of solitons, Springer, Berlin. Friedberg, R., Lee, T. D. and Sirlin, A. (1976). Phys. Rev. D 13 2739. Gisin, N. (1990). Phys. Lett. A 143 1. Gisin, N. and Gisin, B. (1999). Phys. Lett. A 260 323. Gisin, N. and Rigo, M. (1995). J. Phys. A 8 7375. Jordan, T. F. and Sariyianni, Z. E. (1999). Phys. Lett. A 263 263. Karpman, K. I. (1979). Phys. Lett. A 71 163. Konopelchenko, B. G., (1979). Phys. Lett. A 74 189. Konopelchenko, B. G., (1981). Phys. Lett. B 100 254. Konopelchenko, B. G., (1982). Phys. Lett. A 87 445. Konopelchenko, B. G. (1987). Nonlinear integrable equations, Springer, Berlin. Konopelchenko, B. G. (1995). Solitons in multidimensions, World Scientific, Singapore. Lai, Y. and Haus, H. A. (1989). Phys. Rev. A 40 844 and 854. Lee, T. D., et al. (1975). Phys. Rev. D 12 1606. Makhankov, V. G. and Fedyanin, V. K. (1984). Phys. Rep. 104 1. Mielnik, B. (2001). Phys. Lett. A 289 1. Miura, R. M. (1976). Backlund transformations and the inverse scattering method, solitons and their applications, Springer, Berlin. Pang, X. F. and Chen, X. R. (2000). Chin. Phys. 9 108. Pang, X. F. and Chen, X. R. (2001). Commun. Theor. Phys. 35 323. Pang, X. F. and Chen, X. R. (2001). Inter. J. Infr. Mill. Waves 22 291. Pang, X. F. and Chen, X. R. (2001). J. Phys. Chem. Solids 62 793. Pang, X. F. and Chen, X. R. (2002). Commun. Theor. Phys. 37 715. Pang, X. F. and Chen, X. R. (2002). Inter. J. Infr. Mill. Waves 23 375. Pang, X. F. and Chen, X. R. (2002). Phys. Stat. Sol. (b) 229 1397. Pang, Xiaofeng (1983). Physica Bulletin Sin. 5 6. Pang, Xiaofeng (1984). Principle and theory of nonlinear quantum mechanics, Proc. 3rd National Conf. on Quantum Mechanics, p. 27. Pang, Xiaofeng (1985). Chin. J. Potential Science 5 16. Pang, Xiaofeng (1985). J. Low Temp. Phys. 58 334. Pang, Xiaofeng (1985). Problems of nonlinear quantum theory, Sichuan Normal Univ. Press, Chengdu. Pang, Xiaofeng (1990). The elementary principle and theory for nonlinear quantum mechanics, Proc. 4th APPC (Soul Korea) p. 213. Pang, Xiaofeng (1991). The theory of nonlinear quantum mechanics, Proc. ICIPES, Beijing, p. 123.
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Pang, Xiaofeng (1992). Proc. IWPM6 (Shenyang), p. 83. Pang, Xiaofeng (1993). The theory of nonlinear quantum mechanics, in research of new sciences, eds. Lui Hong, Chin. Science and Tech. Press, Beijing, p. 16. Pang, Xiaofeng (1994). Acta Phys. Sin. 43 1987. Pang, Xiaofeng (1994). The theory for nonlinear quantum mechanics, Chongqing Press, Chongqing. Pang, Xiaofeng (1995). Chin. J. Phys. Chem. 12 1062. Pang, Xiaofeng (1995). J. Huanghuai Sin. 11 21. Pang, Xiaofeng (1997). J. Southwest Inst. Nationalities, Sin. 23 418. Pang, Xiaofeng (1998). J. Southwest Inst. Nationalities, Sin. 24 310. Pang, Xiaofeng (2003). Soliton physics, Sichuan Sci. and Tech. Press, Chengdu. Sabatier, P. C. (1990). Inverse method in action, Springer, Berlin. Sulem, C and Sulem, P. L. (1999). The nonlinear Schrodinger equation: selffocusing and wave collapse, SpringerVerlag, Berlin. Toda, M. (1989). Nonlinear waves and solitons, Kluwer Academic Publishers, Dordrecht. Weinberg, S. (1989). Ann. Phys. (N.Y.) 194 336. Weinberg, S. (1989). Phys. Rev. Lett. 62 485. Wright, E. M. (1991). Phys. Rev. A 43 3836. Zakharov, B. E. and Shabat, A. B. (1971). Zh. Eksp. Teor. Fiz. 61 118. Zakharov, B. E. and Shabat, A. B. (1972). Sov. Phys. JETP 34 62. Zakharov, V. and Fadeleev, I. (1971). Funct. Analiz 5 18.
Chapter 4
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics In this chapter, we will look into physical properties of microscopic particles which are described by the nonlinear quantum mechanics. In particular, we focus on the wavecorpuscle duality of microscopic particles and its description in nonlinear quantum mechanics. Whether microscopic particles have wavecorpuscle duality is a key and central issue of quantum mechanics. Unless the nonlinear quantum mechanics can give a correct description for the wavecorpuscle duality of microscopic particles, it cannot be accepted as a proper theory and a different theory from the linear quantum mechanics which only describes the wave feature of microscopic particles properly. Therefore, in this chapter we present an extensive and indepth study on this feature of microscopic particles, from their static state to dynamic properties, based on the fundamental principles of the nonlinear quantum mechanics. Hence, this chapter is the center of this book. In accordance with the significance of the corpuscle feature of microscopic particles, we focus our discussion on the following issues: (1) the mass, momentum and energy of microscopic particles and the corresponding conservation laws; (2) the size, position and velocity of microscopic particles and the laws of motion of microscopic particles in free space, in the presence of an external field, and in a viscous environment; (3) interaction and collision between microscopic particles, or between microscopic particles and other particles or objects; (4) the stability of the microscopic particles under external perturbations; and so on. Prom these discussions, we will see clearly that microscopic particles in the nonlinear quantum mechanics become solitons, which have properties of a classical particle. It can move over a macroscopic distance, retaining its shape, energy, momentum, and remains stable after collisions and interactions with other particles or external fields. We will also examine the wave features of a microscopic particle, including its wavevector, velocity, amplitude, the superposition principle of waves, scattering, diffraction, reflection, transmission and tunneling phenomena. We will see that each microscopic particle has a determinate wave vector, amplitude and width, and obeys the physical laws of scattering, diffraction, reflection, transmission, tunneling, etc. Therefore, the microscopic particles has not only corpuscle but also wave features, i.e., it has evident wavecorpuscle duality in the nonlinear quantum mechanics. 109
110
4.1
Quantum Mechanics in Nonlinear Systems
Invariance and Conservation Laws, Mass, Momentum and Energy of Microscopic Particles in the Nonlinear Quantum Mechanics
We have learned in earlier chapters some conservation laws in a system of microscopic particles which are described by the nonlinear Schrodinger equation in the nonlinear quantum mechanics. In practice, these conservation laws are related to the invariance of the action with respect to groups of transformations through the Noether theorem, which was studied by Gelfand and Fomin (see Sulem and Sulem 1999) and Bluman and Kumei (1989) (See Sulem and Sulem, 1999 and references therein). Therefore, before these conservation laws are given, we first discuss the Noether theorem for the nonlinear Schrodinger equation which was given by Sulem. To simplify the equation, we introduce the following notations: £ = (t, x) — ( l o . l i , • • • ,ia), do = dt,d=
• • ,dd) and $ = ( $ ! , $ 2 ) = (cf>, f).
(do,du
According
to the Lagrangian (3.93) corresponding to the nonlinear Schrodinger equation, the action of the system
5{} = / " f C{
J
now becomes
5 W = f /°°£($,a$K. JD Jx'
(4.1)
Under the transformation T£ which depends on the parameter e, we have £ + £'(£, $, e), $ > $(£,$, e), where  and
JD
/
C(*,d*)d£',
JX'
where d denotes differentiation with respect to £'. The change 5S = S{(p} — S{(j)} in the limit of e under the above transformation can be expressed as
6S =
ID II [ £ ( ^ } " £ ^' 9$) ] d$ + fD /" A*,^) E ^fdt
where we used the Jacobian expansion
d
&>>£d) =1 , y  ^ f r
d(io,,id)
^odiv'
and £(#,5$) in the second term on the righthand side has been replaced by the leading term £($,9$) in the expansion.
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
111
Now define (4.2)
BMi)  9VHO = 0v  dv)*id)+ev[hd)  mi with
* = §*,= (^ + ^ U = a. + ^K We then have
= a*;6**+ww ~c^i+dv [wm s*{ ~d" [m*T)\6**Equation (3.96) can now be replaced by
where we have used
Using the EulerLagrange equation, the first term on the righthand side in the equation of SS vanishes. We can get the Noether theorem. (I) if the action (4.1) is invariant under the infinitesimal transformation of the dependent and independent variables
or
4r I £ * + so^ ( *'^' 6 ' ) 1  0 in terms of 6$i defined above.
(4 3)

112
Quantum Mechanics in Nonlinear Systems
(II) If the action is invariant under the infinitesimal transformation t¥
i=t
+ St(x,t,<j>),
x > x = x 4 Sx(x, t, cf>),
then J
f \^(dt<j>dt [d(t>t
+ Vcf>5x
S<j>) + ^{dt
+ V
dx
is a conserved quantity. For the nonlinear Schrodinger equation in (3.2) with A((j>) — 0, we have dC
=
i
and
Wt 2^'
dC
i
Wt^'^'
where C is given in (3.93). Several conservation laws and invariances can be obtained from the Noether theorem. (1) Invariance under time translation and energy conservation law. The action (4.1) is invariant under the infinitesimal time translation t > t + St with Sx = 6(f> = 8(f>* = 0, then equation (4.3) becomes
dt [V*
f \v
(4.4)
(2) Invariance of phase shift or gauge invariance and mass conservation law. It is very clear that the action related to the nonlinear Schrodinger equation is invariant under the phase shift <j> = elB4>, which for infinitesimal 6 gives 6tfi = iO<j>, with 5t = Sx = 0. In this case, equation (4.3) becomes dt\
h
(4.5)
113
Wave Corpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
where j is the mass current density
J=i(0V0"0*V0. Equation (4.5) is the same as (3.110). (3) Invariance of space translation and momentum conservation law. If the action is invariant under an infinitesimal space translation x t x + 5x with St = 5<j> = 8cf>* = 0, then (4.3) becomes & [ i ( W 
This leads to the conservation of momentum P =i
(V<£*  <j>*V4>)dx = constant.
(4.6)
Note that the center of mass of the microscopic particles is defined by
(x) =
^jx\^dx,
we then have
A T ^ M = f xdt\<j>\2dx =  f xV • [t(0V0* 
= / i(cj)V(t>*  <j>*V
fjdx.
This is the definition of momentum in classical mechanics. It shows clearly that the microscopic particles described by the nonlinear Schrodinger equation has the feature of classical particles. (4) Invariance under space rotation and angular momentum conservation law. If the action (4.1) is invariant under a rotation of angle 58 around an axis / such that St = 8<j> = 6(f>* = 0 and Sx = 561 x x, this leads to the conservation of the angular momentum
M = i I f x (4>*V
114
Quantum Mechanics in Nonlinear Systems
If the action is invariant under the Galilean transformation x » x" = x — vt, *  > * " = t,
St = 0 and 5<j> =
~(i/2)vx(f>(x,t). After integration over the space variables, equation (4.3) leads to the conservation law (4.7) which implies that the velocity of the center of mass of the microscopic particle is a constant. (6) Scale invariance of power law nonlinearities in the nonlinear Schrodinger equation with V(x, t) = A((f>) = 0 If \(j)\2 in the nonlinear term, b\<j>\2()), in the nonlinear Schrodinger equation (3.2) is replaced by 6^>2
X'V'Wx",*2?).
But the action scales as \l2/<'~1ld under the same transformation, where d is the size of the system. In the critical case, a = 2d, the action is invariant under the scale transformation while the nonlinear Schrodinger equation remains invariant. a = 2d also happens to be the critical condition for a singular solution and is usually referred to as the critical dimension. In such a case, the invariance under the scale transformation leads to an additional conservation law. For an infinitesimal transformation, A' = 1 — e, with e < 1, fe = ex, St = 2et and 6(f> = —(e/a)<j>(x,t). Then (4.3) results in f UetjC  ^* (^
where C\ is a constant, E is the energy of the system. (7) Pseudoconformal invariance at critical dimension
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
115
Talanov (see Sulem and Sulem 1999) proved that when the nonlinearity is a power law b\(f>\2(f> at the critical dimension a = 2d, the nonlinear Schrodinger equation has other additional symmetry. This invariance is given here. Obviously, the nonlinear Schrodinger equation (3.2) with V(x, t) = A{
l(t)' d/2
il
il
t*(t*t)' =
Id/2(j,(x,t)eia^2^12
with a
~
dl _t* t dl dt~~ldt'~ t20 '
where to appears as an arbitrary time unit. This shows that there exists certain symmetry in the explicit solution of the nonlinear Schrodinger equation at the critical dimension which is singular at some finite time t*. It is referred to as pseudoconformal invariance of the nonlinear Schrodinger equation at the critical dimension. Expressing " in terms of
H1 = J
[(VT) 2  Y^I
where V" corresponds to x", becomes
H' = j Q/V* + if*2  ~l\
f (\x
dx.
Kuznetsov and Turitsyn (1985) (see Sulem and Sulem 1999) pointed out that the above transformation preserves the action, and the invariance given by the above equation is a result of Noether theorem. This can be seen by considering a transformation close to identity by choosing to = t*, and by assuming that 8A' = 1/t* is infinitesimally small. In such a case Sx = xtSX', St = t26X', and 5<j> = (—dt/2 + i\x\2/A)(j>8X'. Upon evaluation of the space integration, equation (4.3) leads to the conservation of
I [t2 (V«/>2 + ^ J  l ^ + 2 ) + lt2(W 
116
Quantum Mechanics in Nonlinear Systems
Schrodinger equation (in the natural unit system)
where t' = t/h, x' = (V2m/h)x, y' = (y/2rn/ti)y. It is well known from classical physics that the invariance and conservation laws of mass, energy and momentum and angular momentum are some fundamental and universal laws of matters in nature, including classical particles. In this section we demonstrated that the microscopic particles described by the nonlinear Schrodinger equation in the nonlinear quantum mechanics also have such properties. This shows that the microscopic particles in the nonlinear quantum mechanics also have corpuscle feature. Therefore the proposed nonlinear quantum mechanical theory reflects the common rules of motions of matters in nature.
Fig. 4.1 Nontopological solitary wave of nonlinear Schrodinger equation and its frequency spectrum
In the onedimensional case with V(x, t) = A(<j>) = 0, Zakharov et al. found the solution of (3.2) using the inverse scattering method. The solution is a belltype nontopological soliton 4>(x,t) = Aosech  ^ ^
e «..[(..).*]/>»
[(x xo)vet}\
(4.8)
where A
°\l
I{v2e2vcve)m
26
•
Therefore the microscopic particle in nonlinear quantum mechanics is a soliton. Here ip(x,t) = Ao sech.{Aoy/bm[(x  x0)  vet]/h} is the envelope of the soliton, and exp{ivem(x — x0)  vet)]/2h} is its carrier wave. The form of the soliton is shown in Fig. 4.1. The envelope ip(x,t) is a slow varying function and Ao is its amplitude. The width of the soliton is given by W = 2Trh/(y/mbA0). Thus the size of the soliton, A0W = 2nh/Vmb, is a constant. It depends only on the nature of the microscopic particles, but not on its velocity. This shows the corpuscle feature of the microscopic particles. In (4.8), ve is the group velocity of the soliton, and
Wave Corpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
117
vc is the phase speed of the carrier wave. From Fig. 4.1 and (4.8), we see that the carrier wave carries the envelope to propagate in spacetime. The envelope is typical of a solitary wave, its changes in the spacetime are slow and its form and amplitude are invariant in the transport. Thus the frequency spectrum of the envelope has a localized structure around the carrier frequency wo, as shown in Fig. 4.1(b). This figure shows the width of the frequency spectrum of the envelope ip(x,t). For a certain system, ve and vc and the size of the soliton are determinate and do not change with time. This shows that the soliton (4.8) or the microscopic particle moves freely with a uniform velocity in spacetime, and it has certain mass, momentum and energy, which are given by
p= i
((t>*x  (j>(f>*x.) dx' = 2V2Aove
Joo
oo
/
r
= ^y=Nsve 2v2
= const. • ve,
1
hi
\\
= Eo +
Msoiv2e,
where Msoi = Na = 2\/2Ao is the effective mass of the microscopic particles and it is a constant. These results clearly show that the energy, mass and momentum of a microscopic particle are invariant in nonlinear quantum mechanics and they are not dispersed in its free motion. Thus the microscopic particles can move over a macroscopic distance with the group velocity ve and retain its energy and momentum. It shows that the microscopic particle has corpuscle characteristics like a classical particle. 4.2
Position of Microscopic Particles and Law of Motion
Besides mass, momentum and energy of microscopic particles mentioned above, position of a microscopic particle can also be precisely determined in nonlinear quantum mechanics. We know that the solution (4.8) shown in Fig. 4.1, of the nonlinear Schrodinger equation (3.2) with V(x',t') = A((j>) = 0, has the behavior
*+
0 x —x0
Thus we can infer that the position of the soliton is x' = x'o at t' = 0. Next, at \x'\ —> co,
/OO
r
— /
118
Quantum Mechanics in Nonlinear Systems
of definition in (4.7), the center of mass of the microscopic particle at x'o + vet' is given by /•OO
/ (x1) = x'g = ^
/ <j>*<j>dx' Joo
where x' = x/y/h2/2m, t' — t/h. We can thus obtain the law of motion for the center of mass of the microscopic particles described by (3.2) with A((f>) = 0 and its conjugate equation. Under such a condition, we have d
f°°
— at
f°°
J — oo
r°°
J — oo
= i{ I
8
P^tt'dx' "x
Joo
(4.10)
(4>*
(J — oo
°°
f°°
BV
1
/ with /•OO
/
J—oo
/OO
{fj>*<}>x'x'x'  ^I'x'^x^dx'
K'^x'\oo OO
/
= 4>*4>x>x'\°^oo~ I 4>*xi<j>x'x>dx' J—oo
+ f°° Px'
I
pOO
/
QL
(4.11)
\
[b4>*4>2^ + W ) V * ' ] dx =  ^ J W*)2 [j&) ^dx' = °.
where b is constant. Using (4.11) and (4.12), equation (4.10) becomes
*L* t *" f c l ='/.*•»**'•
(412) (4 13)

From (4.7) and (4.9), the acceleration of the center of mass of the microscopic particle can be defined as /•oo
roo
(4.14) J — oo
J—oo
Since /•OO
/ J—OO
(j>*(j>dx' = const,
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
d f°°
— /
r°°
119
f°°
(j)*x'
(fcx'^ + px'^dx1
= 2i /
ffodx'
we can obtain the following from (4.13) and (4.14) (4.15) 7OO
where V = V(x') is the external potential field experienced by the microscopic particle. We now expand dV/dx' in (4.15) around the center of mass x' — (x1) as dV{x') _ dV((x')) dx> ~ d(x>)
^ _ j ^j {
j))
d2V({x')) d{x))d{x'j)
Taking the expectation values on both sides of the above equation, we can get
\dx'/
d3V((x'))
1 ^
ldV\=dV{{x'))
d(x') 2 f c ^
>kdtydtydw
where
Ajlfc = ([(4  (4»(^  (^))]> = (x'A)  (^)<4)For the nontopological soliton solution, (4.8), the position of its center of mass is determined as mentioned above, thus Aj^ = 0. Therefore the above equation becomes /dv(x')\^dv({x')) \ dx' I d(x') ' Thus the acceleration of the center of mass of the microscopic particles in nonlinear quantum mechanics, (4.15), can be expressed as (4.16)
Returning to the original variables, equation (4.16) becomes cP(x)
dV((x))
(4.17)
This is the equation of motion of the center of mass of the microscopic particle in the nonlinear quantum mechanics. It is analogous to the Newton's equation for a classical object. It shows that the acceleration of the center of mass of a microscopic
120
Quantum Mechanics in Nonlinear Systems
particle depends directly on the external potential field. It also states that motion of a microscopic particle in nonlinear quantum mechanics obeys the classical laws. For a free particle, V = 0, thus dP{x)/dt2 = 0. Hence, the center of mass of the microscopic particle moves with a uniform speed, which is consistent with results of earlier discussion. We can find from the definition (4.9) that the speed is the group velocity of the microscopic particle (soliton), ve in (4.8). From (4.9) and (4.13), the velocity of the center of mass of the microscopic particle can be expressed as
M
I
I / PW*'
(4.18)
We have used the following, — / (f>*x
J — OO
/•OO
= t/ = 2t f lim
x'+oo
{tf>V [4>x,t, + bcj>*4>2  Vd>] 
Joo (t>*
cc'>oo
lim
a:'foo
>x =
lim
x'>oo
4>*x,x'<j) = 0,
/•CO
/
cj)*(j)dx' = const.
and the boundary condition
= ve = const.
(4.19)
which is the same as the results obtained from (4.16) above. It shows clearly that for a microscopic particle described by the nonlinear Schrodinger equation with V = 0, not only its velocity of the center of mass is the group velocity of the soliton, but it also moves with a uniform speed ve in the spacetime. The above equation of motion of microscopic particles can also be derived from the nonlinear Schrodinger equation (3.2) with A(
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
121
A(<j>) = 0 takes t h e form of
E=
Vx>
L
P =i
r°°
l^*®2+w&w*]dx>'
(4.20)
(
J — oo
For this system the energy E and quantum number Ns = J^ {^dx' are integral invariant. However, the momentum P is not conserved and has the following property (4.21)
where the boundary condition is
2
dx'o
(4 22j
Ns

where x'o is the position of the center of mass of the microscopic particle. Equation (4.21) or (4.22) is essentially consistent with (4.17) which is in the form of the equation of motion for a classical particle. Indeed, if we write the onesoli ton solution as
(4.23)
It is easy to verify that (4.23) is a solution of (3.2) with A(
P=pNs.
Let p = dx'0/dt' for the center of mass of the particle. Then we can obtain from (4.22) and (4.24) that
^
dt12
= J^., o, J£—g. dx'o
dt2
(4.25)
oxo
This is the same as (4.16) or (4.17) and it is the Newton's equation for a classical particle. Equations (4.22) and (4.25) indicate that the center of mass of the microscopic particle moves like a classical particle in a weakly inhomogeneous potential field V(x'Q). Let the function
in the following Df + 2dx'Dx,  pDx, sin2i?  ptix,Dcos2d + Ddx>x. = 0, (4.135)  D i V  D + Dx>x>  Dti2x, + D3 + 0DX> cos2t?  p&x,Dsm2d
s[x  xo(t)] +g^an{t)$[x
t\
Multiplying the first equation by D, we get dfD2 + dX'[{2tix,  Psin2i9)D2] = 0. It is easy to find a solution for i?^ + 0. In that case, Dv = PDX sin2i9; Di?t< D
+ Dx,x> +D3 + &DX, cos 2i? = 0.
Thus D is a function of z = x' +/3
ft' Jo
sin2i?(r)dT
and a solution for $ may be found in the form of i?t, = u + Pf(z) cos 2tf.
= 0.
168
Quantum Mechanics in Nonlinear Systems
Here the amplitude D satisfies the nonlinear Schrodinger equation (w  1)D + Dzz + D3 = 0, if the condition f{z) = 92(lnD) is fulfilled. Makhankov et al. finally obtained the solution
D(z) = /.osech O^j , where tio =
y/2(lu),
or
4> = Mosech te ^ + ^ cos 2t'^ 1
(4.136)
H#4)''+f^(^)H} which holds when /?//o
/3sin2i? J = const.
Because of the boundary conditions at infinity the constant must vanish. Therefore, the phase •d satifies the stationary SineGordon equation B2
dxixi = — sin4i9,
which has the solution
tf^tanife^'^j+tfi, where tf+= n+ 7r>
( )
d =nn

'
(» = 0 , ± l ,  " ) 
In this case the second equation in (4.135) is reduced to Dx.x> D + D3  /32D sin2 2d  $DX, cos 2t? = 0. 4
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
169
Here the sign of the last term is determined by the phase behavior at co. Fixing t?+ or i?_ as an "initial" phase, we can get Dx,x,
D + D3  /32£>sin2<x ± /3DX, COS2CT = 0.
Hence for /? C 1 we get Dx.x.
D + D3 ±/3Dtanhfix' = 0.
(4.137)
In the first case, the existence of a solution is proved. Equation (4.137) has a countable set of solitonlike solutions Dn, one of which, Do, is nodeless, and the other ones have an increasing number of nodes n. The above results show that the influence of the small term /?*, can lead to interesting results. This means that the structural stability of the nonlinear quantum mechanical systems is a rather delicate problem which must be solved either for a specific system or for a restricted class of systems. 4.8
Demonstration on Stability of Microscopic Particles
In addition to the initial and strutural stabilities discussed above, we can also demonstrate the stability of the microscopic particles in nonlinear quantum mechanics by means of the minimum energy concept. A system of microscopic particles is stable, when the particles are located in a finite range which results in the lowest potential energy. This stability principle is very effective, when the microscopic particles are in an extenally applied field. As a matter of fact, since interaction between the microscopic particles is very complicated, it is not easy to define the behavior of each one individually. We cannot use the same strategies as those used in the discussions of initial and structural stabilities in this case. Instead, we apply the fundamental workenergy theorem of classical physics: a mechanical system in the state of minimal energy is said to be stable, and in order to change this state, external energy must be supplied. Pang applied this fundamental concept to demonstrate the stability of microscopic particles described by the nonlinear quantum mechanics, which is outlined in the following. Let (f>(x, t) represent the field of the particle, and assume that it has derivatives of all orders, and all integrations of it are convergent and finite. The Lagrangian density function corresponding to the nonlinear Schrodinger equation (3.2) with A(<j>) = 0 is given in (3.93). The momentum density of this field is defined as P = dC/d
Eiti)=
IZ \^]\'l^+vix'M2
dx>
 (4i38)
170
Quantum Mechanics in Nonlinear Systems
However, in this case, b and V(x') are not functions of t'. So, the total energy of the system is a conservative quantity, i.e., E(t') = E = const., as given in (4.4). We can demonstrate that when x' > ±00, the solutions of (3.2) with A((p) = 0, and <j>(x',t') = 0 approach zero rapidly, i.e. lim 4>{x',t')= lim  ^ = 0 .
z'>oo
z>oo dx'
Then /•OO
/
J00
lim * v £ * = lim 2£*V = 0.
i'too
dx'
x'».oo 9a;'
The average position or the center of mass of the field (p can be represented as (x1) = x'g, as given in (4.9). Making use of (4.13), the average velocity or the velocity of the center of mass of the field can be written as (4.18). However, for different solutions of the same nonlinear Schrodinger equation (3.2), JLQQ
[°°
J00
^
a i
=«(«{,).
(4.139)
t'=t'o
Now we assume that the solution of nonlinear Schrodinger equation (3.2) with A(<j>) = 0 is of the form (j)(x',t')
= tp(x',t')ei$ix''t').
(4.140)
Substituting (4.140) into (4.138), we obtain the energy of the system
E
11 [{&)'+^ (£)'  b v ' + " H *"• ("41>
Equations (4.139) then become oo
/
xVdx' 1
roo
an
2 J^dx<  ^ = u(t0). / <^dx' J— OO
(4.142)
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
171
Finding the extreme value of the functional (4.141) under the boundary conditions (4.142) by means of the Lagrange multiplier method, we obtain the following EulerLagrange equation,
0P = \v(X')+dtoc^a*'
 «(*{,)]
+C3(t'o) [ 2 ^  «(f0)] + [^j y  6
(4.143) (4,44)
where the Lagrange factors C\, Ci and C3 are all functions of t'. Let C5(t'o)
=\u(t'o).
If rift
we can get from (4.144)
d2e
2 dip _
~9^2
Integration of the above equation yields * = d6
1
ft?  2
U
'
^
or S«
_ £(
a * * = * " ^2
2
'
f4145)
where g(t'o) is an integral constant. Thus,
6(x',t') = 9(t'o) [X%
+ ^x'
z Jo V Here M(t'o) is also an integral constant. Let again
C2G0) = ^(^o)
+ M(t'o).
(4.146)
( 4  147 )
172
Quantum Mechanics in Nonlinear Systems
Substituting (4.145)  (4.147) into (4.143), we obtain
bv3 + ^p.
(4.148)
Letting Cl{Q
=
U
^f^^
+
(4.149)
M(t'o) + p',
where /3' is an undetermined constant, which is i'independent, and assuming Z = x' — u(t'o), then
ay 2 _ ay 2
d(x>) ~ dz
is only a function of Z. In order for the righthand side of Eq. (4.149) to be a function of Z, the coefficients of y>, ip3 and 1/ip3 must also be funciton of Z. Thus, 9(t'o) = 9o = const., and V(x>) + J^x'
+ M(t'0)^.
= V0(Z).
Equation (4.149) then becomes
•0p
= {VW  u(t'o)} + / ? >  bV3 +
9
^.
Since V0(Z) = V0[x'  u(t'o)] = 0 in the present case, (4.150) becomes
g.^v+m.
(4.150)
(4.I5I)
Therefore, ip is the solution of (4.151) for /?' = constant and g(t'o) = constant. For sufficiently large \Z\, we may assume that \ip\ < /3/Z 1 + A , where A is a small constant. However, in (4.151) we can only retain the solution
As a matter of fact, if dO/dt' = ii/2, then from (4.149) and (4.150), we can verify that the solution given in (4.8) satisfies (4.152). In such a case, it is not difficult to show that the energy corresponding to the solution (4.8) of (4.152) has a miminal value under the boundary conditions given in (4.142). Thus we can conclude that the soliton solution of nonlinear Schrodinger equation, or the microscopic particle in nonlinear quantum mechanics is stable in such a case.
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WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
4.9
MultiParticle Collision and Stability in Nonlinear Quantum Mechanics
One of the most remarkable properties of microscopic particles (solitons) in nonlinear quantum mechanics is that elastic collision between them results only in a phase shift, and the phase shift due to the collision with several microscopic particles (solitons) is the sum of partial phase shifts resulting from separate collisions between a given particle and each other microscopic particle (soliton), as discussed in the previous sections. Their stabilities have a similar property. This property is commonly referred to as absence of multisoliton (or "manyparticle") effects in the integrable models. A natural question arises: what about multimicroscopic particle (soliton) collisions in the nonintegrable nonlinear quantum mechanical models? It is commonly believed that the main difference between an integrable model and a nonintegrable model is due to radiation emitted by the interacting microscopic particles (solitons). Prauenkron et al. studied the properties of multimicroscopic particle collisions and demonstrated the existence of nontrivial effects, which do not involve radiation and exist for any value of the perturbation parameter e in multimicroscopic particle collisions, by extended numerical simulations based on a simple integration scheme. These effects are due to energy exchange between the colliding microscopic particles (solitons) and excitation of internal soliton modes which distinguishes multimicroscopic particle collisions in the integrable and the nonintegrable models. To understand the multiparticle effects in multimicroscopic particle collisions in nonlinear quantum mechanics systems, we consider a perturbed nonlinear Schrodinger equation with a small quintic nonlinearity,
«£ + 0 + 2M2tf = <#V
(4153)
where e is the amplitude of the perturbation which is assumed to be small. Equation (4.153) can be used to describe evolution of the electric field of a light in an optical waveguide with an intensitydependent refractive index nni (I = \cj>\2), which slightly deviates from the Kerr dependence. Its solutions will be given in Chapter 8. In the absence of the perturbation (e = 0), the nonlinear Schrodinger equation (4.153) is known to be exactly integrable and it supports propagation of an microscopic particle (envelope soliton) with amplitude Ao and velocity v, as given in (4.8). Three elementary integrals are the norm N, field momentum P and energy E, represented here by iVs = 2A0,
Ps = Ao,
Es = ^Aov2^A3Q
(4.154)
respectively. In addition, equation (4.153) possesses an infinite number of integrals of motion. Unlike higher (nonelementary) integrals of motion, the above three basic quantities remain conserved in the case of perturbed nonlinear Schrodinger equation,
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Quantum Mechanics in Nonlinear Systems
(4.153). As a matter of fact, the effect of conservative perturbation in (4.153) on the microscopic particles in nonlinear quantum mechanics is trivial. This is consistent with the fact that a perturbed equation has an exact solitary wave solution which is a slightly modified nonlinear Schrodinger soliton. However, interaction of these solitary waves for (4.153) differs drastically from the interaction of two solitons described by the integrable nonlinear Schrodinger equation. To study multimicroscopic particle collisions in nonlinear quantum mechanics, Frauenkron et al. integrated (4.153) using the fourthorder symplectic integrator. The symplectic numerical method is much better than traditional numerical schemes, because it allows conservation of the norm within a relative accuracy of 10~ n and conservation of the energy within 10~6 during the integration. They used a grid spacing of dx' = 0.1, with the total length L = [—800,800], and a time step dt' = 0.005. They studied collisions of three microscopic particles (solitons) and two microscopic particles (solitons), respectively, in nonlinear quantum mechanics. In the case of three microscopic particles, a "fast" particle, with a relatively large velocity, was taken as the exact solution of (4.153) to avoid initial oscillation of its amplitude. The other two "slow" microscopic particles were modeled by an exact twosoliton solution of the unperturbed nonlinear Schrodinger equation to avoid radiation due to strong initial overlapping. The microscopic particles were put on the grid at positions a;' = —650 and i a ; ^ = ±45 with initial amplitudes Aj = l/\/2, As\ = 0.35 and velocities Vj = 3.0 and vs\ = ±0.2. These values were selected in such a way that all three microscopic particles collide when the two slow particles are overlapping significantly (see Fig. 4.10).
Fig. 4.10 Triple microscopic particle collision in nonlinear quantum mechanics. The fast microscopic particle with amplitude Aj and velocity Vj collides with two slow microscopic particles of equal amplitudes Asi propagating towards each other with velocities ±vsi, so that the three microscopic particles significantly overlap at the moment of collision.
Frauenkron et al. first studied collision of twomicroscopic particles in nonlinear quantum mechanics. Simulations of the collision were performed using the same
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
175
initial conditions as mentioned above, but with only one slow microscopic particle (with negative velocity). For e < 0.1, the changes in velocities of the microscopic particles after the collision were so small that they could not be measured with a sufficient resolution, and they were expected to be on the order of the numerical error. To understand this, we note that for collision of two microscopic particles, inelastic effects may exist due to emission of radiation. The emitted energy Erad has been calculated analytically in the limit of two symmetric microscopic particles with equal amplitudes A\ = A2 — A and velocities i>i]2 = ±v, where v > A. The result is £racj = C'e2A7[l + F(v/A)}, where C" is a constant. Similarly, it was found that the radiationinduced change of the norm is iVra(i = {A/v2)ET&d. Because of the symmetry, the total momentum of the two microscopic particles was not affected by the radiation. Due to conservations of the norm and the total energy, we have NS = N'S + JVrad
and
Es = E's + £ r a d ,
where N's and E's are given in (4.154). The lefthand sides of the above equations pertain to the microscopic particles before the collisions, whereas the righthand sides take into account the changes due to the radiation emitted. These two balancing equations for the conserved quantities allow us to determine the changes in the amplitudes AA and velocities Av of the microscopic particles, and they are given by •I
AA0 =  j £ r a d ,
nA
and Av = —£Erad.
This shows that the changes in the microscopic particles' parameters are proportional to e2 and inversely proportional to v2. For collision between two microscopic particles with large velocities v, the interaction time is small and therefore the changes in the microscopic particles' parameters are smaller. For two different microscopic particles these changes will be further reduced due to shorter time of overlapping between the microscopic particles. Unlike the case of two microscopic particles in nonlinear quantum mechanics, Frauenkron et al. obtained nontrivial effects which are in the first order of e, for the collision of three microscopic particles. Figures 4.11 show the changes in the velocities of the microscopic particles after the collision for different values of the perturbation amplitude e. It is clear that the change is linear in e. After the collision small oscillations in the amplitudes of the microscopic particles were observed, which is more prominent for larger e. It is obvious that these effects are different in collision of two microscopic particles and that of three microscopic particles. The results also show a nontrivial energy exchange in the first order of the perturbation amplitude e. To find analytical results, Frauenkron et al. considered a collision between a fast microscopic particle with amplitude Aj and velocity Vj, and a symmetric pair of "slow" microscopic particles with equal amplitudes Asi and opposite velocities ±v s i. Assuming Vj 3> vsi ^> Asi, they presented a threesoliton
176
Quantum Mechanics in Nonlinear Systems
Fig. 4.11 Velocities of the microscopic particles after collision between three microscopic particles, for different values of e: (a) the fast particle with Vj = 3.0; (b) one of the slow particle with D S1 = 0.2. Open squares and circles are numerical data. Dashed lines are fitted linear functions.
solution in the form of <j) = (/>j(vj) + <j>ai(vsi) + ^si(vsi), and calculate the changes in the microscopic particles' parameters by means of a perturbation theory based on the inverse scattering method, with the onesoliton Jost functions. If the initial amplitudes of the slow microscopic particles are equal, the microscopic particles parameters after the collision are given by v'j = VJ + AVJ
and ± v'sl = ±vsl ^ Au s i,
At;, = 192e^^G(5) V]
and Avsl = 96e^lG(6). Vj
where (4.155)
Here 6 = Asi(x'^2 ~ x'sii) i s t n e separation between the slow microscopic particles at the moment of collision with the third (fast) microscopic particle. In (4.155), sinh 5 [ tanlr<5
J
(4.156)
which vanishes as 5 —* 0 and S —> oo, and is of the same order as the perturbation. There is no change to the amplitudes of the microscopic particles. Because the amplitudes of the microscopic particles do not change after the collision, they further found that AN = J2i AAj = 0, and the total momentum is conserved up to the order of v~2 due to the symmetry of the problem, whereas, the energy of the microscopic particles changes by AE = (2AS1VSIAVSI+AJVJAVJ). Using the results of (4.155), they got AE = 0. This shows that the results given in (4.155) are consistent with the conservation laws. To analyze the dependence of the energy exchange between the microscopic particles on the separation of the microscopic particles, Prauenkron et al. fixed the
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
177
perturbation amplitude at e = 0.01 and varied the initial separation between the two slow microscopic particles. This results in a variation of the average distance between the two slow microscopic particles at the moment of collision, i.e., effectively 8 in (4.156). To measure the separation, they considered simultaneously the collision between the two slow microscopic particles under the same conditions but without the third microscopic particles. The initial distance between the slow microscopic particles, x'D, was measured at the moment when the fast microscopic particle passed the point x' = 0. The final velocities were measured by the linear regression analysis. The numerical results are summarized in Fig. 4.15, where the relative changes in the velocities of the microscopic particles, AVJ and Awsi, respectively, are shown as functions of x'D. Indeed, the changes in the velocities of the slow microscopic particles are due to the energy exchange during the collision. This effect strongly depends on the separation between the colliding microscopic particles at the moment of collision and it vanishes for larger separation, which is different from what has been predicted by previous theory. It is also noted that the energy exchange vanishes when the centers of the slow microscopic particles (solitons) approximately coincide.
Fig. 4.12 Changes in the velocities of the microscopic particles vs. the extrapolated initial distance between the slow microscopic particles xD. The dotted line shows the change in the velocity of the fast microscopic particle, Auj, and the dashdotted line shows that of the slow microscopic particle, Avs\. Symbols indicate data obtained from direct numerical simulations at e = 0.01.
More detailed analysis indicates that the energy exchange is more complicated and it involves the excitation of an internal mode of the fast microscopic particles. In fact, the internal mode appears as a nontrivial localized eigenmode of the linear
178
Quantum Mechanics in Nonlinear Systems
problem associated with the microscopic particle (soliton) of the perturbed nonlinear Schrodinger equation (4.153). This mode always exists if e > 0, and describes longlived oscillations of the amplitude of the microscopic particles. Therefore, internal mode of solitons plays an important role in the energy exchange between microscopic particles during the three particle collisions. In conclusion, the above discussion shows that, unlike collisions betweeen two microscopic particles, the collision involving three microscopic particles in nonlinear quantum mechanics is accompanied by a radiationless energy exchange among the particles and excitation of internal modes of the colliding microscopic particles. Both effects lead to changes in the soliton velocities which are of first order in the perturbation amplitude e. This effect depends nontrivially on the relative distance betwween the microscopic particles, and vanishes when the microscopic particles are separated by a large distance. 4.10
Transport Properties and Diffusion of Microscopic Particles in Viscous Environment
We have seen that a remarkable feature of microscopic particles in nonlinear quantum mechanics is their ability to maintain the shape, velocity and energy during propagation in free space. But what about propagation of microscopic particles in viscous systems? Can the microscopic particles retain these properties in such systems at finite temperature? This will be the focus of this section. We will discuss the dynamics of microscopic particles in such viscous systems. The motion of microscopic particles in viscous media can be described using two parameters, the damping and the diffusion coefficients. The former is related to the systematic force applied to the microscopic particles by the environment, while the latter describes the effect of fluctuations in the interaction. We will show that the microscopic particles move in the form of low energy Brownian motion in such a nonlinear system due to the scattering of internal excitation. It also exhibits classical features of microscopic particles in nonlinear quantum mechanics. As metioned above, in the nonlinear quantum mechanics the position of the center of a microscopic particle (soliton), being a function of time, is sufficient to describe its dynamics in the classical nonlinear Schrodinger equation. In the classical field theory, we may think that the microscopic particles are field configurations which move in space without changing their shape. In quantum field theory the center of mass of the microscopic particles plays also a special role. It can be viewed as a true quantum dynamical variable and the microscopic particle can be treated as a quasiclassical particle. Thus classical and translational invariant theories with the soliton solutions described by dynamic equations in nonlinear quantum mechanics can be quantized using the collectivecoordinate method. The problem is that at finite temperature not all the degrees of freedom contribute to the formation of the soliton, and the soliton is never free to move as in the classical theory. In other
179
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
words, there is always a residual interaction due to quantum degrees of freedom which changes drastically the dynamical properties of the microscopic particles. This residual interaction results in a damped motion of the microscopic particle (soliton). The microscopic particle thus undergoes a Brownian motion. This problem was studied by Neto et al. using a model of dissipation in nonlinear quantum field theory. In the following, we will follow their approach in our discussion on the transport properties of the microscopic particles described by the nonlinear KleinGordon equation in a viscous system. Let's consider, according to (4.1) or (3.75) and (3.94), the following classical static action in the 1+1 dimension for a nonlinear KleinGordon scalar field under the rescaling
where g is a coupling constant of the fields. The natural unit system is used in the above equation. The extremum value of the action is given by SS/5
Equation (4.158) is similar to (3.77). It is a relativistic dynamic equation of a microscopic particle in nonlinear quantum mechanics. Due to its translational invariance, equation (4.158) must have the soliton solution (j>s = (f>s(x — XQ). We can expand (4.157) near the extremum (4.158) in terms of the coupling constant g, according to (3.78), in the first order approximation, (4.159)
(t>s{x,t) =
which gives f
f
d2
f
S[6
W*,t),
plus some higherorder terms in g. Using (4.158), we get
m = SM = l y > * ( ^ ) 2 ,
(4.160)
which is the classical mass of the microscopic particles. Therefore, up to the leading order in g, the eigenmodes of the system are given by
 T T + C I D ]*n(xx0)=u£*n(xxo). dx2
Vd0 2 /0.J
(4.161)
Using (4.158) in (4.161), it can be shown that the function d
180
Quantum Mechanics in Nonlinear Systems
Suppose we displace the center of the microscopic particle (soliton), x0, by an infinitesimal amount 8x0. Then up to the first order in 6xo, we have <W« = 4>s{xo + Sx0)  <j>s(x0) = zZSxo. oxo
However, since 4>s is a function of the difference x — x0, we have 60<j>s = (d
where N' is a normalization constant. We can then get from (4.159) 00
84>{x,t) =
n
n=0
£ian(t)*n(xxo), , .
00
4>(x,t) = 4>s{x  x0) + •fiiao(t)£+g52an(t)&n(x
 x0).
n0
It can be seen that the center of the microscopic particle is a true dynamical variable. We rewrite the above expansion as 00
(4.162)
 xo(t)],
n—l
where a:oW=*o^70o(t1)
Equation (4.162) is known as the collectivecoordinate method and is consistent with (4.39) in the first order approximation. It has all the physics of this system. The microscopic particle whose motion is represented by that of its center of mass is a collective excitation (since it is a solution of the nonlinear field equation), but it is also coupled to all other modes by the relative coordinate x  x0. The corresponding classical Hamiltonian can be expressed as
(pY,
00
n,»=l
G
**»*
\\ 2 + /
E (f +^t)' °°
/ 2
n=l ^
2 2 \ '
(4 163)

where P is the momentum canonically conjugate to xo, qn and pn are also conjugate parts, and
Gin = Jdxd^*n(x) couples the microscopic particle to the other modes (phonons) in the system.
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
181
This shows that the microscopic particle cannot move freely. Phonons are scattered by the microscopic particle, resulting in the microscopic particle moving like a Brownian particle. Although (4.163) is obtained using a perturbative method, this expression is general and captures the essential physics of this problem. Making use of the commutation relations, [xo,P] — ih, [qn,Pi] = ih&ni, and in the context of quantum dissipation, the quantum model for the Hamiltonian (4.163) can be expressed as H
/
oo
= 2^ : K " ? y
h
\
oo
: + £>„&+&„,
9inbtK
n,i=l
J
(4.164)
n—1
where : • • • : denotes the normal order, and I K
(un\
/„
iPn\
= (2h) [qn + ~n ) '
with [bn,b+) = Smn,
[b+,b+)=O,
and
i r /wr 9ni = 7T: \\
h
[ur\r
bin
is the new coupling constant of the system. Equation (4.164) describes the dynamics of microscopic particles at low energy and can also be used to model other physical systems where microscopic particles are coupled with their environment. Here we are interested only in the quantum statistical features of the microscopic particles and phonons acting as a source of relaxation in the diffusion processes. Consider the density operator of the system consisting of microscopic particles and phonons, p(t). This operator evolves in time according to p(t) = eifltlhp{ti)eitltlh,
(4.165)
where H is given in (4.164), /5(0) is the density operator at t = 0, which is assumed to be decoupled and is given by the product of the density operators of the microscopic particles, p s (0), and the phonon, /3R(0), respectively, i.e., p(0) — PS(0)PR(Q). Neto et al. studied this problem in which they assumed that the phonons are in thermal equilibrium at t = 0, that is, Pfl(0) = where
z
,
182
Quantum Mechanics in Nonlinear Systems
and HR is the Hamiltonian of the free phonons which is given by the last term in (4.164). We define a reduced density operator ps(t) — trn[p(t)], which contains all the information about the system when it is in thermal contact with a reservoir. Projecting p(t) in the coordinate representation of the microscopic particle system x o\i) — Q\Q) and in the coherent state representation for the phonons bn\an) = otn\on) We have
Ps(x,y,t)= Jdx" f
dy'J(x,y,t;x",y',0)ps(x",y',0),
where J is the superpropagator of the microscopic particle (soliton) which can be written as
J= f dx [Vdyei(s°Ws°WkF{x,y], Jx"
(4.166)
Jy'
where
W={r[5ffl]
(4.167)
is the classical action for the free microscopic particle. F is the influence functional,
F[x,y] = l ^ 0 J
WpR0j.f^W'lflV^I'/a
D2a /°'JD27e5il».«]+sri».7]
x f"
h
h*
(4.168)
where j8' denotes the vector (ft,/J2> • • • , JSJV) and S\ is a complex action related to the reservoir and the interaction,
*[«,«] = I clt" [I ( a  f  «• • £ )  j ( ^  x.0] ,
(4169)
with oo
oo
HR = ^J/ia; n a*an, n=l
/ii = 2 J ^9nmO*mann,m=l
Now we expand the action (4.169) around the classical solution of its EulerLagrange equation. After integrating (4.168), we can get 00
F[x,y}='[[(lrnn[x,y]nn)1, n=l
where
?nm = W*nm[y\ + Wmn[x] + J2 WL[y}Wln[x], 1=1
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics oo
r
/
Wnm[r] = 6mn+J2
183
dt"Knn,(t")Wnlm(t"),
n'=l J°
Knm([x],T) = ignmXir'y^^', ft nn = ( e ^ «   l )  1 . Here iiTnm is the kernel of the integral equation, Wnm[r] is the scattering amplitude from mode k to mode j . The terms that appear in the summation represent the virtual transitions between these modes. If the velocity of the microscopic particle is small, we can use the Born approximation. In matrix form, this can be written as W = (1  W0)^0
&W° + W°W°.
In such a case, the terms Tnm are small. Then we can write approximately F[x, y] « exp J ] T Tnm[x, y}nn \ . ln=l J
(4.170)
If the interaction is turned off (F »• 0) at T = 0, the functional (4.170) is unity, and, as we would expect, the microscopic particle moves like a free classical particle. Using the Born approximation for W in (4.170), Neto et al. obtained the following.
J = J*Dx j " Dy exp j lS[x, y) + \G\X, y] J ,
(4.171)
where
§ = j T * " { x [*2(O  f(t")] + [±(*")  y(t")) = J df'T^t"  t'")[x(t") + y(t'")}V
(4.172)
G= f dt" f dt1" {[TR(t"  t'")][x{t")  y(t")][x(t'")  y(t'")}} ,
(4.173)
and Jo
Jo
oo
TR(t) = H&(t) ^
Olm^n COs(wn  Um)t,
n,n=l oo
rl(t) = m'(t) Y^ 9lmnnsin(u;n  ujm)t. n,n=l
184
Quantum Mechanics in Nonlinear Systems
Here ®'(t) is the theta function defined as ,
ri, ( j
~\0,
if*>o if t < 0 •
Introducing the following new variables
X = Z±*,
Y = xy,
the equations of motion for the action (4.172) can be written as X{T) + 2 / dt"7(T  t')X{t") = 0, Jo r*
(4.174)
Y(T)  2 / dt"j(t"  T)Y(t") = 0, Jo
where 77lQ Ut
or l{t) =
h®'(t) °° — Y\ glmnn(u!n  wm) cos(wn  ojm)t
(4.175)
n,m=l
which is the damping function. The two equations in (4.174) have the same form as that obtained in the case of quantum Brownian motion by Calderia and Leggett. In the limit of the time scale of interest being much greater than the correlation time of the phonon variables, we can write *y(t) = j(T)S(t), where 7(T) is a damping parameter which is temperaturedependent and 6{t) is the Dirac delta function. The above form of *f{t) is the same as that in the Markovian approximation in these systems. If we use (4.174) and expand the phase of (4.171) around this classical solution, we can get the wellknown result for the quantum Brownian motion provided that the damping parameter 7 (temperature independent) is replaced by j(T) and the diffusive part is replaced by (4.173). Neto et al. gave the diffusion parameter in momentum space J2T>
D(t) = h^f = H2&(t)
°°
J2 92nmnn(Un  um)2 c o s K  um)t. n,m=l
As discussed before, in the Markovian limit, we can write D(t) in the Markovian form, D(t) = D(T)S(t), where D{T) and j(T) obey the classical fluctuationdissipation theorem at low temperature. If we define the scattering function as 00
S(w  J)  S(CJ'  u) = ^2 glm6(LJ  um)2 cos(u'  w m ) , n,m=l
Wave Corpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
185
D(t) can be written as h2 f°° r°° D(t) = —z6(t) / dcu / dJS(u  J){J  u)2{n(ujn) + n(wm)] COS{LJ  u')t. * Jo Jo In the above discussion, we established that the Hamiltonian (4.164) leads to Brownian dynamics. That is, a microscopic particle in nonlinear quantum mechanics moves like a classical particle at low energy in a viscous environment where its relaxation and diffusion are due to scattering by the phonons. These phonons are the residual excitations created by the presence of the microscopic particle. We showed that the damping and the diffusion coefficients of the microscopic particle depend on temperature, since phonons must be thermally activated in order to scatter off the microscopic particle. Therefore at absolute zero temperature the microscopic particle moves freely, but its mobility decreases as the temperature increases. This shows again the classical feature of microscopic particles in nonlinear quantum mechanics. By expanding the perturbation in powers of temperature in the systems, Dziarmaga et al. developed a perturbative method to obtain an explicit expression for the diffusion coefficient of microscopic particles (solitons) moving in a dissipation systems with thermal noise. For a microscopic particle described by the (jAequation in one spatial dimension, we have (in natural units) r4H = 4>zx+2(l
(4.176)
where F is the dissipation coefficient and r)(t, x) is a Gaussian white noise with correlation (r)(t,x)) = 0, and (r1(t1,x1)ri(t2,X2)) = 2KBTTS(t1  t2)S(Xl  x2).
(4.177)
The system is coupled to an ideal heat bath at temperature T. At a finite temperature the field 4>{x,t) performs random walk in its configuration space. If the noise is absent at T = 0, equation (4.176) admits a static kink solution (p(t, x) = 4>s(x) = tanh(a;), and an antikink is given by (j>s{—x). Small perturbations around the kink take the form 4> — e~7t/rcp(x). Linearization of (4.176) with respect to tp{x), for r](t, x) = 0, gives axz
[
cosh (x)\
(4.178)
Its eigenvalues and eigenstates are 7 = 0,
7 = 3,
ip(x) = cj>sx(x) =
j—, cosh {x)
(4.179)
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Quantum Mechanics in Nonlinear Systems
A , ;2 7 = 4 +ft2,
, \ , \ ikx\i 3ifcmtanh(a:)  3tanh 2 (x)l
where k is the momentum of the microscopic particle. The zero mode (7 = 0) is separated by a gap from the first excited state (breather mode, 7 = 3). The continuum states are normalized so that /
J—00
dx
The field in the one kink sector for the theory (4.176) can be expanded using the complete set (4.179) as 4>{t, x) = 4>,[x m\ + £ An(t)yn[x  mi
(4180)
Inserting (4.180) into (4.176) and projecting on the orthogonal basis (4.179) gives a set of stochastic nonlinear differential equations, TNoi(t)  r & ) £ An(t)Mn0 + J2 Ai(t)Aj(t)pHo+ £
Ai(t)Aj(t)Ak(t)Rijk0=ilQ(t), +£
TNnAn(t) + lnNnAn{t)  Ti{t)Y,Ai{t)Min £
Ai(t)Aj(t)Ak(t)Rijkn
Ai(t)Aj(t)Pikn+ (4.181)
= Vn(t),
»,j,t#o
where the coefficients are /•oo
Nn=
J—oo
/•oo
dx(pn(x)(p*n(x),
J—oo
dxtpnx(x)(p*(x),
J—oo /CX)
/>OO
Pmj=6/
Mni=
dx
J—oo
Vn(t)=
f J—oo
dxipn(x)ipi(x)ipj(x)ip*k(x),
dxr){t,x)ip*n{x).
Applying (4.177) and the orthogonality of the basis (4.179), we get (Vn(t)) = 0,
« ( t i )»?*(«)> = 2KBTTNn5ni5{t1
t2).
(4.182)
The natural parameter of expansion at low temperature is \/KBT. The projected noises, (4.182), are of the same order. Dziarmaga et al. introduced the rescaling y/KsT —> E\/KBT. Terms in (4.181) can then be expanded in powers of e and a power series solution of the equation can be obtained, e was set back to 1 at the end of the calculation. With the expansion of the collective coordinates,
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
187
£(t) = e£tt) + £(2)^(2) + • • • and An = e^ 1 * + £ ( 2 Ul 2 ) + • • •, equations in (4.181) become, to the leading order in e, IW0£(1) W = ijb(t),
TNnAgHt) + lnNnA^(t)
(4.183)
= r,n{t).
In this approximation the zero mode £ ^ ( t ) and the excited modes An (t) are uncorrelated and stochastic process driven by their mutually uncorrelated projected noises. f^(*) represents a MarkovianWiener process whose only nonvanishing sinwhich gle connected correlation function is
dp_ av at de' where the diffusion coefficient is given by K
BT FN0
=
3KBT
4r '
Other terms in the expansion of collective coordinates can be recursively worked out and they are
42)(*) = E ^ fdre^^i^{r)Af\r)

E^r^^^M^w^w,
»,^o
l iVn
70
e(3)W = £ K ( 1 ) (*)4a)W + (i« 2 ) ] ^ 
^[^(^fw+do^^i
E ^w^w^w^. The equilibrium correlations of ^(t), up to one order higher than the leading order, are
(mm)«£2(^(1)(*K(1)(i))+£4(^(2)(^(2)©)+e4[<4(1)(*)^(3)(*)>
= <^>w^>(«)> i + ^
E
+ a «• 3)]
^+^ E ^ ^ M )
+....
188
Quantum Mechanics in Nonlinear Systems
The integrals over the continuous part of the spectrum and the summation over the breather mode yield
(i(t)£®) = ^ f p ^ C  *)[1 + 1.8164**71 + O[(KBTf]. Finally, Dziarmaga et al. gave the diffusion coefficient in the equilibrium state, which is given by the average over time much longer than the relaxation time of the breather mode. It is given by £>(oo) = ^ ^ ( 1 +
I&VMKBT)
+ 0[{KBT)3].
(4.184)
Dziarmaga et al. performed numerical simulation of (4.181) with T — 1 for a range of temperature values. The numerical results were consistent with the prediction of the perturbative theory, (4.184). 4.11
Microscopic Particles in Nonlinear Quantum Mechanics versus Macroscopic Point Particles
Prom previous sections of this Chapter, we understand that a microscopic particle in nonlinear quantum mechanics is a soliton which exhibits many features of a macroparticle. However, the microscopic particle cannot be regarded as a macropoint particle. We already know from sections 4.9 and 4.10 that the excitations of the internal mode of soliton  a longlived oscillation of the soliton amplitude participates in inelastic and radiationless energy exchange among particles during the threesoli ton collisions, and the residual excitations (phonons) created by the presence of the microscopic particles result in damping and diffusion of the particles in viscous systems. These phenomena do not occur in system of macropoint particles. In this section, we will further explore these phenomena and address the point particle limit of microscopic particles in nonlinear quantum mechanics, following the work of Kalbermann. A simple coupling between the center of the microscopic particle (soliton) and a potential will be introduced, which is useful in the discussion of differences and similarities between a microscopic particle in nonlinear quantum mechanics and a point particle in classical physics, as well as the pointlike limit of the microscopic particles. Kalbermann considered the center of mass of a microscopic particle (soliton) as a candidate for the location of the particle, and coupled it to an external potential. He took a kink mode supplemented by a potential and a Lagrange multiplier to force the interaction with the center of mass. This procedure treats the center of the microscopic particle (soliton) as the collective coordinate, as it is usually done in the case of topological defects such as those by Kivshar et al. and Neto et al, given in Section 4.10. In this approach, certain parameters of the microscopic particle, related to the symmetries or zero modes, are identified and coupled to external sources. In this case, translational invariance yields the corresponding
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
189
symmetry. In other cases, internal symmetries can be identified and the generators of the symmetries can be treated as the collective coordinates of the microscopic particles as a whole. This enables the separation of the bulk characteristics of the microscopic particle from its internal excitation. In this way, the coupling between the center of the microscopic particle and the collective coordinate corresponding to translational invariance breaks the symmetry, which allows a clear distinction between internal excitations and bulk behavior of the microscopic particles as a whole. As our aim is to demonstrate the importance of the extended character of the microscopic particles on its interaction with external sources, the above method seems to be the most straightforward approach. Kalbermann used the following Lagrangian in his study of the problem
C = \d^dv<$> + ^A ( V  ?p}  V(x0) + x(s, t)4>S(x  x0),
(4.185)
where x{x> i) is the Lagrange multiplier that enforces the coupling between the center of the microscopic particle, XQ, and the external potential, U(XQ). For kinks with topological charge =±1 for which
U{xo) = \ j dxV{x)\U2T^\ ,
(4.186)
which represents interaction between a microscopic particle and an impurity. The integrand is zero except in the region around the center of the microscopic particle where 4> « 0. If V(x) is a smooth potential (in contrast to the 6 spike impurity), then to the lowest order of the potential, the above equation becomes U(x0) = C'V(XQ), where c' is a constant depending on the initial parameters of the microscopic particle. The choice for the interaction corresponds to the case in which the microscopic particle propagates in a smooth medium and the impurity is similar to a background metric. If the microscopic particle can be regarded as topological soliton, then the present treatment would correspond to the classical propagation of a microscopic particle in a background potential. In order to see clearly this correspondence, Kalbermarnn chose the interactions which consist of introducing a nontrivial metric for the relevant space time. The metric carries the information of the medium. The Lagrangian in the scalar field theory for the (lll)dimensional system immersed in a background with the metric g^v is
C = y/g \gw\d^dv
 U(4>)] ,
(4187)
where g is the determinant of the metric, and U is the selfinteraction that makes the existence of the soliton possible. For a weak potential we have g0Q « 1 + V(x),
gu =  1 , fl_u = (?!_! = 0,
(4.188)
190
Quantum Mechanics in Nonlinear Systems
where V(x) is the external spacedependent potential. The equation of motion of the microscopic particle in the background becomes (4.189)
This equation is identical, for slowly varying potentials, to the equation of motion of a microscopic particle interacting with an impurity V(x). The constraint in (4.185) only insures the proper identification of the center of the microscopic particle in the present case. In the following, we will let A = m2 = 1 in (4.185). Variations of the field 4> as well as the collective coordinate x0 in (4.185) yield the equation of motion of the microscopic particle constrained by the Lagrange multiplier, (4.190)
where
(4.191)
B(x,t) = 8(xxo)(j£)
is the new source term that denotes the coupling between the center of the microscopic particle and the external potential. The limit of a pointlike object is achieved, when the width of the microscopic particle is negligible as compared to the characteristic scale of the potential. This method provides a clean separation between the particulate behavior of the microscopic particle and its wavelike extended character in the nonlinear quantum mechanics. For the potential in (4.187), Kalbermann used a bellshaped function, U(x0) = A sech2 (^—^) >
(4192)
where A is the height (depth) of the potential located at r, and a is related to the width, W, of the interaction potential by a = W/(2n). Using a 5function distribution for p(x), 6(x)=p(x) = \imp((x),
pe(x) = eM~^/e),
(4.193)
•
(4194)
B(x,t) can be expressed as
B(x,t)=
P(X
Z^L
w
J p(x — xo)(dcp/dx)dx In actual calculations the limit of e * 0 is achieved when e « dx, the spatial step, is small enough compared to the size of the microscopic particle. In solving the partial differential equation of motion, the spatial boundaries are taken to be  2 5 < x < 25, with a grid of da; = 0.04 and a time lapse of dt = 0.02 up to a maximal time of t = 200 (10000 time steps). The upper time limit of t = 200
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
191
is sufficient for the resonances to decay, to define clearly the asymptotic behavior of the microscopic particle (soliton) while at the same time to avoid reflection from the boundaries. Kalbermann started with a free microscopic particle impinging from the left, at distance from the potential range, with an initial velocity v, (4.195)
with xo(t — 0) = — 5. The total and kinetic energies of the free microscopic particle are E(v) =
! ^ _ , 2vl  v
Ek(v) = E{v)  E(0),
(4.196)
respectively. The value of A in (4.192) was chosen such that there are visible results for the kink with m = 1. Kalbermann used A = 0.1 and A = —0.2. The former allowes transition through the barrier for a pointlike particle with velocity v w 0.43, and the latter provides enough strength to trap it in a reasonable range of the initial velocity. The location of the potential was fixed at r=3. For the attractive potential, it was found that there were no islands of trapping between transmissions. The microscopic particle either is trapped, when its velocity is below a certain threshold, or passes through the impurity. For a barrier depth of A = —0.2 and W = 1 which is comparable to the size of the microscopic particle, the critical velocity is v — 0.583. At this initial velocity, the microscopic particle drifts extremely slowly through the potential well, and lags behind the free propagation by almost an infinite time, but it eventually goes through. If the velocity is below this critical value, the microscopic particle is always trapped. Contrary to the case of a point particle, there appear to be bound states for velocities below the threshold for positive kinetic energies. These states may be referred to as bound states in the continuum, extraneous to classical particulate behavior. This feature is due to the finite extent of the microscopic particle (soliton) in nonlinear quantum mechanics. One way to show that such a phenomenon is indeed due to space extension of the microscopic particle is by increasing the well width. In order to confirm this assertion, Kalbermann selected an initial velocity of the microscopic particle which was well below the threshold transmission velocity, v — 0.2, and a well depth of A = —0.2 as before. Starting from W = 1 he gradually increased the well width and found that transmission occured when the well width became W = 5, which was much wider than the extent of the microscopic particles. When the velocity was above the threshold velocity, the microscopic particle slowly drifted through the well in an effectively infinite time. It remained within the barrier range for almost t — 200 and then started to emerge from the opposite side. Figure 4.13 shows the asymptotic velocity of the microscopic particle (soliton), plotted versus the potential width W. In the case of the microscopic particle being trapped, the asymptotic
192
Quantum Mechanics in Nonlinear Systems
velocity vanishes. The microscopic particle oscillates with zero mean velocity inside the well. It never emerges from it. Its amplitude decreases when the particle moves farther from the center of the well during the oscillatory motion. The energy of the microscopic particle is conserved in all cases. When the microscopic particle is trapped, the initial kinetic energy transforms into deformation and oscillation energies as well as radiation that damps the oscillations.
Fig. 4.13 Asymptotic velocity of the microscopic particle (soliton) as a function of the width of the attractive potential.
When the microscopic particles is transmitted, there is no radiation due to the topological conservation of winding number that forces the microscopic particle (soliton) to be exactly the one that impinged onto the potential. This result was borne out in the case of multisoliton collisions in section 4.8 for which the dominant effect is the inelastic nonradiative exchange of energy between the microscopic particles (solitons). In the present case, there is no possible recipient of such energy. Therefore, the microscopic particle (soliton) will emerge elastically. The asymptotic velocities of the reflected and transmitted microscopic particle were calculated using the actual motion of its center, and so were the theoretical expressions for the kinetic and total energies of the free microscopic particles. This comparison proved energy conservation and served as a measure of numerical accuracy. For the repulsive potential, the microscopic particle (soliton) is reflected even for initial kinetic energies above the barrier height. When A = 0.1, the corresponding initial velocity whose kinetic energy would yield classically to transmission is v ~ 0.43. However, this does not occur. For a barrier of width W = 1, the microscopic particles goes through, only when the initial velocity is v = 0.52, corresponding to an initial kinetic energy of Ek = 0.16, well above the barrier height. The particulate behavior will be restored only when the barrier width is larger than the extent of the microscopic particle (soliton), such that it looks as if almost pointlike. In particular for v = 0.43, transmission starts for barrier widths greater than W = 19. Fig. 4.14 shows the asymptotic velocity of the microscopic particle as a function of the barrier width. The microscopic particle is repelled by sharp barriers and transmits through wider barriers, while trapping never occurs.
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
193
Fig. 4.14 Asymptotic velocity of microscopic particle (soliton) as a function of the repulsive barrier width.
Prom the above discussion we see that a microscopic particle in nonlinear quantum mechanics behaves as a particle only when the width of the potential is large compared to the size of the particle so that the microscopic particles (soliton) appears pointlike. In such a case the microscopic particle obviously has corpuscle feature of a classical particle. In contrary cases, the coupling between the center of the microscopic particle and a repulsive or an attractive potential can result in bound states or trapped states in the continuum, which does not occur for pointparticle objects. Kalbermann simulated numerically the behaviors of the microscopic particles in terms of collective coordinates. The simulation results showed also chaotic behavior depending on the initial position of the center of the microscopic particle. The initial states can transform a trapped state into a reflected or transmitted state. This exhibits clearly the wave feature of the microscopic particle. Therefore, a microscopic particle in nonlinear quantum mechanics has not only corpuscle but also wave features and it is necessary to study its wave feature which will be the topics of discussion in the following sections. 4.12
Reflection and Transmission of Microscopic Particles at Interfaces
As mentioned above, microscopic particles in nonlinear quantum mechanics represented by (3.1) have also wave property, in addition to the corpuscle property. This wave feature can be conjectured from the following reasons. (1) Equations (3.2)  (3.5) are wave equations and their solutions, (4.8), (4.56) and (4.195) are solitary waves having features of travelling waves. A solitary wave consists of a carrier wave and an envelope wave, has certain amplitude, width, velocity, frequency, wavevector, and so on, and satisfies the principles of superposition of waves, (3.30), (3.43) and (3.44), although the latter are different from classical waves or the de Broglie waves in linear quantum mechanics. (2) The solitary waves have reflection, transmission, scattering, diffraction and
194
Quantum Mechanics in Nonlinear Systems
tunneling effects, just as that of classical waves or the de Broglie waves in linear quantum mechanics. Some of these properties of microscopic particles will be described in the following sections. In this section, we consider first the reflection and transmission of microscopic particles at an interface. The propagation of microscopic particles (solitons) in a nonlinear and nonuniform medium is different from that in a uniform medium. The nonuniformity here can be due to a physical confining structure or at the interface of two nonlinear materials. One could expect that a portion of a microscopic particles that was incident upon such an interface from one side would be reflected and a portion would be transmitted into the other side. Lonngren et al. observed the reflection and transmission of microscopic particles (solitons) in a plasma consisting of a positive ion and negative ion interface, and numerically simulated the phenomena at the interface of two nonlinear materials. In order to illustrate the rules of reflection and transmission of microscopic particles, we introduce here the work of Lonngren et al. Lonngren et al. simulated numerically the behaviors of microscopic particle (soliton) described by the nonlinear Schrodinger equation. They found that the signal had the property of soliton. These results are in agreement with numerical investigations of similar problems by Aceves et al., and Kaplan and Tomlinson. A sequence of pictures obtained by Lonngren et al. at uniform temporal increments of the spatial evolution of the signal are shown in Fig. 4.15. From this figure, we note that the incident microscopic particles propagating toward the interface between the two nonlinear media splits into a reflected and a transmitted soliton at the interface. From the numerical values used in producing the figure, the relative amplitudes of the incident, the reflected and the transmitted solitons can be deduced.
Fig. 4.15 Simulation results showing the collision and scattering of an incident microscopic particle (soliton) described by the nonlinear Schrodinger equation (top) onto an interface. The peak nonlinear refractive index change is 0.67% of the linear refractive index for the incident microscopic particle, and the linear offset between the two regions is also 0.67%.
They assumed that the energy carried by the incident microscopic particle (soli
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
195
ton) is all transferred to either the transmitted or the reflected solitons and none is lost through radiation. Thus •Sine = Eref + £ t r a n s .
(4.197)
Lonngren et al. gave approximately the energy of a microscopic particle (soliton) A* &C
where the subscript j refers to the incident, reflected or transmitted solitons. The amplitude of the soliton is Aj and its width is Wj. The characteristic impedance of a material is given by Zc. Hence, (4.197) can be written as ^Winc
= ^Wre{
+ ^«Wtrans.
(4.198)
Since AjWj = constant, for the microscopic particles (solitons) of the nonlinear Schrodinger equation (see (4.8) or (4.87) in which Bj is replaced by Aj), we obtain the following relation between the reflection coefficient R = Ave{/Ainc and the transmission coefficient T = A trans /Ai nc
l = R+4±T>
( 4  199 )
for microscopic particle (soliton) of the nonlinear Schrodinger equation.
Fig. 4.16 Sequence of signals detected as the probe is moved in 2 mm increments from 30 to 6 mm in front of the reflector. The incident and reflected KdV solitons coalesce at the point of reflection, which is approximately 16 mm in front of the reflector. A transmitted soliton is observed closer to the disc. The signals at 8 and 6 mm are enlarged by a factor of 2.
196
Quantum Mechanics in Nonlinear Systems
Lonngren et al. conducted an experiment to verify this idea. In the experiment, they found that the detected signal had the characteristics of a KdV soliton. A sequence of pictures as shown in Fig. 4.16 was taken by Lonngren et al. using a small probe at equal spatial increments, starting initially in a homogeneous plasma region and then progressing into an inhomogeneous plasma sheath adjacent to a perturbing biased object. In Fig. 4.16, we can clearly see that the probe first detects the incident soliton, and some time later the reflected soliton. These signals are observed, as expected, to coalesce together as the probe passed through the point where the microscopic particle (soliton) was reflected. Beyond this point where the density started to decrease in the steadystate sheath, a transmitted soliton was observed. From Fig. 4.16, the relative amplitudes of the incident, the reflected and the transmitted solitons can be deduced. For the KdV solitons, there is also AjWf = constant (see (4.92)). Thus, !
=
#3/2
+
cLr3/2>
Zcn for the KdV soliton. The relations between the reflection and the transmission coefficients for both types of solitons are shown in Fig. 4.17, with the ratio of characteristic impedances set to one. The experimental results on KdV solitons and results of the numerical simulation of microscopic particles depicted by the nonlinear Schrodinger equation are also given in the figure. Good agreement between the analytic results and simulation results can be seen. The oscillatory deviation from the analytic result is due to the presence of radiation modes in addition to the soliton modes. The interference between these two types of modes results in the oscillation in the soliton amplitude. In the asymptotic limit, the radiation will spread, damp the oscillation, and result in the reflectiontransmission coefficient curver falling on the analytic curve. The above rule of propagation of the microscopic particles in nonlinear quantum mechanics is different from that of linear waves in classical physics. Lonngren et al. found that a liner wave obeys the following relation
1 = R2 + f^T 2 .
(4.200)
This can also be derived from (4.198) by assuming linear waves. The widths of the incident, reflected and transmited pulses Wj are the same. For linear waves Zcn — Zc\ =~5 T~5~'
n R
and
2Zcn ~5—T^~'
T=
and equation (4.200) is satisfied. Obviously, equation (4.200) is different from (4.199). This shows clearly that the microscopic particles in nonlinear quantum mechanics have wave feature, but it is different from that of linear classical waves and the de Broglie waves in the linear quantum mechanics.
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
197
Fig. 4.17 Relationship between the reflection and transmission coefficients of a microscopic particle (soliton), Eq.(4.199). The solid circles are experimental results on KdV solitons, and the open circles represent results of Nishida. The solid triangles indicate numerical results of Lonngren et al. for microscopic particles(solitons) described by the nonlinear Schrddinger equation.
However, the above results are based on two assumptions: (1) no radiation modes are excited through the interaction of the incident soliton with the interface; (2) the product of the interaction of the incident soliton with the interface is a single transmitted and a single reflected soliton. If these conditions are not satisfied, the above relation will not hold. As a matter of fact, the reflection of a microscopic particle (soliton) in nonlinear quantum mechanics is complicated. Alonso discovered a shift in the positions of microscopic particles which occurs in the case of nonzero reflection coefficient. He used the inverse scattering method to obtain the phase shifts of the microscopic particles in (4.40) with 6 = 2. As it is known, the inverse scattering method for solving the nonlinear Schrodinger equation is based on the resolution of the integral equations of the Marchenko type (4.51), which can be written as Fl{t',x',y)
l°° K*(t',x' + y + z)FZ(t',x',z)dz = K*(t',x' + y), Jo
fO°K(t',x'+y + z)F1(t',x',z)dz^0, Jo where the kernel K is determined from the set of scattering data Ft(t',x',y)+
(4.201)
S(t') = {0 = & + irij, btf) = bt exp(4iC^'), j = 1, • • • , TV; in the form of K(t',x') = 2^2bj(t')e2i^x> j
1
+
f°°
"^ JOO
c(t',C)e2iCx'dC.
Solution of (4.40) is given by 4>{t',x') = F1{t',x',+0).
(4.202)
As t' > ±oo, cj>{t',x')
198
Quantum Mechanics in Nonlinear Systems
evolves into a superposition of N freely moving solitons with velocities Vj = 4£,, and a radiation component which decays as I*'!"1/2. The parameters q^ which characterize the asymptotic trajectories qf(?) « qf + Vjt' for the solitons as t' 4 ±00 remain to be determined. It is known from Section 4.1 that the nonlinear Schrodinger equation is a Galileaninvariant Hamiltonian system and its corresponding generator for the pure Galilean transformation can be written as "
J—oo
=  £ > M
(4.203)
where J
VW
P
llcC;
L2vr 7 . ^ g  C  iO q\'
'
2
[ p(Q = ln(l + c(C) ), F(0 = c(Oo(C),
(4.204)
Qv = 0.
From (4.203) and (4.204), the following alternative expression for G can be obtained
°E[^5>&£/"i^P« +
5r£"«)a< h^+s'jC^*]*
(4205)
where P denotes the principle value. We observe that for a pure onesoliton solution (j>so\, the position of its center is q(t) = (2r])~1 ln[\b(t)\/2rj\, and therefore, G[&oi(*)] = 2nq(t). As a consequence of the Galilean invariance of the nonlinear Schrodinger equation, we can define the mass and momentum functionals which have the following forms in terms of scattering data
M= /
0W = 2 j >  — / p{QdC,
* J00
P = i
Z7r
j
Pfa'dx' = 8^2^ ^oo
j
J00
+ / n
OKCR
J00
According to these expressions, one may think that the field of microscopic particle depicted by the nonlinear Schrodinger equation appears as a Galilean system composed of N particles with masses 2TJJ and velocities Vj = 4fj (solitons), and a has continuous mass distribution with density —p(()/2n and velocity vg(Q = 4£
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
199
(radiation). This velocity spectrum associated with the scattering data can be understood through the asymptotic analysis of the nonlinear Schrodinger equation field as follows. Prom (4.201) we know that the nonlinear Schrodinger equation field 0 at a given point (t',x'o) depends only on the restriction of the kernel K(t',x') to the interval (x'0,+oo). If the evolution of the scattering data is inserted into (4.202), the modulus of the jth term in the summation propagates with velocity Vj — 4£,, while the group velocity of the Fourier modes in the integral term coincides with vg(C) = 4(. Hence, given arbitrary values x'o and v0, as t' >• ±oo the restriction of K to intervals of the form / ± (t') = [x'o + (v0 ±e)t',+oo] where € is an arbitrary positive number, depends only on those scattering data with velocity v such that ±(v  v0) > 0. In fact, it can be shown that the contribution to K due to the remaining scattering data has a L2 norm on / ± (t) which vanishes asymptotically as t' > ±oo. We now consider the motion of the Ith soliton and let us denote by ± the parts of the nonlinear Schrodinger equation field propagating to the right of the soliton as t' —> ±oo. Then the relevant kernels for characterizing
K±(t', x') = 2^20[±(Vj  vftbiW**' +  r 9{±[vg(Q vtMt'^y^'dC, where 6 is the step function. The sets of scattering data related to <j>± are S±(t') = {Q,bj(t'),j such that ± (Vj  vi) > 0\B(±[v9(Q 
vt})c(t\(),teR}.
The scattering data S±(t'){Q, h(t')} correspond to the parts <j>'± of the nonlinear Schrodinger equation moving with velocity v such that ±(v — vi) > 0 as t' > ±oo. In other words, <j>'± result from the addition of the Ith soliton to <j>±. Hence, as a result of the dispersive character of the radiation component and the localized form of the soliton, it is clear that as t' >• ±oo the difference between the values of the functional Gs for
The phase shift of the Ith microscopic particle (soliton) as it interacts with both
200
Quantum Mechanics in Nonlinear Systems
the other microscopic particles and the radiation component is given by
qf ~ 9f =  S
si
S n ^ ~ v^ln
TZ^
Prom these results we see that the microscopic particles experience a phase shift when they are reflected due to interaction with other microscopic particles and with the radiation component. It shows again the wave feature of the microscopic particles. 4.13
Scattering of Microscopic Particles by Impurities
A wave can be scattered by an obstruction, and scattering is an essential feature of waves. In this section we discuss scattering of microscopic particles by impurities, and propagation of microscopic particles in nonlinear disordered media in the framework of nonlinear quantum mechanics. It is well known that in linear systems, disorder generally creates Anderson localization, which means that the transmission coefficient of a plane wave decays exponentially with the length of the system. However, the nonlinearity may drastically modify properties of propagation of microscopic particles in disorder systems. We will find that besides reflection and transmission, there are also excitations of internal modes and impurity modes in the systems, which again reflects clearly the wave nature of microscopic particles. These properties of microscopic particles depend on the nature of the impurity and its velocity. As the first step towards a full understanding of these phenomena, we start by discussing scattering of a microscopic particle by a single impurity, which was studied by Kivshar et al, Malomed, Zhang Fei, et al, and others. In such a case, the dynamics of the perturbed microscopic particle in nonlinear quantum mechanics is described by the following nonlinear Schrodinger equation
i
(4206)
where e is a real, small parameter, f(x') is a localized potential function arising from the impurity in an otherwise homogeneous medium, which satisfies f{x' > oo) = 0. If e = 0, equation (4.206) has a soliton solution which is given by (4.56), with the amplitude and velocity of the soliton given by 2r\ and +4£, respectively. Equation (4.56) can be interpreted as a bound state of N = 4TJ quasiparticles with a binding energy Eb = Ar)2.
We now consider the scattering of the microscopic particle (soliton) by the impurity described by (4.206) using the Born approximation. We assume that the velocity of the microscopic particle does not change during the scattering under the condition e7j
201
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
scattered microscopic particle, can be calculated following the inverse scattering method of Zakharov et al., and the result is given in Section 4.3, n(A) « TT""1 6(A)2, where 6(A)(6(A)2
^P
= 4i\2b(\,t>) + £  ° ° dx'f(x') [0(i',O*i(^t ; , A]^V,f, A)  0V,t / )¥ / 2(s'.t / MV,t / >A')],
(4207)
where *i,2(x', t', A) and ^>i,2(a:', t', A) are the components of the Jost functions. For the onesoliton solution (4.56), these functions were obtained by Karpman. Kivshar et al. assumed that before the scattering, i.e. at t' >• — oo, all the "quasiparticles" were in the bound state, or the soliton state (4.56). This means that the initial condition for (4.207) should be in the form of 6(A, t' = oo) = 0. Then, the total density rcraci(A) of the linear waves emitted by the microscopic particles during the scattering is nrad(A) = i&(A,t' = +oo) 2
 e2*£i(A) [°° dxi [°° dx'f{xl)f{x')ei^^x'\ J— oo
J— oo
(4.208)
where
0(X) =
l
^ '
V
.
(4.209)
The number of "quasiparticles", N, emitted by the microscopic particle (soliton) in the backward direction (i.e. the reflected "quasiparticles") can be calculated from Nr=
r°°
n rad (A)dA.
Joo
The reflection coefficient R of the microscopic particle can be defined as the ratio of Nr to the total number of "quasiparticles", N = 47?
R=±
[
nrad(A)dA.
(4.210)
Kivshar et al. studied the scattering of a microscopic particle by an isolated point impurity in which f(x') is a deltafunction. Prom (4.208), we get the emitted
202
Quantum Mechanics in Nonlinear Systems
spectral density nren = e 2 n^(A). The reflection coefficient can then be written as fl(D =
_![£L
(*'+D2+*2
r
dx,
(4 211)
where a = 77/^. At a = 0, i^ 1 ' reduces to that of the linear wave packet, i.e., R{o] « £2/4feo = £2/16£2. For small a, R^X^/R^ first increases slowly to 1.004 (at a = ac sa 0.178) and then rapidly decreases to zero, so that for a » 1 the reflection coefficient approaches zero exponentially.
R{1)
7/2
" i^^ 1 ) e " W 2 )
(a>>1)

(4 212)

The fact that R^ differs from i?g shows that the wave nature of the microscopic particles in nonlinear quantum mechanics differs from linear wave packet. The number of transmitted "quasiparticles" Nt, i.e., those emitted by the microscopic particle (soliton) in the forward direction, may be defined by Nt= /°°nrad(A)dA. Jo Note that for a C 1 this quantity is essentially less than Nr, since it has the asymptoic expansion Nt w e2a4/120£2iV. However, for a > 1, Nt asymptotically approaches iVr. We define the density, erad(A) = 4A2nrad(A), of the emitted energy £rad= r' dx>(\4>*\2\4>\*) J—oo
by means of the inverse scattering method. The total energy emitted by the microscopic particles in the forward (Et) and backward (Er) directions can be found by r°° dAe rad (±A). Et,T = / Jo
For the microscopic particles described by the nonlinear Schrodinger equation (4.206), Kivshar et al. found that Er = e2r, (l  la2  ~aA V
Ert*Et
=£
2
m
Et = ^e2r,a\
9/2 n
e~na/2,
for a « 1,
for a > 1.
Kivshar et al. also considered scattering of a microscopic particle (soliton) by two point impurities separated by a distance a. The perturbation ef(x') take the
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
203
form of ef{x') = e\S(x') + S2S(x' — a). It can be shown that
nrad(A) = n£i (£i  e 2 ) 2 + 4 £l e 2 cos2 [^a/3(A)J  , where n ^ and fi(\) have been denned in (4.209). In this case the reflection coefficient of the microscopic particle, R^2\ is determined by two factors, corresponding to the soliton scattering by an isolated point impurity with the effective intensity (ei  £2), and to the resonant effect during the scattering, respectively, and it is given by 28a^2 70
cosh 2 [7r(a; 2 +a 2 l)/4a]
[4
J
(4.213)
where d = 2a^. When a < 1 and a 2 d < 1, equation (4.213) reduces to
RW = 2RP [l + . l^ u
L
,, cos(2d)l .
sinh(2ad)
(4.214)
J
Fig. 4.18 The reflection coefficient of the microscopic particle described by the nonlinear Schrodinger equation scattered by two identical point impurties, as a function of d = 2a£. (a) a < 1; (b) a > 1; R^ is the reflection coefficient for scattering by a single isolated impurity.
The dependence of i? (2) on d for small a is shown in Fig. 4.18(a). If the size of microscopic particle (« if"1) exceeds considerably the distance between the two impurities (i.e. ad w or) < 1), we can distinguish the interference phenomena which manifest themselves as oscillation of i? (2) against the parameter d. Different from the linear approximation, i? (2) does not approach zero, because x' < sinhx' for x' ^ 0. When d = dmin = {I~2a2/3)d°n, where dPn = ir/2+mr (n = 0,1, • • •), R{2) has the minimum values: R{^m = 2(ad min ) 2 /3 « (2a 2 /3)O/2+n7r) 2 . The maximum values, R&lx = 4i?[)1)[(l  na7r/3)2], which occur at d = rfmax = (1  2a2/3)n7r, decrease as d increases. When the size of the microscopic particle becomes comparable to
204
Quantum Mechanics in Nonlinear Systems
the separation of the impurities, a (i.e. ad m 1), the interference decays and for ad » 1, RW approaches the value 2R§' (see Fig. 4.18(a)). That is, the scattering by each impurity is independent of the other. Thus, the microscopic particle shows both wave (ad
( a ) fl + (1 + c') 1/4 cos (\o?d + y'\ 1 ,
(4.215)
where d = 2ar)/K, y' = tan~ 1 (c')/2, and R^(a) is given in (4.212). The dependence of RW on d for a 3> 1 is shown in Fig. 4.18(b). For large values of d (when ad sa d ~S> 1), the reflection coefficient becomes equal to 2Rl1\ which characterizes the nonresonant scattering of the microscopic particle in which the scattering intensities by separate impurities are simply added. For da < 1 (d < 1), the reflection coefficient oscillates with increasing d and the period of the oscillation is approximately a~2 •C 1. The frequency of the internal oscillation of the microscopic particle for a » 1 is then « 7j2, and the resonant condition is a2d ss 1. For a C 1, the period of the internal oscillations is of the order of £~2, and the resonant condition is a£ « d m 1. However, in the case of e\ = Zi = s, scattering of linear wave is known to be resonant and the reflection coefficient RQ in the Born approximation oscillates as a function of d = afeg = 2a£ as R^' = 4R^' cos2 d, where R{01] = £2/16£2. When ffl = 7r/2 + n7r (n = 0,1, • •), R^ = 0, which corresponds to the situation where the energy of the scattered "quasiparticle",fcg= 4£2, coincides with the energy of the resonant state between the two deltafunction potentials. The maximum value, 4/?Q of RQ , corresponds to scattering of a linear wave by a single effective impurity with twice the intensity (e > 2e). Therefore, we see that scattering of microscopic particles described by the nonlinear Schrodinger equation is different from that of linear waves. The above results can be easily generalized to the case of a periodic system of N point impurities, with separation a between adjacent impurities. Tedious but straightforward calculation leads to the following when Na2d
(Nl)/2
RW = J#> (N + 2 £ ^
fc_l
C2Nk+1C%+1 £ (_i)c fc "L 1 2 2 ( 7V  m  1 '
fc,/#0
xsump=1
m=0
C 2{JV _ m _ 1)
sinh{2adp)
cos(Zpd)J ,
where C§ are the binomial coefficients and ( ) stands for the integer part. In the case of aNd 2> 1, the microscopic particle is scattered as a corpuscle and it can be shown that RW « NRg\ However, transmission always occurs in the scattering process of the microscopic particles by the impurities except in the case of complete reflection, the condition
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
205
for which is given above. The transmission coefficients are defined by T^ = Et/Ei for the energy and T ^ = Nt/Ni for the excitation, where £* and Nt are incident energy and number of microscopic particles, respectively, Et and Nt are the transmitted energy and number, respectively. Because Ef = Et + Er and Ni = Nt + Nr, then T(E
Ej+^EjT^iE^Nj), NJ+1=NiTJIf){EJ,Nj) and AEj+1 = EJ+1 Et = EjRW
ANj+1 = Nj+1  Nd =
(EJ ,Nj),
(4.216)
NjRfHEjNj).
When a < 1, 1
{X}
~
N(0) ~ E(Q) ~
where Ao
iee2(o) _
i
 ~ ^ ~  ^RW '
RW is the reflection coefficient due to a single impurity. This shows that the transmission coefficent decays exponentially. When a = T]/£ > 1, Kivshar et al. found that the asymptotic change in rp(N,E)f^z _ x'/x'o), where x'o = 64/npe2, depends essentially on the parameter a(0) = »?(0)/£(0) which is related to the nonlinearity of the incoming microscopic particle. The greater a is, the larger the number of excitations in the microscopic particle (soliton) becomes, and the smaller its spatial extension. On the contrary, if a is small, the microscopic particle looks very similar to a linear wave packet. Therefore, for initial conditions a(0)
206
Quantum Mechanics in Nonlinear Systems
These results show that strong nonlinearity can completely inhibit the localization effect stimulated by the disorder. This effect appears over a threshold nonlinearity. Below this threshold, the transmission coefficient decreases to zero as the size of the system increases, either exponentially throughout or exponentially after a short transient. Above the threshold value this model shows undistorted motion of the microscopic particle in the disordered systems, i.e., the transmission coefficient does not decay and localization does not occur because of the small width of the microscopic particle compared to the large values of a. In the above discussion, the only effect of the impurity is to give rise to an effective potential on the microscopic particle. In particular, a microscopic particle (soliton) may be trapped by an attractive potential due to the impurity as a result of loss of its kinetic energy through radiation. However, the impurity is not a "hard" object, and it may support a localized oscillating state, or the socalled impurity mode. As a consequence, a microscopic particle may be able to transfer part of its kinetic energy to the impurity and thus excite the impurity mode during an inelastic interaction. Hence, there is a different type of interaction between a microscopic particle and an impurity when the impurity supports a localized impurity mode. The microscopic particle can be totally reflected in the case of an attractive impurity if its initial velocity lies in certain resonance "windows". We now discuss these problems. Let us considered the SineGordon model in the 1 + 1 dimensional case, including a local impurity 4>tt  <j>xx + sin (j> = eS(x) sin cf>,
(4.217)
where 8(x) is the Dirac <5function. The natural unit system is assumed in (4.217). It is known that the SineGordon model supports a topological soliton (kink) at e = 0, given by
^=4tan 1 exp[^^],
(4.218)
where X(t) = vt is the kink coordinate and v = X is its velocity. For e > 0, the impurity in (4.217) creates an effective attractive potential well for the microscopic particle (kink). Kivshar et al. studied the scattering of a microscopic particle (kink) by a pointlike impurity, and integrated (4.217) using a conservative numerical scheme. Simulations were carried out in the spatial interval (40,40) which was discretized by a step size of Ax — 2Ai = 0.04. The Dirac <5function was approximated by 1/Ax at x — 0, and zero elsewhere. The initial conditions were chosen such that the microscopic particle (kink) always centers at X = — 6, moving toward the impurity with an initial velocity v, > 0. Fixed boundary conditions, (40) = 0 and (40) = 2TT, were used in the simulation. Only the atractive interaction by the impurity, i.e., e > 0 were considered in the work of Kivshar et al. Depending on whether the microscopic particle passes through the impurity, is
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
207
captured by the impurity, or is reflected from the impurity, there are three different windows of the initial velocity of the incoming microscopic particle, and a critical velocity vc. If the initial velocity of the microscopic particle is greater than vc, it will pass through the impurity inelastically and escape to the positive direction, losing part of its kinetic energy through radiation and excitation of an impurity mode during the process. In this case, there is a linear relationship between the squares of the initial velocity Vi and the final velocity ve of the microscopic particle, i.e., ve = a'(u?  vl), where a' is a constant. For e = 0.7, the critical velocity is approximately 0.2678, and a' « 0.887.
Fig. 4.19 The kink coordinate X(t) versus time for initial velocity, Vi, of the microscopic particle in different windows: passing through (solid line, Vi = 0.268), being captured (dotted line, Vi — 0.257), and being reflected (dashed line, Vi = 0.255).
If the initial velocity of the incoming microscopic particle is smaller than vc, the microscopic particle cannot escape to infinity from the impurity after the first interaction, but will stop at a certain distance and return, due to the attractive force of the impurity, and to interact with the impurity again. For most of the velocity values, the microscopic particle will lose energy again in the second interaction and eventually get trapped by the impurity (see Fig. 4.19). However, for certain values of the initial velocity, the microscopic particle may escape to the negative infinity after the second interaction, i.e., the microscopic particle may be totally reflected by the impurity (see Figs. 4.19 and 4.20). This effect is very similar to the resonance phenomena in the kinkantikink collision which was explained by the resonant energy exchange mechanism proposed by Campbell et al.. The reflection of the particle (kink) is possible only if the initial velocity of the particle is within some resonance windows. By numerical simulation, Campbell et al. found eleven such windows. The details are shown in Fig. 4.21. Similar resonance phenomena for the microscopic particle (kink)  impurity interaction can also be observed in the 4model, 4>ttcf>xx[le5{x)\{
which is given in the natural unit system. The inelastic interaction of the micro
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Quantum Mechanics in Nonlinear Systems
Fig. 4.20 4>(0, t) versus time in the case of resonance (vi = 0.255). Note that between the two interactions there are four small bumps which show the oscillation of the impurity mode, and after the second interaction the energy of the impurity mode is resonantly transferred back to the microscopic particle (kink).
Fig. 4.21 Final velocity of the microscopic particle (kink) as a function of the initial velocity for e = 0.7. Zero final velocity means that the microscopic particle is captured by the impurity.
scopic particle with an impurity was first studied by Belova et al. However, they ignored the impurity mode, and attempted to explain the resonance effects based on energy exchange between the translational mode of the microscopic particle and its internal mode. Later, Kivshar et al. studied again the (/>4kink  impurity interaction by intensive numerical simuiation and found that both the internal mode and the impurity mode take part in the resonant interaction. For example, at e = 0.5, they found six resonance windows below the critical velocity vc w 0.185. The resonance structure in the 4kink  impurity system has an internal mode which also can be considered as an effective oscillator. These problems can be solved using the collectivecoordinate approach taking the three dynamical variables into account. To summarize, we described here a new type of microscopic particle  impurity interaction when the impurity supports a localized mode. In particular, we demonstrated that a microscopic particle can be totally reflected by an attractive impurity if its initial velocity is in a certain resonance window, which shows that the microscopic particle has similar properties as a classical particle. These resonance phenomena can be explained by the mechanism of resonant energy exchange between the translational mode of the microscopic particle and the impurity mode.
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
209
Other phenomena, for example, the transmission, the soliton modes and the impurity modes which occur in the interactions, can be understood by the wave feature of the microscopic particle. Therefore, the scattering of microscopic particles in nonlinear quantum mechanics by impurities also exhibit wavecorpuscle duality. 4.14
Tunneling and Fraunhofer Diffraction
In the last section we discussed the resonant behaviors of microscopic particles through trapping, reflection and excitation of impurity modes, when an microscopic particle is scattered by impurities with an attractive ^function potential well. The use of the <5function impurity potential allowed us to investigate the dependence of reflection and capture processes by the impurities on the initial velocity of the incoming microscopic particle. In these discussions, however, we did not consider the effects of finite width of the potential well or barrier. If this is taken into consideration, tunneling of the microscopic particle through the well or barrier would occur due to its wave property in the nonlinear quantum mechanics. Kalbermann studied numerically the interaction of an microscopic particle (kink) in the SineGordon model with an impurity of finite width, and attractive, repulsive or mixed interactions, respectively. In these studies, the Lagrangian function of the system was represented by (in natural units)
£ = d^9^ + iAU2— J ,
(4.219)
where g is a constant, A = g + U(x), and U(x) is the impurity potential which is given by
U(x) = h! cosh'2 f ^
1
) + ht cosh"2 i^1—)
•
(4220)
Here hi, h2 and xit x2 are the strengths and positions of the two impurities, respectively, a\ and a2 are some constants. If hi > 0 and h2 < 0, this potential describes a repulsive impurity and an attractive impurity. The equations of motion were solved by Kalbermann using a finite difference method. A microscopic particle (soliton) was incident from x — — 3 with an initial velocity v0, on an impurity located at x = 3. The spatial boundaries were taken to be 40 < x < 40, with a grid of dx = 0.04. A time step of dt = 0.02 was used and the simulation was carried out for a total elapsed time of t = 200 (or 10000 time steps), which was sufficient to allow for resonant tunneling to decay, permitting a clear definition of the asymptotic behavior of the microscopic particle. The asymptotic velocities for the reflected and transmitted cases were calculated using the actual motion of the center of the microscopic particle. In his calculation, Kalbermann used g = m2 in (4.219), and 0.7, 1.0, and 1.5 respectively for m, which correspond to the cases where the size of the microscopic
210
Quantum Mechanics in Nonlinear Systems
particle, « 1/ro, is larger, comparable, and smaller, respectively than the barrier width, « a/6, where a is the parameter in the argument of U{x) in (4.220). A width of ai = 1 was assumed for the repulsive barrier while the attractive barrier has a width of «2 = 03 These values were chosen to illustrate all effects in a reasonable range of velocity values around v0 « 0.25. These considerations, plus trial and error, led to the choices of hi = 1 and h2 =  6 . Fig. 4.22 shows the impinging microscopic particle (soliton) as well as the barriers. Figs. 4.23  4.25 show the final velocity v' as a function of the initial velocity v0 for the repulsive {hi = 1 and h2 = 0 ) , attractive {hi = 0 and h2 = 6), and attractiverepulsive cases, respectively.
Fig. 4.22 From top to bottom: Kink with m = 1 impinging from left onto a repulsive barrier; Kink with m = 1 impinging from left onto an attractive impurity; Kink with m = 0.7 impinging from left onto an attractiverepulsive system; Kink with m = 1.5 impinging from left onto a repulsiveattractive arrangement.
In the repulsive case (Fig. 4.23), the microscopic particle is reflected if the final velocity of the particle v' < 0, up to a certain value of the initial speed for which the effective barrier height becomes comparable to the kinetic energy, and then a sudden jump to transmission occurs. In all three cases shown in Fig. 4.23, the transmission starts at the same kinetic energy, with minor differences due to the effective barrier. In the attractive case there are islands of reflection between trappings and resonant behavior in which the microscopic particle remains inside the impurity and oscillates, exciting the impurity mode as mentioned in the previoius section. Again the higher the mass, the smaller the critical velocity for which transmission starts.
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
211
Fig. 4.23 Final velocity v' versus the initial velocity vo for rn = 0.7 (upper panel), m = 1 (middle panel) and m = 1.5 (lower panel) for the repulsive barrier.
The details of the reflection islands depend strongly on the parameters, but the general trend is similar for all three cases of different masses. In the case of combination of attractive and repulsive impurities shown in Fig. 4.25, the reflection dominates at low velocities which is induced by the repulsive impurity. Trapping and resonant behavior occur with islands of reflection. At high velocities, the transmission occurs which is essentially dictated by the same impurity. The repulsiveattractive case is similar to the repulsive case for velocity below transimission, and the critical speed here is determined mainly by the attractive impurity that can drag back the microscopic particle (soliton) after it passes through the barrier. It appears that the larger the mass (the thinner the microscopic particle), the more the attractive impurity is capable of trapping, thereby producing a somewhat counterintuitive behavior in which a massive microscopic particle needs a higher initial velocity in order to traverse them. Concerning the duration of the microscopic particle inside the barrier, there is always a time delay in the impurities, in contrast to the quantummechanical Hartmann effect. Furthermore, the energy of the microscopic particle is conserved in the scattering. The above discussion addressed the tunneling effect of microscopic particles in the onedimensional case. It shows again the wave property of microscopic particles in the nonlinear quantum mechanics. However, in order to relate more closely to actual tunneling of microscopic particles in nonlinear quantum mechanics, one has to consider higher dimensions, such as the 0(3) twodimensional case, and eventually including rotations of the microscopic particle (soliton), and other effects such as
212
Quantum Mechanics in Nonlinear Systems
Fig. 4.24 Same as Fig. 4.23 but for the attractive case.
fluctuations. Moreover, actual barriers can be dynamic. There is then a need to allow for more flexibility in the modeling of impurities as well as the possibility of energy dissipation. In the following, we consider the Praunhofer diffraction of microscopic particles described by the nonlinear Schrodinger equation (4.40) with a small scale initial phase modulation and a rather large initial intensity. These conditions allow us to use a perturbution approach based on multiscale expansions to investigate such problems, as was done by Konotop. Following the approach of Lax and the inverse scattering method, the linear spectral problem corresponding to the nonlinear Schrodinger equation satifies the ZakharovShabat equation (4.42), where A is the spectral parameter, <po(x') = cj)(x',t' = 0) being the initial condition for the nonlinear Schrodinger equation (4.40). Konotop assumed the following for for <j)o(x'), 4>Q{x') = b'g(x')f(x'),
(4.221)
here g(x') is a random and statistically homogeneous function < = (g(x')g*(x'1)) = Qdl(x'x'1), (9(x')9(x'i)) = Gd2(x'
(4.222)
x'i),
where (• • •} denotes the average over all realizations of the random function g{x'), di and d% are correlation radii (from now on, we assume d\ = d2 = d for simplicity),
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
213
Fig. 4.25 Same as Fig. 4.23 but for the attractiverepulsive case.
f(x') is a regular function varying on a unit scale, \f(x')\ ~ g(x')l ~ 1> &' represents the amplitude of the initial pulse and is a nonlinearity parameter. The solution of the ZakharovShabat equation is known to be of the form * = (t1)
= C(i')^i(x',A) + D{x>)th&,V,
(4223)
with /piXx'\
Vi(x',A)=^e0 y
f
0
M^)=(eixx>)
\
( 4  224 )
C(x') and D{x') in (4.223) satisfy the "initial" conditions: C(x' = oo) = C o , D(x' = —oo) = 0. The reflection coefficient R(\) was determined by Konotop, which is given by R(X)=
lim r{x')e2iXx\
r(x') = ?^e2iXx'.
(4.225)
Prom (4.221), r(x') satisfies the Riccati differential equation  ^ = 2i\r + i(j>o(x')b'  i^x'y"2.
(4.226)
Konotop restricted the "potential"
214
Quantum Mechanics in Nonlinear Systems
and r(x' = oo) = r(L). Equation (4.226) then becomes
^ = it [Ar + r(dZ')g'*(O  fidt'WiOr2} ,
(4.227)
where
A=A,
e = db>,
g(e) = g(x'), £' = £
For small correlation radius d
1
, . • • , ) + •••
(4.228)
l
where & = e £' a n d correspondingly
i
=
W + €Wi+""
(4 229)
"
In the case of f(d£) = f(xi) (i.e. V = 1 and e = d), from (4.227)  (4.229), Konotop obtained 0r(o)  ^ T = 0,
(4.230)
~ ^ T +«Ar(0) +i/*^)5"(e')  i / ^ i V ^ ) ^ ] 2 ,
(4231)
and
^T
=
where ^', ^i (i = 1,2, • • •) are independent variables. Equation (4.230) implies that r(o) = r(°)(£i). An expression for r^ follows directly from (4.231)
r(1) = e^
+ iM'r™ + it'nWit')
 t£7(&)F(£')[r(0)]3.
(4.232)
Here F(£') represents the "average" F
(O = J, t dx'9{x'), ? Jo
introduced by Konotop. r^ (^i) is then determined by requiring the secular terms on the right hand side of (4.232) to vanish. That is ^ ~
= iArW + ir&)F*(L)  if(^)F(L)[r^}2
(4.233)
where L = L/e. In the case under consideration the average reflection coefficient is given by r{L) = r^(L) + O(e2). Now let G{ti,x')=ir(Z1)F*{x')if(ti)F{x>)[rM]2,
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
215
where \3G(M=G2)
and
cjr(i)=rW.
Then, (4.232) may be rewritten in the following form r(i1)(O =
Z'[Gi(Si,OGi(S1,L)}.
In the case of stochastic initial conditions, i.e., g(x') is a random function, the reflection coefficient r^ (£') is a random function too. Hence, there is e(r1)(e',6))«l,
(4.234)
e V / iy[rj 1 ) (e',Ci)]«l,
(4.235)
and
where W[r] = (r2) — (r) 2 is the dispersion, and it is given by
W [rf >(£')] =W[eGi(t1,?)] + iw [LGitfuL)]
(4.236)
~T [(S'Gi{Si,aLGifo,L))  itCMi,?)) (LGi(ti,L))] . Since our discussion is limited to the statistically homogeneous function g(x') with finite dispersion \Q(x')\ » W(a;') « 1, an estimate for W[r] follows from (4.236), W[r] « L. Therefore, the inequalities (4.234) and (4.235) may be rewritten in the form of L <$i e~l. If we further assume that g{x) is an ergodic process, the function F(L) reduces to (g(L)) + 0{eL~l) and
= iArW + iffoW
 if(Zi)9(x')[rM}2.
(4.237)
Konotop also studied Praunhofer diffraction of an microscopic particle (soliton) which has features of a phase modulated noncoherent wave through a slit. The diffraction pattern detected on the screen is also affected by the nonlinearity of the medium behind the screen which was taken into consideration by Konotop. The corresponding initial condition for the nonlinear Schrbdinger equation in this diffraction is taken to be a square potential with a random phase, , , M 0ola:J
_ / cj>oeie^')+ie", ~\O,
for 0 < x1 < L, for *' < 0 or x' > L.
where 0(x') is a Gaussian noise with statistical characteristics {0(x)} — 0, (0(x)6(x')) = Q{x'  ar'j), Q(0) = a2. For 0 < x' < L, it is easy to show that
(4>o(z')> = foe'212 and (0o(O<M*i)) « Qtf  A)
(4238)
216
Quantum Mechanics in Nonlinear Systems
If Q{x') > 0 as x' >• oo, the process
_
ie
(j)(*+D/^ o tanh [JV+WL&B]
e
°2iXL
(*) ( < !+1)/2 Atanh[i(' 5 + 1 )/ 2 LA i ]+jA 5
'
(4 239)
'
where A , = ^4>2oes2
+8X2,
(<5 = ± 1 ) .
(4.240)
We can also determine the property of the Fraunhofer diffraction in such a case. In fact, in the diffraction problem, 6 = 1 and 5 = —1 correspond to focusing and defocusing medium, respectively. The diffraction picture in the Fraunhofer zone is determined by the Manakov's formula l
(4.241)
Prom (4.238), (4.240) and (4.241), it follows that the diffraction picture of the phase modulated microscopic particle is similar to that of the regular wave with a smaller amplitude. As far as the phase modulations are concerned, there are no limitations on the phase fluctuation dispersion, i.e., (el\e(x')+0(x'i)'\) < 1 for any
Q(x'x[). Similar results may be obtained for a wave with an amplitude modulation of the type j . , ,,
MX)
f
\O,
forO<x'
for z ' < 0 or z ' > L.
The solution takes the form of (4.238), with — a2 substituted in the place of a2. But for the case of a focusing medium one must take the possibility of creation of a microscopic particle (soliton) into account. This leads to an upper bound for a2 for applicability of (4.241) in the case of 5 = 1. Equation (4.233) for r^ does not contain L explicitly. However, if we extend the above results to infinite initial conditions with high decreasing speed, the time interval in which the initial condition cf>o(z') is essentially nonzero can be used as parameter L. From the above discussion we see that the Fraunhofer diffraction phenomenon takes place when a microscopic particle described by the nonlinear Schrodinger equation moves in a perturbed systems with random initial condition and a small correlation radius. This phenomenon also occurs for pulse waves with slowly varying initial phase modulations (amplitude and phase modulations) which propagate in a dispersive medium. Once more, this shows the wave nature of microscopic particles described by the nonlinear Schrodinger equation. Fraunhofer diffraction of microscopic particles, described by (4.40), from a belt in a nonlinear defocusing medium, was also studied by Zakharov and Shabat, under
Wave Corpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
217
the initial conditions
[ 1,
for a:' > a'.
y
'
Here t' and x' are regarded as the longitudinal and transverse coordinates, and a' is the halfwidth of the belt required for the diffraction, the amplitude of the infinite wave is set equal to unity. In accordance with the results of Section 4.3, it is necessary to solve the eigenvalue problem (3.21) and (4.42) with the function q = 4>\V=Q. We have
Cl [j]e^'+c 2 [°]e^', * = < c2 Cl
f 1 A2
( 
i)
(for \x'\ < a'),
1/2 A
 LvTrFx')
i(lAa)1/a~A
e VT=JSX
~
'>
(fsxaj>a,)t
(4.243)
(for i ' <  a ' ) ,
where Ci and ci are the Jost coefficients. Requiring the solution (4.243) to be continuous at x' = ±a, we obtain the following eigenvalue equation after some elementary transformations cos2Aa' = A.
(4.244)
Equation (4.244) has a set of zeros, ±An, which are symmetric and approaching zero, and An < 1, as expected. For sufficiently small a', there is only one pair of zeros. For large a', the number of pairs of zeros, N, can be estimated from N « 2a'/n. Local darker bands in the diffraction pattern, given by the zeros of (4.244), can be found which correspond microscopic particles (solitons) that propagate along straight lines in the (x',t') plane. The microscopic particle (soliton) with the eigenvalue An moves in a direction that makes an angle 0n = tan 1 (2A n ) with the direction of propagation of the wave. The minimum number of such bands is two. Thus, the diffraction of a microscopic particles by a belt in a nonlinear medium differs in principle from the diffraction in a linear medium in that it can be observed at an arbitrarily large distance from the belt, whereas in a linear medium the diffraction picture becomes "smeared out" at large distances. As a matter of fact, the Fraunhofer diffraction for superconducting electron (soliton) was experimentally observed in the superconducting junctions (See Fig. 2.3). Therefore, it is confirmed beyond doubt that Fraunhofer diffraction can occur for microscopic particles in nonlinear quantum mechanics.
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Quantum Mechanics in Nonlinear Systems
4.15
Squeezing Effects of Microscopic Particles Propagating in Nonlinear Media
Interferometer experiments of Wu et al. showed that propagation of a laser field along an optical fiber produces a squeezing effect, which refers to the reduction of narrowband quantum fluctuation in one of the quadrature phases of the field to a level below the usual limit for a coherent state. The squeezing of a laser field can be induced by fourwave mixing, resulting from quantum noise. Calculation on the quantum fluctuations in a dispersionless medium predicted the squeezing effect in one of the quantum phases of a propagating continuous wave (CW) field that is consistent with experimental results. In nonlinear quantum mechanics such squeezing phenomena also occur for quantum microscopic particles described by the nonlinear Schrodinger equation (3.62), i.e., propagation of a microscopic particle in a dispersive nonlinear medium can lead to a wideband squeezing of quantum fluctuations in the vicinity of the microscopic particle (soliton). This problem was studied by Carter et al.. In their work, source of quantum noise was added in the original nonlinear Schrodinger equation as a stochastic field, and the nonlinear Schrodinger equation becomes the following stochastic form
iW£Jl
=
1 \i ± ^ _ j ftfj)
_ 101*0(^,0 +i][Ir1(x',t')ct>(x',t'), (4.245)
where n — k" /bt'Q is the average number of excitation of the particle field, or photon number in the CW field, k" = d2k/duj2\ko is the dispersion of overall medium group velocity, t'o = ^/w9A;"/(2Aw) is the time scale, b is the effective nonlinearity, vg is the group speed of the microscopic particle (soliton). The real and stochastic field r\ and rj+ are introduced here with (r,(xi,*i)r,(x'2:t'2))
= (V+ (x[,t[)
V+
(x'2,t'2)) =S(x[
x>2)8(t[ 
t'2),
(4.246) (v+(x'1,tl1)v(x'2,t'2))=0, where the (+,  ) signs in (4.245) correpond to the cases of normal (k" > 0) and anormalous (k" < 0) dispersions, respectively. Equation (4.245) can be derived from quantum theory of propagation in dispersive nonlinear media in which quantum fluctuations are handled via the coherentstate positiveP representation of Carter et al. In (4.245), the stochastic part provides the quantum fluctuations in the problem, and is the source of inherently squeezed quantum noise. This term is easily modified to include thermal phasemodulation noise. Carter et al. first studied the squeezing effect of a CW field. They first defined the spectrum of quadrature fluctuations in the CW field at location x' with a phase
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
219
angle 8, following the approach of Gardiner et al, S(u,,x',6) = ^ ° 5 K
\e2ie(A^D,x')A^(u,,x'))
+ {AJ>(Q,x')A4>+{u,x'))] •
(4247)
A similar equation for A<£+ can be obtained 1 rT/2t'o &<j>(Q,x') =  = / A^VK*' dt', y/2K JT/2t'o
A
1  costs') 5
ism(jx)
, /?[cos(73/)  1 ] ' 2 '
, . (4.248)
where /? = l±w 2 ; 7 = wVw2 ± 2. This expression can also be derived using operator techniques, and in the nondispersive limit it gives a rigorous basis. The spectrum, in the anomalousdispersion case, is shown in Fig. 4.26. It has an exponential fluctuation reduction at finite frequency (Q2 < 2), with perfect squeezing in the limit of a;' > 00. This implies exponential growth in the complementary quadratures, so that linearization will not be valid for sufficiently large x'. Similar modulation instabilities are known to exist for propagation of a classical CW. In the case of a microscopic particle (soliton) in nonlinear quantum mechanics, an identical computational technique can be used. Carter et al. extended the above study to the case of a known classical soliton solution. Here the fluctuations are defined relative to a timedependent solution of <j>o{t') = sech(i') Aw is chosen to give a characteristic time scale, t0, corresponding to the size of the microscopic particle. The peak power is hu>o\k"\/btlQ. The linearized equations are similar to
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Quantum Mechanics in Nonlinear Systems
Fig. 4.26 Graph of the logarithmic spectrum ln[l 4 S(x',u>)ma.x] of maximum squeezing in the case of anomalousdispersion.
those given above in the case of CW, except for the timedependent fourwave mixing and stochastic terms. These can be treated numerically as a set of coupled linear differential equations in x'. The maximumsqueezing curves are shown in Fig. 4.27. The results clearly demonstrate a wideband squeezing over the spectral width of the input microscopic particle (soliton). The phase angle of the largest squeezing is approximately TT/4.
Fig. 4.27 Graph of the spectrum of maximum squeezing in the case of anomalousdispersion microscopic particle (soliton).
Note that quantum limits in the CW and microscopic particle in nonlinear quantum mechanics are somewhat different. The fluctuations are now localized over a time interval ss 2to near the mass center of the microscopic particle. This correponds to squeezing in a localized mode of the radiation field copropagating with the microscopic particle. Hence the pulse duration used here to normalize the spectrum is Tp = 4£Q, which is approximately the size of a classical microscopic particle (soli
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
221
ton). This is necessary in order to obtain a finite result for a pulsed field. Squeezing can also be defined relative to more general mode functions. In summary, the quantum noise source is added into the general nonlinear Schrodinger equation as a stochastic field, which reduces the quadrature fluctuations in one quantum at a phase of TT/4 relative to the carrier. This intrinsic squeezing initially increases linearly with distance. At high relative frequencies, the effects of linear dispersion cause a phase rotation of the squeezing generated at different spatial locations, which causes interference in the spectrum, and reduced squeezing. For long propagation distances the quantum noise undergoes further modifications due to fourwave mixing and nonlinear dispersion. In the normaldispersion regime, this reduces the extent of the squeezing. In the anomalousdispersion regime, there is a range of frequencies where the linear dispersion counterbalances the nonlinear dispersion. This results in enhanced squeezing and exponential noise reduction with distance. These general comments of the squeezing effect hold for both CW and the microscopic particle (soliton) inputs. This squeezing effect of the solitons shows that a microscopic particle has wave property in nonlinear quantum mechanics, because only wave is known to have the squeezing property. When the input is CW, the results are well known. In the case of a coherent microscopic particle (soliton) input, the results were not obtained before. These demonstrate the squeezing of quantum fluctuations below the vacuum level in a microscopic particle (soliton). Therefore, this result suggestes that the squeezing is an universal property of propagation of microscopic particles described by the nonlinear Schrodinger equation in quantum limit.
4.16
Wavecorpuscle Duality of Microscopic Quasiperiodic Perturbation Potential
Particles
in a
The behaviors of microscopic particles subject to arbitrary perturbations are very complex in general. However, for certain types of perturbations, sufficiently simple behavior can be expected even for relatively strong perturbations, which allow an analytic description in terms of a small number of collective variables. In nonlinear systems there always exist inhomogeneities or disorder, and the interactions involved will certainly have an anharmonic component. These can give rise to interconversion of excitations. Thus the microscopic particle (soliton) can emit radiation which can selffocus to form coherently localized excitations. Meanwhile the microscopic particles can be scattered inelastically or be broken up. In this section, we examine the behaviors of microscopic particles in nonlinear quantum mechanics subject to a spatially quasiperiodic perturbation potential in nonlinear systems. This problem was studied by Scharf et al. The perturbed nonlinear Schrodinger equation (3.2) in such a case can be written
222
Quantum Mechanics in Nonlinear Systems
as i
(4.249)
where V(x') = cos(kx') and e is a small parameter. As mentioned above, for a single localization of a microscopic particle moving in the presence of the perturbation, we make a collective variable ansatz
(4.250)
where q is the location of the microscopic particle (soliton). Equation (4.249) possesses two integrals of motion, the norm N and energy E. For
\4>\2dx' = 4T?,
(4.251)
[<M2  M 4 + e2 cos(fcz')] & '
(4.252)
N= J — oo
and /•+OO
E= / = 7
T " T 1 7 J + sinh(fe7r/4,) C ° S ( f c g ) i
respectively. The conservation of the norm of 0 leads to 77 = const. + O(e2). The time dependence of the phase 6 is decoupled from the time dependence of the position q of the microscopic particle, which is the relevant dynamic variable. The total energy given above depends only on the position q(t) of the microscopic particle. Conservation of the total energy E leads to an equation of motion for q which can be derived from the following effective singleparticle Hamiltonian Hi. = — + , £ * 7r , . cos{kq). v ; 277 2 sinh(ifc7r/4r?)
v
(4.253)
The effective potential of the particle is Veff(q) = ekir csch I — j cos(kq),
(4.254)
which is derived from
VeS(q) = hfe j
J—00
sech[27](a;  q)]V(x).
Scharf et al. found the following timedependent equation for the collective coordinate q(t) = ~V:Aq), where Ve'ff =
dV^/dt.
(4255)
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
223
Equation (4.255) resembles (4.17) and (4.25). It shows that the microscopic particle satisfies the classical equation of motion, i.e., the microscopic particle has corpuscle property just as classical particles, and the mass of the particle is 2TJ in such a case. Equation (4.254) shows also that a periodic potential with short wavelength becomes an effective potential which is exponentially decreasing for large k/rj in such a case. We now neglect the perturbation (e = 0) and focus on a collision between two microscopic particles (solitons). When the two microscopic particles collide their centers of inertia, q,, as well as their phases &i, suffer a shift as mentioned earlier. As the perturbed nonlinear Schrodinger equation (4.249) is U(l) invariant, we neglect the dynamics of the phases. The shifts of the positions qi ("space shifts") of the microscopic particles amount to an attractive interaction between the microscopic particles. For example, the microscopic particle that was on the right hand side at t )• co and on the left hand side at t ~> too has the form lim \
For the other microscopic particle, the shift a should be replaced by —a. The shift a is given by o=lln[
16
fa+^)a
+
^^
a
1,
(4.256)
2 [ (vi v2)2 J where v\ and v2 are the asymptotic velocities of the two separated microscopic particles and r]i and r\i their amplitude parameters [see (4.250)]. A simple calculation shows that the following twoparticle Hamiltonian gives rise to the same space shifts as (4.256)
* = £ + £ ***(* + m) h 2 p ^  ^ l •
(4257)
Combining the singleparticle Hamiltonian Hi with the attractive twomicroscopic particles interaction in (4.257), we can find the following effective Nparticle Hamiltonian: (4.258)
l
l
" < 3  l
This Hamiltonian is similar to that describing the motion of N anharmonically coupled nonlinear pendula which becomes nonintegrable for N = 2. The microscopic particle (soliton) depicted by (4.250) has a spatial width and thereby selects a certain length scale. Thus two parameters, the size of the microscopic particle, Ls, and the width of the perturbation potential, Lp, respectively
224
Quantum Mechanics in Nonlinear Systems
can be introduced. Two situations, Ls > Lp and Ls < Lp, can be distinguished. In the former, the microscopic particle (soliton) covers many wiggles of the potential, while in the latter the microscopic particle experiences only negligibe potential differences over its size. In the intermediate case, Ls ss Lp, the behavior of the microscopic particle is much more complicated and bears no resemblance to the unperturbed dynamics. Scharf et al. carried out numerical simulation of (4.249)  (4.258) which gave rich and fascinating behaviors of the microscopic particle. In summary, if the width of the microscopic particle (soliton) is small compared to the length of perturbation potential, it is natural to neglect all degrees of freedom of the microscopic particle besides its center of mass motion and therefore treat it as a particle. In contrast to the full dynamics this reduced dynamics will still be integrable for the nonlinear systems. Its properties are governed by an effective Hamiltonian (4.258) or (4.253) and (4.257). If the width of a microscopic particle is large compared to the typical lengthscale of the perturbation, then one can expect the effect to be an average over the fine spatial details of the potential. Thus the microscopic particle shows obvious wave property. This was verified by detailed investigations. Starting from (4.249), with V(x') = cos(kx') and with a large microscopic particle (77
(4.259)
The bare soliton <po(x',t') of the unperturbed equation acquires a dressing in the presence of the perturbation. It can be shown that the dressed microscopic particle (soliton) fulfills an unperturbed effective nonlinear Schrodinger equation with renormalized parameters. The dressed microscopic particles behave like bare solitons when subject to additional longwavelength perturbation. Again, their center of mass motion can be described by an effective single particle Hamiltonian. When two dressed microscopic particles collide they reemerge essentially unchanged thereby illustrating the nearintegrability of the dynamics. As can be seen from (4.259) there are two types of dressed microscopic particles, namely the slow and the fast particles depending on whether q C k or q 3> k. The slow dressed microscopic particles are spatially modulated, with the maxima appearing at minima of the perturbing potential, while the maxima of the fast dressed microscopic particles occur at the maxima of the potential. These two regimes are devided by a "phase resonance" leading to a destruction of the microscopic particles. This analysis has been extended to more general potentials with a quasiperiodic
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
225
part containing many short wavelengths and an arbitrary long wavelength part. The resulting solitary solutions are dressed microscopic particles (solitons) of more complicated forms moving like particles in a long wavelength effective potential. Lengthscale competition in general leads to complicated behavior in space and time. In the case of the nonlinear Schrodinger equation this can happen either through the "phase resonance" ( q sa k ) as mentioned above or a "shape resonance" {q » k). The effective particle approximation for the microscopic particles in the nonlinear quantum mechanics shows that the nonintegrability of the perturbed nonlinear Schrodinger equation can manifest itself through "microscopic particle (soliton) chaos" induced by the longwavelength part of the perturbation. The effective decoupling of a few degrees of freedom leads to simple dynamics of the microscopic particle through nonintegrable behavior in space and time. Inelastic effects can show up in processes involving only few effective coordinates or many degrees of freedom. Larger curvature of the longwavelength potential (i.e., V"(a;') ss TJ2) can lead to a decay of the microscopic particles through radiation. The radiative power can be calculated using perturbation theory. Collisions of two microscopic particles with nearly vanishing relative velocity in a perturbing potential show effects no longer described by the effective twoparticle Hamiltonian He$ given in (4.258). The soliton parameters rn which played the role of masses in the collective variable description are no longer constant, but become dynamic variables themselves. As the amount of radiation generated might still be negligible an extended collective variable description seems to be feasible. Simultaneous collision of three and more microscopic particles in the presence of a perturbation lead to effects which in general can no longer be treated by effective twoparticle interactions. The power of radiation generated by "dressed microscopic particles" appears to be of higher order than r\jk, as concluded from numerical simulation. Collisions of two "dressed microscopic particles" in the case of the nonlinear Schrodinger equation as well as kinks and antikinks for SineGordon equation lead to a noticible increase in the power of radiation, probably to the order of 77/fc. These phenomena are due to the wave feature of the microscopic particles. Therefore, the above discussion gave evidence to the wavecorpuscle duality of microscopic particles in the nonlinear quantum mechanics. In order to justify this statement, Scharf et al. carried out further studies on the properties of microscopic particles described by the following perturbed SineGordon equation (using the natural unit system) <j>tt 
sin <j> = 0.
(4.260)
This equation of motion is generated by the Hamiltonian r+00
H=
r 1
1
1
dx\ $ + 
(4.261)
J00 [2 2 ) They used two different kinds of exact solutions of the unperturbed SineGordon
226
Quantum Mechanics in Nonlinear Systems
equation (e = 0) as initial conditions, the breathers and the kinkantikink (KK) solutions. The breather at rest has the form «/>">,*) = 4tan 1 ( t a n M S i n « ;  ' ° ^ ) , [ cosh[(a;xojsin^J J
(4.262) '
where fj, is the parameter governing the breather shape and size. As /x —j 0, the breather becomes shallower, its frequency, given by w&r = cosfi, grows, and it can be effectively described by the nonlinear Schrodinger equation. On the other hand, when fi^ n, the breather frequency goes to zero and it is actually very close to the KK pair. As for the KK solution, its expression at rest is given by
***(x,t) = 4tan ( S i n h ; ] f  ^ 1 ,
(4.263)
1 v cosh[7(x  x0)] J with v1 < 1, 7 = l / \ / l — v2. Equation (4.263) can also be obtained from (4.262), by letting fi  ir/2 + ia', where ea> = 7(1 + v). Finally, both (4.262) and (4.263) can be derived using standard methods which will be discussed in Chapter 8. The energy of an unperturbed breather at rest, as can be obtained from (4.260) and (4.261), is EQT = 16 sin/i, whereas the same computation for an unperturbed KK solution with its center of mass at rest is E$~K = I67. In the absence of any perturbation the energies for the breather and the KK solutions at rest obviously fulfill EQT < 16 < Eff'R.
The perturbation may shift this borderline between the
breather and the KK solutions. Of relevance to this analysis will be the potential energy of an excitation in the absence of any perturbation ^
dx^<j>l
+
lcos<j>).
The amount by which the total energy is changed in the presence of the perturbation is given by r+00
VeS=
dx'ecos(kx'){lcos
(4.264)
j00
For a breather at rest at to = 0, Sharf et al. obtained _n E°
8 tanh 2 z = 8sinM+ — :
p
sin/*
82 sin u(l  cot 2 a sinh 2 z) ^ % ', sinhzcosh 3 z
and trbri \ VZZ(xo,z) =
47resinhzcos(fea;o) [sin(Kz) . , . 1 5 —^—' + Kcos(Kz)smz\ , eff sin/xcosh 2 z sinh(/s:7r/2) L coshz J' where K = fc/sin/z and z, denned by sinhz = tan fi sin(t cos n), is a measure of the distance between the kink and the antikink bound in the breather.
WaveCorpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics
227
For a KK solution at rest at t0 = 0, they found correspondingly _ , _ , , 8 7 z ( l + i; 2 S inh 2 /) — T5 , •^pot  °7 H sinh z cosh z 0
and
47resinh2cos(fcx0) \s\n( Kz) ,T N 1 2 . * ' , ; + # c o s # z sinz , 4.265 J 7CoslTz smh(Jff7r/2) L cosh 2 where if = fc/7 and 2, denned by sinh.z = v'1 sinh^ut), is a measure of the distance between the kink and the antikink. The effective potential Veff depends on the distance between the (virtual) kink and antikink, z, and on the center of mass, XQ, of the KK solution or the breather. For nonrelativistic velocities (7 « 1), Veff (xo,z) can be used to calculate the influence of the potential on the motion of the center of the excitation as well as on the relative distance between the kink and the antikink in an adiabatic approximation, assuming that the parameter z can be considered as a second collective variable. A breather is a bound state solution of the SineGordon equation which oscillates around <j> = 0 with a period T — 2n/cos^x. For sufficiently strong perturbations the total potential energy J5pot = E^ot + Vea can have other minima besides 0 = 0 around which the solution can oscillate. ^rKRi \ Kff ^o,^=
Fig. 4.28 Schematic "phase diagram" for the microscopic particle in the perturbed SineGordon equation.
Scharf et al. introduced two characteristic length scales, which are the breather width (As) and the perturbation period (Ap) and carried out a number of numerical simulations spanning large intervals of the parameters, namely, the length ratio (As/Ap) and the potential strength (e). They found that there are basically three possible behaviors for the breather, largely independent of its amplitude. If AB < Ap (see Fig. 4.28) the breather can indeed be considered as a particle in the external periodic potential, and if AB > Ap, this is still valid except that now the effective potential in which the particle moves is not the original but a renormalized one. On the contrary, if AB ~ Ap (competing lengths), this particlelike behavior ceases to be true even for small e values, and the breather rapidly breaks up, either into a kindantikink (KK) pair (if its amplitude is large enough) or into two or more
228
Quantum Mechanics in Nonlinear Systems
breathers, involving always a great amount of radiation. Interestingly, they also found that breather breakup happens also for noncompeting lengths when e is above a certain threshold e T , which depends on A B / A P . They observed nonradiative splitting of largeamplitude breathers into KK pairs for large A B / A P , which seems to have its origin in energetic considerations and not in the length competition. All these phenomena are smoothed out by dynamic effects, when the breathers are moving, as observed in defail in the nonlinear Schrodinger equation. Therefore, the features shown in Fig. 4.28 indicate that microscopic particles described by the SineGordon equation and the nonlinear Schrodinger equation have wavecorpuscle duality because they have either corpuscle feature in one case, or wave feature in other case. Thus it gives us a satisfactory answer to the question of this chapter whether microscopic particles have the wavecorpuscle duality in nonlinear quantum mechanics.
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Kivshar, Yu. S. and Malomed, B. A. (1989). Rev. Mod. Phys. 61 763. Kivshar, Yu. S., Fei, Z. and Vazquez, L. (1991). Phys. Rev. Lett. 67 1177. Kivshar, Yu. S., Gredeskul, S. A., Sanchez, A. and Vazquez, L. (1990). Phys. Rev. Lett. 64 1693. Klyatzkin, V. I. (1980). Stochastic equations and waves in randomly inhomogeneous media, Nauka, Moscow (in Russian). Konotop, V. V. (1990). Phys. Lett. A 146 50. Lai, Y. and Haus, H. A. (1989). Phys. Rev. A 40 844 and 854. Lamb, G. L. (1980). Elements of soliton theory, Wiley, New York, pp. 118125. Lawrence, B., et al., (1994). Appl. Phys. Lett. 64 2773. Lax, P. D. (1968). Comm. Pure and Appl. Math. 21 467. Leedke, E. and Spatschek, K. (1978). Phys. Rev. Lett. 41 1798. Lifshits, I. M., Gredeskul, S. A. and Pastur, L. A. (1982). Introduction to theory of disordered systems, Nauka, Moscow. Lonngren, K. E. (1983). Plasma Phys. 25 943. Lonngren, K. E., Andersen, D. R. and Cooney, J. L. (1991). Phys. Lett. A 156 441. Lonngren, K. E. and Scott, A. C. (1978). Solitons in action, Academic, New York, p. 153. Louesey, S. W. (1980). Phys. Lett. A 78 429. Makhankov, A. and Fedyamn, V. (1979). Phys. Scr. 20 543. Makhankov, V. (1980). Comp. Phys. Com. 21 1. Makhankov, V. G. (1978). Phys. Rep. 35 1. Makhankov, V. G. and Fedyanin, V. K. (1984). Phys. Rep. 104 1. Marchenko, V. A. (1955). Dokl. Akad. Nauk SSSR 104 695. Miles, J. W. (1984). J. Fluid Mech. 148 451. Miles, J. W. (1988). J. Fluid Mech. 186 718. Miles, J. W. (1988). J. Fluid Mech. 189 287. Nakamura, Y. (1982). IEEE Trans. Plasma Sci. PS10 180. Nishida, Y. (1982). Butsuri 37 396. Nishida, Y. (1984). Phys. Fluids 27 2176. Pang, Xiaofeng (1985). J. Low Temp. Phys. 58 334. Pang, Xiaofeng (1989). Chin. J. Low Temp. Supercond. 10 612. Pang, Xiaofeng (1990). J. Phys. Condens. Matter 2 9541. Pang, Xiaofeng (1993). Chin. Phys. Lett. 10 381, 417 and 570. Pang, Xiaofeng (1994). Phys. Rev. E 49 4747. Pang, Xiaofeng (1994). The theory for nonlinear quantum mechanics, Chongqing Press, Chongqing. Pang, Xiaofeng (1999). European Phys. J. B 10 415. Pang, Xiaofeng (2000). Chin. Phys. 9 89. Pang, Xiaofeng (2000). European Phys. J. E 19 297. Pang, Xiaofeng (2000). J. Phys. Condens. Matter 12 885. Pang, Xiaofeng (2000). Phys. Rev. E 62 6898. Pang, Xiaofeng (2003). Phys. Stat. Sol. (b) 236 43. Pang, Xiaofeng (2003). Soliton physics, Sichuan Sci. and Tech. Press, Chengdu. Perein, N. (1977). Phys. Fluids 20 1735. Peyrard, M. and Campbell, D. K. (1983). Physica D9 33. Pitaerskii, L. P. (1961). Sov. Phys. JETP 13 451. Rytov, S. M. (1976). Introduction to statistical radiophysics, Part I. Random Processes, Nauka, Moscow. Sanchez, A., Scharf, R., Bishop, A. R. and Vazquez, L. (1992). Phys. Rev. A 45 6031. Sanders, B. F., Katopodes, N. and Bord, J. P. (1998). United pseudospectral solution to
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water wave equations, J. Engin. Hydraulics of ASCE, p. 294. Satsuma, J. and Yajima, N. (1974). Prog. Theor. Phys. (Supp) 55 284. Scharf, R. and Bishop, A. R. (1991). Phys. Rev. A 43 6535. Scharf, R. and Bishop, A. R. (1992). Phys. Rev. A 46 2973. Scharf, R. and Bishop, A. R. (1993). Phys. Rev. A 47 1375. Scharf, R., Kivshar, Yu. S., Sanchez, A. and Bishop, A. R. (1992). Phys. Rev. A 45 5369. Shabat, A. B. (1970). Dinamika sploshnoi sredy (Dynamics of a continuous Medium, No. 5, 130. Shelby, R. M., Leverson, M. D., Perlmutter, S. H., Devoe, P. S. and Walls, D. F. (1986). Phys. Rev. Lett. 57 691. Skinner, S. R, Allan, G. R., Allan, D. R., Andersen, D. R. and Smirl, A. L. (1991). IEEE Quantum Electron. QE27 2211. Skinner, S. R., Allan, G. R., Andersen, D. R. and Smirl, A. L. (1991). in Proc. Conf. Lasers and ElectroOptics, Baltimore, (Opt. Soc. Am.). Snyder, A. W. and Sheppard, A. P. (1993). Opt. Lett. 18 482. Sulem, C and Sulem, P. L. (1999). The nonlinear Schrodinger equation: selffocusing and wave collapse, SpringerVerlag, Berlin. Takeno, S. (1985). Dynamical problems in soliton systems, Springer Serise in synergetics, Vol. 30, Springer Verlag, Berlin. Tanaka, S. (1975). KdV equation: asymptotic behavior of solutions, Publ. RIMS, Kyoto Univ. 10 pp. 367379. Tan, B. and Bord, J. P. (1998). Phys. Lett. A 240 282. Tappert, F. D. (1974). Lett. Appl. Math. Am. Math. Soc. 15 101. Taylor, J. R. (1992). Optical solitontheory and experiment, Cambridge Univ. Press, Cambridge. Thurston, R. N. and Weiner, A. M. (1991). J. Opt. Soc. Am. B8 471. Tominson, W. J., Gordon, J. P., Smith, P. W. and Kaplan, A. E. (1982). Appl. Opt. 21 2041. Tsuauki, T. (1971). J. Low Temp. Phys. 4 441. Wright, E. M. (1991). Phys. Rev. A 43 3836. Wu, L., Kimble, H. J., Hall, J. L. and Wu, H. (1986). Phys. Rev. Lett. 57 252. Yajima, N., Dikawa, M., Satsuma, J. and Namba, C. (1975). Res. Inst. Appl. Phys. Rep. XXII70 89. Zabusky, N. J. and Kruskal, M. D. (1965). Phys. Rev. Lett. 15 240. Zakharov, B. E. and Shabat, A. B. (1972). Sov. Phys. JETP 34 62. Zakharov, V. E. (1968). Sov. Phys.JETP 26 994. Zakharov, V. E. (1971). Sov. Phys.JETP 33 538. Zakharov, V. E. and Manakov, S. V. (1976). Sov. Phys.JETP 42 842. Zakharov, V. E. and Manakov, S. V. (1976). Sov. Phys.JETP 44 106. Zakharov, V. E. and Shabat, A. B. (1973). Sov. Phys.JETP 37 823. Zhou, X. and Cui, H. R. (1993). Science in China A 36 816.
Chapter 5
Nonlinear Interaction and Localization of Particles As mentioned in Chapter 4, in the nonlinear quantum mechanics, microscopic particles have wavecorpuscle duality. Obviously, this is due to nonlinear interactions in the nonlinear quantum systems. However, what is the relationship between the nonlinear interactions and localizations of microscopic particles? What are the functions and related mechanisms of the nonlinear interactions? These are all important issues, which are worth further exploration. We will discuss these problems in this chapter.
5.1
Dispersion Effect and Nonlinear Interaction
We consider first the effects of the dispersion force and nonlinear interaction, from the evolution of the solutions of the linear Schrodinger equation (1.7) and the nonlinear Schrodinger equation (3.2), respectively. In the linear quantum mechanics, the dynamic equation is the linear Schrodinger equation (1.7). When the external potential field is zero (V = 0), the solution is a plane wave, * (f, t) = I*) = AeWEtVh.
(5.1)
It denotes the state of a freely moving microscopic particle with an eigenenergy of E=
=
^{P*+PZ+P*h
(™
(52)
This is a continuous spectrum. It states that the particle has the same probability to appear at any point in space. This is a direct consequence of the dispersion effect of the microscopic particle in the linear quantum mechanics. We will see that this nature of the microscopic particle cannot be changed with variations of time and external potential V. In fact, if a free particle is confined in a rectangular box of dimension a, b and c, then the solutions of the linear Schrodinger equation are standing waves * (*,y,z,t)=Asin ( ^ )
sin
233
(^)
sin ( ^ )
e~iBt'\
(5.3)
234
Quantum Mechanics in Nonlinear Systems
In such a case, there is still a dispersion effect for the microscopic particle, namely, the microscopic particle still appears with a determinate probability at each point in the box. However, its eigenenergy is quantized, i.e.,
E = —— 4 + if + 4
5.4)
•
v 2m \a2 b2 c2 J ' The corresponding momentum is also quantized. This shows that microscopic particles confined in a box possess evident quantum features. Since all matters are composed of different microscopic particles, they always have certain bound states. Therefore, microscopic particles always have quantum characteristics. When a microscopic particle is subject to a conservative timeindependent field, V(f, t) = V(r) ^ 0, the microscopic particle satisfies the timeindependent linear Schrodinger equation
^V2rP
(5.5)
+ V(f)i; = E^,
where * = ^{r)eiEt'h.
(5.6)
Equation (5.5) is referred to as eigenequation of the energy of the microscopic particle. In many physical systems, a microscopic particle is subject to a linear potential field, V = F • f, for example, the motion of a charged particle in an electrical field, or a particle in the gravitational field, where F is a constant field independent of f. For a one dimensional uniform electric field, V{x) — —eex, the solution of (5.5) is
i> = AVSH$(le^,
(f=f + A)
(5.7)
where H^(x) is the Hankel function of the first kind, A is a normalized constant, I is the characteristic length and A is a dimensionless quantity. The wave in (5.7) is still a dispersed wave. When £ —>• oo, it approaches
^(0 = AT 1 / 4 e 2 * 3 / 2 / 3 ,
(5.8)
which is a damped wave. If V(x) — Fx2 (such as the case of a harmonic oscillator), the eigen wavevector and eigenenergy of (5.5) are r/;(x) = Nnea***/2Hn(ax),
En^(n+^\hw,
( n = 0,1,2, •••)
(5.9)
respectively, where Hn(ax) is the Hermite polynomial. The solution obviously has a decaying feature. We can keep changing the form of the external potential field V(f), but soon we will find out that the dispersion and decaying nature of the microscopic particle
Nonlinear Interaction and Localization of Particles
235
persists no matter what form the potential field takes. The external potential field V(f) can only change the shape of the microscopic particle, i.e., its amplitude and velocity, but not its fundamental property such as dispersion. This is due to the fact that the kinetic energy term, (H2/2m)V2 = p2/2m itself is dispersive. Because microscopic particles are in motion, the dispersion effect of the kinetic energy term always exists. It cannot always be balanced and suppressed by the external potential field V(f, t). Therefore, microscopic particles in linear quantum mechanics always exhibit the wave feature, not the corpuscle feature. However, nonlinear interactions present in nonlinear quantum mechanics, for example, in the nonlinear Schrodinger equation (3.2), and such nonlinear interactions depend directly on the wave function of the state of the microscopic particle. It becomes possible to change the state and nature of the microscopic particle. If the nonlinear interaction is so strong that it can balance and suppress the dispersion effect of the kinetic term in (3.2), then its wave feature will be suppressed, and the microscopic particle becomes a soliton with the wavecorpuscle duality. This can be verified using the nonlinear Schrodinger equation (4.40). Both kinetic energy dispersion and nonlinear interaction exist in the nonlinearly dynamic equation and the equation has soliton solutions (4.8) and (4.56) with constant energy, shape and momentum. This demonstrates clearly the suppression of the dispersion effect by the nonlinear interaction, and localization of the microscopic particle. As can be expected, the localization of the microscopic particle cannot be changed by altering the external potential field V(x) in (3.2) at A(4>) = 0 . As a matter of fact, when V{x) is a constant, as in (2.148) for the superfluid, its soliton solution is given by (2.150). Here sech(a;) is a stable function. If V(x') = eex, as in (2.122) in a superconductor, its soliton solution is given by (2.141) and (4.26), respectively. If V(x') — a2x'2, its soliton solution is given in (4.28). These solutions are all of the form sech(a;), which has localized feature and stable. For a more complicated external potential V(x), for example V(x) = kx2 + A(t)x + B(t), the soliton solution of (3.2) with A(<j>) = 0 can be written as (t> =
(5.10)
where inn
(f (x  u(t)) = y —sech (ax  «(*)),
0(x,t) = \2Vksm(2Vki +/?) + y ] * ~ [ { [uo(t')  2kcos(2Vk¥ + 0)^ +B(t') + [ y  2Vk sin(2v/H7 + /?)] \dt' + Xot + g0. and u(t) = 2cos(2\/ki + /3) +uo(t).
236
Quantum Mechanics in Nonlinear Systems
When A{t) = B(t) = 0, u(t) = 2cos(2Vkt) + uo(t),
6(x, t) =  I  [fc(2 cos(2v^ + u0(*)]2 (2Vksin(2Vkti^ + —' 1 dt' 2V^sin (2Vki + —'a:) + g0. This solution is similar to (4.28) and is still a sech(a;)type soliton, and is stable, although it oscillates with a frequency which is directly proportional to A;. It will be discussed in more details in Chapter 8. The localization feature or wavecorpuscle duality of a microscopic particle cannot be changed by changing the form of the external potential V(x) in nonlinear quantum mechanics. This shows also that the influence of the external potential V(x) on soliton feature of microscopic particles is secondary, the fundamental nature of microscopic particles is mainly determined by the combined effect of dispersion forces and nonlinear interaction in nonlinear quantum mechanics. It is the nonlinear interaction that makes a dispersive microscopic particle a localized soliton. To further clarify these, we consider in the following the effects and functions of the dispersion force and nonlinear interaction. What exactly is a dispersion effect? When a beam of white light passes through a prism, it is split into beams with different velocities. This phenomenon is called a dispersive effect, and the medium in which the light or wave propagates is referred to as a dispersive medium. The relation between the wave length and frequency of the light (wave) in this phenomenon is called a dispersion relation, which can be expressed as OJ — u(k) or G(LJ, k) = 0, where
det
aS^ 0 '
or d2k2 T ' in onedimensional case. It specifies how the velocity or frequency of the wave (light) depends on its wavelength or wavevector. The equation determines wave propagation in a dispersive median and is called an dispersion equation. The linear Schrodinger equation (1.7) in the linear quantum mechanics is a dispersion equation. If (5.1) is inserted into (1.7), we can get w — Hk2/2m from (5.2), here E = Hu, p = hk. The quantity vp = w/k is called the phase velocity of the microscopic particle (wave). The wave vector k is a vector designating the direction of the wave propagation. Thus the phase velocity is given by ve = (ui/k2)k. This is a standard dispersion relation. Therefore the solution of the linear Schrodinger equation (1.7) is a dispersive wave.
Nonlinear Interaction and Localization of Particles
237
But how does the dispersive effect influence the state of a microscopic particle? To answer this question, we consider the dispersive effect of a wavepacket which is often used to explain the corpuscle feature of microscopic particles in linear quantum mechanics. The wavepacket is formed from a linear superposition of several plane waves (5.1) with wavevector k distributed in a range of 2Ak. In the one dimensional case, a wavepacket can be expressed as •i
/fco+Afc
rp(k,t)e^kx^dk.
*(*.*) = 5  / 27r
(5.11)
JkoAk
We now expand the angular frequency <j(k) at kQ by (5.12)
If we consider only the first two terms in the dispersive relation, i. e., fdu\
&
where £ = Afe = k — ko, then,
(ko)ei{koXWDt)
fAk
JAk =
d^[*{d»Idk)kot]i
(513)
2^(fc o ) S i n { [ X " ( ^ / / ^ o t ] A f c } e^°^), x  (du/dk)kot
where the coefficient of e*(fco»—">ot) m (5.13) j s the amplitude of the wavepacket. Its maximum is 2ip(ko)Ak which occurs at x = 0. It is zero at x = xn = nn/Ak (n = ± l , ± 2 , .  . ) . Figure 5.1 shows that the amplitude of the wavepacket decreases with increasing distance of propagation due to the dispersion effect. Hence, the dispersion effect results directly in damping of the microscopic particle (wave). We can demonstrate from (1.11) and (1.13) that a wavepacket could eventually collapse with increasing time. Therefore, a wavepacket is unstable and cannot be used to describe the corpuscle feature of microscopic particles in the linear quantum mechanics. When nonlinear interactions exist in a system, such nonlinear interaction can balance and suppress the dispersion effect of the microscopic particle. Thus the microscopic particle becomes a localized soliton as mentioned repeatedly, and its corpuscle feature can be observed. Therefore, microscopic particles have wavecorpuscle duality as described in Chapter 4. As the nonlinear interaction increases, the corpuscle feature of microscopic particles becomes more and more obvious. The localization nature of the microscopic particle can always persist no matter what form the externally applied potential field takes. Therefore, the localization and wavecorpuscle duality are a fundamental property of the microscopic particles in nonlinear quantum mechanics.
238
Quantum Mechanics in Nonlinear Systems
Fig. 5.1 Amplitude of a plane wave propagating along the xdirection.
5.2
Effects of Nonlinear Interactions on Behaviors of Microscopic Particles
What would be the effects of nonlinear interactions on microscopic particles? To answer this question, we first consider carefully the motion of water wave in a sea. When a wave approaches the beach, its shape varies gradually from a sinusoidal cross section to triangular, and eventually to a crest which moves faster than the rest, as shown in Fig. 5.2.
Fig. 5.2
Changes of the shape of a water wave approaching a beach.
This is a result of nonlinear nature of the wave. As the water wave approaches the beach, the wave will be broken up due to the fact that the nonlinear interaction is enhanced. Since the speed of wave propagation depends on the height of the wave, this is a nonlinear phenomenon. If the phase velocity of the wave, vp, depends weakly on the height of the wave, h, then, vp = r = vpo + Sih,
(5.14)
where dh
h=h0
ho is the average height of the wave surface, Vpo is the linear part of the phase velocity of the wave, 5i is a coefficient denoting the nonlinear effect. Therefore, the nonlinear interaction results in changes in both the form and the velocity of the
Nonlinear Interaction and Localization of Particles
239
waves. This effect is similar to that of dispersion, but their mechanisms and rules are different. When the dispersive effect is weak, the velocity of the wave can be denoted by V
P =  = V'PO + S2k2,
(5.15)
where wp0 is the dispersionless phase velocity, and 2
^
2
fc=fe0
is the coefficient of the dispersion of the wave. Generally speaking, the term proportional to k in the expansion of the phase velocity gives rise to the dissipation effect, and the lowest order dispersion is proportional to k2. To further explore the effects of nonlinear interaction on the behaviors of microscopic particles, we consider a simple situation
(5.16)
where the time evolution is given solely by the dispersion. In (5.16), if><j>x is a nonlinear interaction. There is no dispersive term in this equation. It is easy to verify that an arbitrary function of the variable x — (f>t, <j> = $'(x
<j)t)
(5.17)
satisfies (5.16). Fig. 5.3 shows (j> as a function of x for various values of t, obtained from (5.16) and (5.17). In this figure, each point (j> proceeds with the velocity 4>. Thus, as time elapses, the front side of the wave gets steeper and steeper, until it becomes a triplevalued function of x due to the nonlinear interaction, which does not occur for a normal wave equation. Obviously, this wave deformation is the result of the nonlinear interaction.
Fig. 5.3 Form of the wave given in (5.16).
If we let 4> = $ ' = COSTTX at t — 0, then at x = 0.5 and t — n'1, cf> = 0 and 4>x = oo. The time te = n~1 at which the wave becomes very steep is called destroy period of the wave. The collapsing phenomenon can be suppressed by adding a dispersion term <j>xxx, as in the KdV equation. Then, the system has a
240
Quantum Mechanics in Nonlinear Systems
stable soliton, sech2 (X) in such a case. Therefore, a stable soliton, or a localization of particle can occur only if the nonlinear interaction and dispersive effect exist simultaneously in the system, so that they can be balanced and their effects cancel each other. Otherwise, the particle cannot be localized, and a stable soliton cannot be formed. However, if 4>xxx is replaced by (f>xx, then (5.16) becomes 4>t + 4>4>x=v
(5.18)
This is the Burgur's equation. In such a case, the term v<j)xx cannot suppress the collapse of the wave, arising from the nonlinear interaction (jxpxx Therefore, the wave is damped. In fact, using the ColeHopf transformation
= 27 (iogi//)
^~ £
'
equation (5.18) becomes dip1 _
d2ip'
This is a linear equation of heat conduction (the diffusion equation), which has a damping solution. Therefore, the Burgur's equation (5.18) is essentially not a equation for solitary wave, but a onedimensional NavierStokes equation in viscous fluids. It only has a damping solution, not soliton solution. This example tells us that the deformational effect of nonlinearity on the wave can only be suppressed by the dispersive effect. Soliton solution of dynamic equations, or localization of particle can then occur. This analysis and conclusion are also valid for the nonlinear Schrodinger equation and the nonlinear KleinGordon equation with dispersive effect and nonlinear interaction. This example also demonstrated that a stable soliton or localization of particle cannot occur in the absence of nonlinear interaction and dispersive effect. In order to understand the conditions for localization of microscopic particles in the nonlinear quantum mechanics, we examine again the property of nonlinear Schrodinger equation (3.2). Prom (3.17)  (3.19), we get E{p) KE0bp
= E0 + {bp),
(5.19)
where EQ is the eigenenergy of Ho, corresponding to the linear Schrodinger equation. To be more accurate, it is the energy of the microscopic particle depicted by the linear quantum mechanics. If we consider E(p) as the energy of a microscopic particle in nonlinear quantum mechanics, then it should depend on the mass and state wave function of the microscopic particle. From (5.19), we know that the energy is lower than that of the particle in the linear quantum mechanics by bp. This indicates that in the nonlinear quantum mechanics, the microscopic particle is more stable than that in linear quantum mechanics. Obviously, this is due to the localization of the microscopic particle. Thus an energy gap between the normal
Nonlinear Interaction and Localization of Particles
241
and the soliton states exists in the energyspectrum of the particles which is seen in superconductors in Chapter 2. The term bp (< 0) here is the binding energy or localization energy of the microscopic particle (soliton). Since the energy decrease is related to the state wave function of the particle, —b\4>\2, this phenomenon is called a selflocalization effect of the particle, and this energy is referred to as selflocalization energy. If there is no nonlinear interaction, i.e., 6 = 0, then the selflocalization energy is zero, and the microscopic particle cannot be localized. Therefore, the nonlinear interaction is a necessary condition for the localization of the microscopic particle in nonlinear quantum mechanics. Known that nonlinear interaction is essential for localization of microscopic particle, the next question would be how could a microscopic particle be localized? To answer this question, we look further at the potential function of the system. Equation (3.17) can be written as  ~ V 2 < ^ + [ F ( f ) E]
b\<j>\2<j> = 0.
(5.20)
For the purpose of showing the properties of this system, we assume that V[f) and b are independent off. Then in onedimensional case, equation (5.20) may be written as (5.21)
with V,«{4>) = \b\4>\i\{VE)\<j>\2.
(5.22)
When V > E and V > 0, the relationship between Ves{4>) and 4> is shown in Fig. 5.4. From this figure we see that there are two minimum values of the potential, corresponding to the two ground states of the microscopic particle in the system, i.e.,
This is a doublewell potential, and the energies of the two ground states are — (V — E)2 /4b < 0. It shows that the microscopic particle can be localized because it has negative binding energy. The localization is achieved through repeated reflection of the microscopic particle in the doublewell potential field. The two ground states limit the energy diffusion. Obviously, this is a result of the nonlinear interaction because the particle is in the normal, expanded state if b — 0. In that case, there is only one ground state of the particle which is
242
Quantum Mechanics in Nonlinear Systems
Fig. 5.4 The nonlinear effective potential in (5.22).
the medium and the particles. In the first mechanism, the attractive effect is due to interactions between the microscopic particle and other particles. This is called a selfinteraction. A familiar example is the BoseEinstein condensation mechanism of microscopic particles, which is due to attraction among the Bose particles, and is also referred to as selfcondensation. In the second mechanism, the medium has itself anomalous dispersion effect (i. e., k" = d2k/dui2\UJo < 0) and nonlinear features resulting from the anisotropy and nonuniformity. Motion of the microscopic particles in the system are modulated by these nonlinear effects. This mechanism is called selffocusing. The third mechanism is referred to as selftrapping. It is produced by interaction between the microscopic particles and the lattice or medium. In the subsequent sections, these mechanisms will be discussed in more details. Prom (5.22), we know that when V > 0, E > 0 and V < E, or V > E, E > 0 and V < 0 for b > 0, the microscopic particle may not be localized by the mechanisms mentioned above. On the other hand, we see from (5.20)  (5.22) that if the nonlinear selfinteraction is of a repelling type (i.e., b < 0), then, equation (3.2) with A() = 0 becomes ih^t+^^\b\\4>\24>
= V(x,t)4>.
(5.23)
It is impossible to obtain a soliton solution, with full matter features, of this equation. However, if V(x,t) = V(x) or a constant, solution of kink soliton type exists. Inserting (3.15) into (5.23), we can get
^ 0   % > 3 + [ £  W ] V = O.
(5.24)
If V is independent of x and 0 < V < E, equation (5.24) has the following solution
^ggEii^yEEn^^
(5.25)
This is the kink soliton solution when \V\ > E and V < 0. In the case of V(x) = 0, Zakharov and Shabat obtained dark soliton solution (see Section 4.3) which was
Nonlinear Interaction and Localization of Particles
243
experimentally observed in optical fiber and was discussed in the BoseEinstein condensation model by Bargeretal. Equation (5.24) may be written as E
= 164)!tf, (6.73) where r 0 is a spacing between two neighboring lattice points, j labels the discrete lattice points, J is the total number of lattice points in the system. The vector form of the above equation is iK
(P2) = ^AW
+ 32V2A2r,e.
(6.58)
The standard deviations of position Ax' = \/{x'2) — {x')2 and momentum Ap = y/ip2)  (p)2 are given by
(Ax')2 = A2 [ ^ + 4 ^ ( 1  4 ^ A 2 ) ] = ^ , (Ap)2 = 32V2A2 [ir,3 + tfa(l  4 ^ ^ ) ] = i, 2 ,
(6.59)
295
Nonlinear versus Linear Quantum Mechanics
respectively. Thus we obtain the uncertainty relation between position and momentum for the microscopic particle, (4.56), in nonlinear quantum mechanics, (6.60)
Ax'Ap = ~. 0
This result is not only valid for microscopic particle (soliton) described by the nonlinear Schrodinger equation, but holds in general. This is because we did not use any specific property of the nonlinear Schrodinger equation in deriving (6.60) n in (6.60) comes from the integral constant l/^/2n. For a quantized microscopic particle, n in (6.60) should be replaced by nh, because (6.56) is replaced by
4>s{p,t') = ±= f°° V2nh Joo
dx'U^,t')eipx>/h
The corresponding uncertainty relation of the quantum microscopic particle (soliton) in the nonlinear quantum mechanics is given by
(6.61)
AxAp=f = ±.
The relations (6.60) or (6.61) are different from that in linear quantum mechanics (6.54) i.e., AxAp > h/2. However, the minimum value AxAp = h/2 has not been observed in practical systems in linear quantum mechanics up to now except for the coherent and squeezed states of microscopic particles. The relation (6.61) cannot be obtained from the solutions of the linear Schrodinger equation. Practically, we can only get AxAp > h/2 from (6.54), but not AxAp = h/2, in linear quantum mechanics. As a matter of fact, for an onequantum coherent state, oo
n
\a) = exp(ab+  a*b)\0) = e~^2 Y
°
ntjVi1
6+n0),
which is a coherent superposition of a large mumbler of microscopic particles (quanta). Thus , .„, .
„.
I h~ .
(aa;a) = y 7 ^ ( « + a),
. ,
(a\p\a)
.
.
hmcj. „
= i\j  ^ — (a* 
.
a),
and
(a\x2\a) = ^—{a*2
+ a2 + 2aa* + 1),
(a\p2\a) = ^(a*2
+ a2  2aa*  1),
where /
h
ft
i+\
,
. I hum
/,
t
\
and b+ (b) is the creation (annihilation) operator of the microscopic particle (quantum), a and a* are some unknown functions, u is the frequency of the particle, m
296
Quantum Mechanics in Nonlinear Systems
is its mass. Thus we can get {AX)
(A^ = — '
= W
(Ax)2(Ap)^T>
and Ax _ 1 Ap urn' or Ap = {um) Ax.
(6.62)
For the squeezing state, /3)=exp[/J(6+ 2 b 2 )]0), which is a twomicroscopic particle (quanta) coherent state, we can obtain = ^
/
^
=
fc^e*
using a similar approach as the above. Here /? is the squeezing coefficient and /3 < 1. Thus A
h
A
AxAp=, ^ 2'
Ax
1 ofl
—— = e8^1, A p mw '
or Ap = Ax(ojm)e80.
(6.63)
This shows that the momentum of the microscopic particle (quantum) is squeezed in the twoquanta coherent state compared to that in the onequantum coherent state. From the above results, we see that both the onequantum and the twoquanta coherent states satisfy the minimal uncertainty principle. This is the same as that of the nonlinear quantum states in nonlinear quantum mechanics. We can conclude that a coherent state is a nonlinear quantum state, and the coherence of quanta is a nonlinear phenomenon, instead of a linear effect. As it is known, the coherent state satisfies classical equation of motion, in which the fluctuation in the number of particles approaches zero, i.e., it is a classical steady wave. According to quantum theory, the coherent state of a harmonic oscillator at time t can be represented by
\a,t) = eifltlh\a) = eM*+M/2)ta) = c*,t/2Ma/a f^ ""C^V) n=0 2
iut
= e^ \ae ),
(n> = (6+)»0».
^
297
Nonlinear versus Linear Quantum Mechanics
This shows that the shape of a coherent state can be retained during its motion, which is the same as that of a microscopic particle (soliton) in the nonlinear quantum mechanics. The mean position of the particle in the timedependent coherent state is (a,t\x\a,t)
{a\eiHtlhxeiHtlh\a)
=
= (a x^[x,H}+{^[[x,H},H} = (a x + ^  ^t2u2x = (a x cos u)t H
+ ^a) (6.64)
+ ••• a) sin ujt a)
mui = \ \a\ cos(u>t + 9), V mu> where 6 = tan" 1 (  ) ,
x + iy = a,
and
[x,H\ = —,
ihmu2x.
\p,H} =
Comparing (6.64) with the solution of a classical harmonic oscillator, rcosM + 0),
x=\
2
V mu we find that they are similar, with
E= £ + moj2x2, 2m 2
E = hua2 = (a\H\a)  (0\H\0),
H = hw fb+b + M .
Thus we can say that the center of the wave packet of the coherent state indeed obeys the classical law of motion, which happens to be the same as the law of motion of microscopic particles in nonlinear quantum mechanics discussed in Section 4.2. We can similarly obtain (a,t\p\a,t) = —V2mEuJ\a\sm(ujt + 9),
(a, t\x2 \a, t) = — fH 2 cos2 (wt + 6) + ]] , 4J
urn \_
(a,t\p2\a,t) [Ax{t)]2
= 2rnHu \\a\2sin2(ut =
2 ^ '
[Ap{t)]2
+ 0) +  , =
\
m u K
298
Quantum Mechanics in Nonlinear Systems
and Ax(t)Ap(t) = \ . This is the same as (6.62). It shows that the minimal uncertainty principle for the coherent state can be retained at all time, i.e., the uncertainty relation does not change with time t. The mean number of quanta in the coherent state is given by n = (a\N\a) = (a\b+b\a) = a2,
(a\N2\a) = a 4 + \a\2.
Therefore, the fluctuation of the quantum in the coherent state is
An = \J(a\N2\a)((a\N\a))2
= \a\,
which leads to An n
1 \a\
It is thus obvious that the fluctuation of quantum in the coherent state is very small. The coherent state is quite close to the feature of soliton or solitary wave. These properties of coherent states of microscopic particles described by the nonlinear Schrodinger equation, the (j>4equation, or the SineGordon equation in nonlinear quantum mechanics are similar. In practice, the state of a microscopic particle in the nonlinear quantum mechanics can always be represented by a coherent state, for example, the Davydov wave functions, both £>i) and \D2) (see (5.72)), the wave function of excitonsolitons in protein molecules and acetanilide (see Chapter 9), and the BCS wave function (2.28) in superconductor, etc. Hence, the coherence of particles is a kind of nonlinear phenomenon which occurs only in nonlinear quantum mechanics. It does not belong to systems described by linear quantum mechanics, because the coherent state cannot be obtained by superposition of linear waves, such as plane wave, de Broglie wave, or Bloch wave, which are solutions of the linear Schrodinger equation in linear quantum mechanics. Therefore, the minimal uncertainty relation (6.63) as well as (6.60) and (6.61) are only applicable to microscopic particles in nonlinear quantum mechanics. In other words, only microscopic particles in nonlinear quantum mechanics satisfy the minimal uncertainty principle. It reflects the wavecorpuscle duality of microscopic particles because it holds only if the duality exists. This uncertainty principle also suggests that the position and momentum of the microscopic particle can be simultaneously determined in certain degree and range. A rough estimate for the size of the uncertainty is given in the following. We choose f = 140, 77 = x/300/0.253/2%/2 and x0 = 0 in (4.56) or (6.56), so that
299
Nonlinear versus Linear Quantum Mechanics
(6.60). These results show that the position and momentum of a microscopic particle in nonlinear quantum mechanics can be simultaneously determined within a certain approximation. Finally, we determine the uncertainty relation of the microscopic particle described by the nonlinear Schrodinger equation, arising from the quantum fluctuation effect in the nonlinear quantum field. The quantum theory presented in Section 4.6 was discussed by Lai and Haus based on the nonlinear Schrodinger equation. As discussed in Chapters 3 and 4, a superposition of a subclass of the bound states n, P) which are characterized by the number of Bosons, such as photons or phonons, n, and the momentum of the center of mass, P, can reproduce the expectation values of the field of the microscopic particle (soliton) in the limit of a large average number of Bosons (phonons). Such a state formed by the superposition of \n,P) is referred to as a fundamental soliton state. In quantum theory, the field equation is given by (3.62), with the commutation relation (3.63). The corresponding quantum Hamiltonian is given by (3.65). In the Schrodinger picture, the time evolution of the system is described by (3.66). The manyparticle state \tp') can be built up from the nparticle states given by (3.69). The quantum theory based on (3.69) describes an ensemble of Bosons interacting via a ^potential. Note that H preserves both the particle number.
N = f ci>+(x)4>(x)dx
(6.65)
and the total momentum
P=i
\J [^WiW ~ t+Wj^fa)] dx
( 6  66 )
Lai et al. proved that the Boson number and momentum operators commute, so that common eigenstates of H, P and N can exist in such a case. In the case of a negative b value, the interaction between the Bosons is attractive and the Hamiltonian (3.63) has bound states. A subset of these bound states is characterized solely by the eigenvalues of N and P. The wave functions of these states are
( where
oo
,
oo
\
ip^xj +  Y^ kiZjl j=l
l
,
(6.67)
I
Thus /„(*!, •"*»,*) = /^n(p)/ n ,p(zi,z n ,*)e~ i B ( n ' p ) t ,
(6.68)
300
Quantum Mechanics in Nonlinear Systems
where fcrfW."~.
and
9 ( P )^HP^/[2(Ari']},
Using / n i P given in (6.67), we found that \n, P) decays exponentially with separation between any pair of Bosons. It describes an nparticle soliton moving with momentum P — hnp and energy E(n,p) = np2  \b\2(n2  l)n/12. By construction, the quantum number p in this wave function is related to the momentum of the center of mass of the n interacting Bosons, which is now defined as X = lim (xcj>+(x)(t>{x)dx{e + N)'1,
(6.69)
with
[x,P] = ih. The limit of e » 0 is introduced to regularize the position operator for the vacuum state. We are interested in the fluctuations of (6.65), (6.66) and (6.69) for a state \ip'(t)) with a large average Boson number and a welldefined mean field. Kartner and Boivin decomposed the field operator into a mean value and a remainder which is responsible for the quantum fluctuations.
4>(x) = W(0)\j>+(x)W(0)) + 4>i(x),
[Mx),$t(x')]=6(xx'), [jn(x)MJ)]=0
(670)
Since the field operator <j> is time independent in the Schrodinger representation, we can then choose t = 0 for definiteness. Inserting (6.70) into (6.65), (6.66) and (6.69) and neglecting terms of second and higher order in the noise operator, Kartner et al. obtained that N = no + An,
P = hnopo + hno^p,
X — XQ ( 1
I + Ax,
with n 0 = [dx(4>+{x))(j>(x)),
po = — [ dx(fc{x))($(x)), n0 J
If xo = — dxx(4>+{x))(cf>(x)), n0 J
An= (dx{4>+{x))4>x{x) +c.c,
Ap=— [dx(4>i(x))Mx)+c.c, n0 J
1 f Ax = — / dxx{
where Ax is the deviation from the mean value of the position operator, An, Ap, and Ax are linear in the noise operator. Because the third and fourthorder correlators of 4>i and
301
Nonlinear versus Linear Quantum Mechanics
Note that Ap and Ax are conjugate variables. To complete this set we introduce a variable conjugate to An,
[(i+(x)} + x(4>+(x))] pox(4>+(x))}Mx) + c.c
Ad = ±Jdx{i
As it is known, if the propagation distance is not too large, then to the first order, the mean value of the field is given by the classical soliton solution
with 4>o,nQ(x, t)   ^ — exp [iClni  ip\t + ipo(x  x0) + i0o] XSeCh
\ ^ ( x xo
2pd*)l .
(671)
where the nonlinear phase shift flnj = no6 2 t/4. If po = XQ = 60 = 0, we obtain the following for the fluctuation operators in the Heisenberg picture,
Afi(t) = Jdx[f_n(xyF^ + c.c], A0(t) = Jdx{f_g(x)*F^+c.c), Ap(t)= Jdx[fp(x)*F^+c.c}, Jdx[f_x(xyF^+c.c],
Ax(t)= with
Kt=e«inl4>1(x,t), and the set of adjoint functions /n(x) = —^—sech(a; no ), fe(x)
= — — \sech{xno) + xno[
&
fp(x)
=
fx(x)
=
f
*
1
ax
na
o,xno
sech(a;no),
7 =a; n o sech(a; 7 1 o ),
noVl^l
where 1 ... Zn0 = 2 n o \b\ x
sech(x no ) , j
302
Quantum Mechanics in Nonlinear Systems
For a coherent state defined by 4>(x)\$0,n0) =
or
0l(aOl0O,no) = 0 ,
where
$o >no ) = exp y dx [^no(x)4>+(x)
 <^,no(*)<£+(x)] } o>,
and (/>o,no has been given (6.71), Kartner and Boivin further obtained that
(A^)=n O l
<APg> = 3 ^ ,
(A^) =
 ^
J
where TQ = 2/n o 6 is the width of the microscopic particle (soliton). The uncertainty products of the Boson number and phase, momentum and position are, respectively, (Ah20){A92) = 0.6075 > 0.25,
and n2Q(Ap2)(Axl) = 0.27 > 0.25.
Here the quantum fluctuation of the coherent state is white, i.e.,
{4>i(x)My)) = (it(x)My))
= o.
However, the quantum fluctuation of the soliton cannot be white because interaction between the particles introduces correlations between them. Thus, Kartner and Boivin assumed a fundamental soliton state with a Poissonian distribution for the Boson number
and a Gaussian distribution for the momentum (6.68) with a width of (Apg) = n o 6 2 /4/i, where fj, is a parameter of the order of unity compared to n 0 . They finally obtained the minimum uncertainty values ..22.
0.25
{n20Ap2}
0.25 I"
n0
_ / 1 \]
[
\noj\
up to the order of l/n 0 , for the corresponding initial fluctuations in soliton phase and timing. Thus at t = 0 the fundamental soliton with the given Boson number and momentum distributions is a minimum uncertainty state in the four collective variables, the Boson number, phase, momentum, and position, up to terms of
303
Nonlinear versus Linear Quantum Mechanics
O(l/no),
(Ang)(A^02) = 0.25[l + o ( ^ ]
!
ng(Apg> = 0.25 [l + O ( £ ) ] .
(6.72)
These are the uncertainty relations arising from the quantum fluctuations in nonlinear quantum field of microscopic particles described by the nonlinear Schrodinger equation. They are the same as (6.61)  (6.63). Therefore, we conclude that the uncertainty relation in the nonlinear quantum mechanics takes the minimum value regardless a state is coherent or squeezed, a system is classical or quantum. Pang et al. also calculated the uncertainty relation of quantum fluctuations and studied their properties in nonlinearly coupled electronphonon systems based on the Holstein model but using a new ansatz which includes correlations among onephonon coherent and twophonon squeezing states and polaron state. Many interesting results were obtained. The minimum uncertainty relation takes different forms in different systems which are related to the properties of the microscopic particles. Nevertheless, the minimum uncertainty relation (6.63) holds for both the onequantum coherent state and twoquanta squeezed state (see Jajernikov and Pang 1997, Pang 1999, 2000, 2001, 2002). These work enhanced our understanding of the significance and nature of the minimum uncertainty relation. In light of the above discussion, we can distinguish the motions of particles in the linear quantum mechanics, nonlinear quantum mechanics, and classical mechanics based on the uncertainty relation. When the motion of the particles satisfy AxAp > h/2, the particles obey laws of linear quantum mechanics, and they have only wave feature. When the motion of the particles satisfy AxAp = h/12 or TT/6, the particles obey laws of motion in the nonlinear quantum mechanics, and the particles are solitons, exhibiting wavecorpuscle duality. If the motion of the particles satisfy AxAp = 0, then the particles can be treated as classical particles, with only corpuscle feature. The nonlinear quantum mechanics introduced here thus gives a more complete description of physical systems. Therefore, we can say that the nonlinear quantum mechanics bridges the gap between the classical mechanics and the linear quantum mechanics.
6.4
Energy Spectrum of Hamiltonian and Vector Form of the Nonlinear Schrodinger Equation
Like in the linear quantum mechanics, it is useful to obtain the energy spectrum of the Hamiltonian operator for a given system in the nonlinear quantum mechanics. The energy of a microscopic particle satisfying the nonlinear Schrodinger equation
304
Quantum Mechanics in Nonlinear Systems
can be obtained from /•OO
E= /
Hdx,
J — OO
where H. is the classical Hamiltonian density which depends on the wave function
General approach
As discussed in Chapter 3, the wave function of a microscopic particle can be quantized by the creation and annihilation operators of the particle in nonlinear quantum mechanics. The Hamiltonian of a system described by the wave function
=
lH
~2^{lt>i+1
\ih%   ^ I
Ob
TflTn
~ 24>i + ^l)
~ b^2(t>i
+ V
^ ^
{j
 V(j, t)} 4> = eM4>  &diag.(0i 2 , \<j>2\2 • • • \4>a\2)4>, J
(6.74)
where <j> is a column vector, ^ = Col.(<j>i,2 • • • <Pa) whose components are complex Equation (6.74) is a vector nonlinear Schrodinger equation with a modes of motion. In (6.74), b is a nonlinear parameter and a is the number of modes that exist in the
305
Nonlinear versus Linear Quantum Mechanics
systems, M = [Mnt] is an a x a real symmetric dispersion matrix, e = H2/(2mrl). Here n and t are integers denoting the modes of motion. The Hamiltonian and the particle number corresponding to (6.74) are
H = J2 {^o\K\2  U<j>n\4)  e J2 Mne<j>nct>t, n=l V
l
/
(6.75)
r#l
N=^2\
(6.76)
n=l
where
We have assumed that V(j, t) are independent of j and t. In the canonical second quantization theory, the complex amplitudes (4>*n and
,
with the Boson commutation rule BnB+  B+Bn — 1. Equations (6.76) and (6.76) then become
H = J2[(*»o \b) (B+Bn + ^  \bB+BnB+B^ ej^MraB+Bt, (6.77) N = JT (B+Bn + 1) . 71=1 ^
(6.78)
'
Prom now on, we will use the notation \m\m2 • • • ma] to denote the product of the number states mi)m2) • • • \ma). The stationary states of the vector nonlinear Schrodinger equation (6.74) must be eigenfunctions of both N and H. Consider an mquantum state (i.e., the mth excited level, m — mi + rri2 H— • rrij), with m < a.
306
Quantum Mechanics in Nonlinear Systems
A n eigenfunction of N c a n b e established a s  * m > = C i [ m , 0 , 0 ,    , 0 ] +  + C l 2 [ 0 , m , 0 , 0 ,    , 0 ] +  + C i [ 0 , 0 , 0 ,    ,m] + ••• + Ci+1[(m
 I),1,0,
•• ,()] + ••• +
(6.79)
Cp[0,0,  , 0 , 1 , 1 ,   , ! ] • (m times)
The number of terms in (6.79) is equal to the number of ways that m quanta can be placed on the a sites, which is given by P
_ (m + a — 1) ~ m!(al)! '
The wave function $ m ) in (6.79) is an eigenfunction of N for any values of the C'as. Thus we are free to choose these coefficients such that H\*m)
= E\*m).
(6.80)
Equation (6.80) requires that the column vector C = Col.(Ci,C2, • • Cp) satisfies the matrix equation \H  IE\C = 0,
(6.81)
where H is a p x p symmetric matrix with real elements. / is a p x p identity matrix, E is the eigenenergy. Equation (6.80) is an eigenvalue equation of the quantum Hamiltonian operator (6.77) of the system. We can obtain the eigenenergy spectrum Em of the system from (6.81) for given parameters, e, w0, and b. Scott, Bernstein, Eilbeck, Carr and Pang et al. used the above method to calculate the energyspectra of vibrational excitations (quanta) in many nonlinear systems, for example, small molecules or organic molecular crystals and biomolecules. These results can be compared with experimental data, and will be discussed in Chapter 9. 6.4.2
System with two degrees of freedom
We now discuss a simple instance and consider a system with two degrees of freedom (or two modes of motion). This was studied by Scott et al. and Pang. It takes the form
H^m
<>
The corresponding N and H, in such a case, are given by JV = £ + ! ? ! + 5 ^ 5 2 + 1, H = (tkj0  ^ N   (B+^B+Bi
+ B+B2B+B2)  e (B+B2
(6.83) + BXB%) ,
307
Nonlinear versus Linear Quantum Mechanics
respectively. We seek eigenfunction $) of both the operators. For the ground state, $o) = 0)0), which is the product of the ground state wavefunctions of the two degrees of freedom, we have iV$0>  #o>,
H\$o) = Eo\*o),
where JEJo = fkoo 
.
For the first excited state, we have 1$^ = Cx jl)O)+C2O)l) with Ci 2 + C 2  2 = 1. Thus iV$i) = 2$i). From #  $ i ) = Ei$i), we can get {2huo  Ei) b
e
e
(CA_
{2HuoE1)h
C2
\ J
The first excited state splits into a symmetric and an antisymmetric states, I*I.> = ^ (  1 >  0 > + 0>1»,
l*la> = ^ (  l )  0 )   0 >  l ) ) , with Eis = ^o + ^wo — b — e, Ela = E0 + tlLJo b + E.
respectively. For the second excited state, we have $ 2 ) = Ci2)0) + C21)1) + C30)2), with
i=l r
A $ 2 )=3$ 2 ),
H*2> = ^*2>,
and y/2e
0
V2e
3^wo  E2 + ^b
y/2e
0
V^e
3hujoE2 + bJ
(3hwoE2
K
+ lb
\
.„
N
\ C2 V
= 0.
3/
Diagonalizing this matrix equation, we get the eigenenergies with E2a = 2ftu>o + Ei + 76/2 corresponding to the antisymmetric state Ca = (1,0,  1 ) / A / 2 , and # 2 ± = 2/kj0 + ^i + 36 ± 3\/b2 + 16^2 corresponding to the symmetric state
308
Quantum Mechanics in Nonlinear Systems
C± = (e, Vb2 + 16£2/2, y/2e), respectively. In the limit of b » e, C+ = (0,1,0), C = (l/\/2,0,l/\/2), and the state C a is a localized mode which has all the energy and the particle has equal probability to be in each of the two sites. In this case, C± have the same structure, but a different relative phase. Linear combinations of these states will first localize all the energy on a particular site. For the mth excited state, we have \*m) = Cim>0> + C2\m  1)1> + • • • + Cm\l)\m  1) + C m+1 0)m)
(6.84)
with m
#* m ) = (m+l)* m >,
H\$m)=Em\*m),
and [(m + l)fiuo Emnm]C
= 0,
here C = Col.(Ci,C2,• • • ,C n +i), ^m is an (m + 1) x (m + 1) tridiagonal matrix. For an odd number n, we have (D{1) Q(l) 0 ••• g ( l ) I>(2) 0(2) ••• 0 Q(2) •••
n
~
\ 0
0 0 0
••• ••• •••
0 0 0
•••0 ••• 0 •••()
0 0
0 0
0 0
• • • D[(m + l)/2] • • • Q[(m + l)/2] • • • • • • Q[(m + l)/2] • • • D[(m + l)/2] • • •
0 0
0 0 0
0 0 0
••• ••• •••
0 0 0
••• ••• •••
0 0 0
0 0 0 0 0
0 \ 0 0 0 0
0
• • • 0 Q(2) 0 ••• 0 ( 2 ) D(2) 0 ( 1 ) • • • 0 0(1) £>(!)/
where
D{i) = h [m + 1 + (m + 1  if + (t  I) 2 ] , Q{i) = ey/i(m +
li).
0
'
309
Nonlinear versus Linear Quantum Mechanics
For an even number m, we have / D ( l ) Q(l)
0 •••
0 ( 1 ) D(2) Q(2) • • • 0 0 ( 2 ) 0 •••
Qn
\
0
0
0
•••0
0
0 0
0 0
0 0
••• •••
0 0
0 0
0
0
0 0 0
0 0 0
0 0 0
••• 0 • • • Q(m/2) ••• 0
0 0 0
0 0 0
0 0 0
••• ••• •••
0 0
Q(m/2) 0 ••• 0 D(m/2 + 1) Q ( m / 2 ) • • • 0 Q(m/2) 0 •••0
0 0 0
0 0 0
0 \
0 0
0 0
0 ••• 0 Q(2) 0 0    0 ( 2 ) D(2) 0 ( 1 ) Q(1)D(1)J 0 ••• 0
where „ (m
D
+i
\
b(
m+i+
(2 ) = 2{
m2\
r)>
_ /m\
frnrrn
«(TWT(T
~\ +1
)
We can see that if (m + l)ftu>o — E\ is one of the (ra + 1) real eigenvalues of fin, and C1 is the corresponding eigenvector, the wave function for the mth excited state can be established by (6.84). For example, for a localized mode with its energy concentrated on a single degree of freedom, the classical system (6.82) has harmonic solutions of the form 4>i = 4>'ie~%ut • For iV > 2e, there is a local mode branch with hw = hu0  bN. Along this branch, E = Hu)0N  bN2/2  e2 jb. In the quantum case here (let e < 7), there are two states corresponding to the localized modes. One is symmetric with C\ = Cm+i = 0(1), C2 = Cm = O(e/b), C3 = Cmi = O(e2/b2), etc. The other is antisymmetric with C\ = — Cm+i = 0(1), C2 = Cm = 0(e/6), C3 =  C m _ i = O(e2/62), etc. The gap between them is about Ems — Ema = O(em/bm~1). To the order of e2, the energy of these modes is
Ema Eo= Ems  Eo = [Y/^o  \ b \ m  ±bm2 
" ^ J.
(6.85)
When m > 4, there could be overtone spectra at integer values of m. Asm> 00, it approaches the classical limit given above except that the frequency is reduced by a factor of 6/2. Therefore, the energy of the local mode remains constant on a particular state in the classical case, but oscillates between the two states with a period r = 0 (b"1"1 /em) in the quantum case. When m —> 00, r —> 00, it returns to the classical result. 6.4.3
Perturbative
method
From (6.73)  (6.76), we know thatfoujocontains the applied potential V(x), which in general depends on x. Strictly speaking, the method discussed above is only
310
Quantum Mechanics in Nonlinear Systems
applicable to the case of V(x) = 0 or a constant. If V(x) depends on x, we cannot use the above method to obtain the eigenvalue spectrum, but we can use perturbation method to find an approximate eigenenergy spectrum, when \V(x)\
(6.86)
At the same time, we assume that the eigenvectors and eigenvalues of H can be written as C = C o + e C i + £ 2 C 2 + ••• , E = EQ + i E i + e 2 E 2 + ••• .
(6.87)
Inserting (6.87) into (6.81) we can get LC0 = 0, LCy = (Ei  V)C0, LC2 = (Ex  V)Ci + E2C0,
(6.88)
LC3 = (Ei  V)C2 + E2Cr + E2C0, with L = Ho — EQ, where the matrix Ho is diagonal and its first a elements are equal. Since we are mainly interested in the dependence of the lowest a eigenvalues on e, Co can be considered the eigenvector corresponding to the eigenenergy Eo. Then L has a null space of dimension a which we must take into consideration as we seek solutions of (6.88). Solution of this problem was given by Scott et al.. The first excited state (m = 1) has a wave function of the form  * 0 = Ci[l,0,0, • • • ,0] + • • • + C a [0, • • • ,0,1]. Measuring energy with respect to the ground state Ho = (hw0  6)diag.(l, 1,1,1, • • • , 1).
(6.89)
Thus Eo = (huj0 — b),L = [0] and the first equation in (6.87) places no constraints on the selection of CoNext, Scott et al. defined an inner product p (v,Q)

^Vi'^i'
Nonlinear versus Linear Quantum Mechanics
311
Since the CV 's may be complex, and L is selfadjoint, from the Fredholm alternative theorem, we know that (6.88) has solutions if and only if the right hand side of the second equation is orthogonal to (a dimensional) null space of L, as pointed out by Strang. Assuming this null space is spanned by the orthogonal set [(CQ )], i' — 1,2, • • • , a and that Co = CQ , the second equation in (6.87) has a solution if and only if
Ei (ciil},€P)  (CP,MC^) = 0,
(6.90)
and
* (CP,CP)  (Cp,MCP) = 0, for k> / i>.
(6.91)
Condition (6.90) is satisfied if we choose
(cfUc<<'>)
(6.92)
For this value of E\, condition (6.91) is satisfied if CQ is an orthogonal eigenvector of M. Then C\ = O leads to Ci — 0, etc. and the exact result of the perturbation theory is C\ = (% and E = Eo + eE\. This is essentially a direct determination of the eigenvectors and eigenvalues of E — (huj0 — b)I + eM. Next, we consider the second excited level (m = 2). With the wave function n) constructed as in (6.79), Ho = diag.{EQ,E 0 , , £ Q , h a + i , h a + 2 ,    ,hp) a times
(6.93)
(pa) times
and
\ O G] }a [CP R\}(pa).
(694)
Similarly \O
[O a
O]}a
Aj}(pa). p—a
where A = diag.(di,d2 ,d p _ a ).
(695)
312
Quantum Mechanics in Nonlinear Systems
All elements of A are nonzero. To be consistent with the partitioning in (6.94) and (6.95), Scott et al. defined
Ci=\WA\™ [Zi\
y
(6.96)
}(pm)
Prom the first equation in (6.88) and the structure of L, we have Zo = O. For solvability of the second equation in (6.88) and the structure of V, the Fredholm alternative theorem requires that E\ = 0. Thus (6.88) can be expressed as the following two sets Zi = A^GjfWb, A~1(GTW1+RZ1), Z2 = Z3 = A1(GTW3 1
T
ZA = A (G W3
+ RZ2  E2Z1),
(6.97)
+ RZ3  E2Z2  EkZx)
and TW0 = 0, TWi = GA~1RZ1 l
TW2 = GA (E2Z1 TW3 = GA^EsZ!
 E3W0,
 RZ2)  E3Wi  E 4 W 0 ,
(6.98)
+ E2Z2  RZ3)  E3W2  E±WX  E5W0,
with T = GA~1GT + E2. Scott et al. gave the following procedure for solving (6.97)  (6.98). (1) Choose Wo as an eigenvector of GA~1GT and —E2 the corresponding eigenvalue, to satisfy the first equation in (6.98), assuming that E2 is not a multiple eigenvalue. (2) Obtain Z\ from the first equation in (6.97). (3) Since Eo is not a multiple eigenvalue, the null space of T is simple WQ . From the Fredholm alternative theorem, a necessary and sufficient condition for the second equation in (6.98) to have a solution is 3
(Wo, GA^RZ,) [Wo,W0) •
(4) Solve the second equation in (6.98) for W\ with the requirement of (W\, WQ) = 0 . (5) Obtain Z2 from the second equation in (6.97).
Nonlinear versus Linear Quantum Mechanics
313
(6) The solvability of the third equation in (6.98) requires 4
[Wo,GA1(E2Z1RZ2)} (Wo,Wo)
Following this procedure, we can find the solutions and the corresponding energy spectrum of the system under the perturbation eV in (6.86). 6.4.4
Vector nonlinear Schrodinger equation
Since we are dealing with colum vectors in (6.74), it would be useful to have a clear understanding on properties of the vector nonlinear Schrodinger equation. This equation can often be written as t/#t =   ^ x x  & ( # ) 0 ,
(699)
where
is a column vector of ncomponents, describing the "isospace" state. The corresponding Lagrangian and Hamiltonian densities of the system are given by
C =  ( # t  M) ~ ^ (&*») + \ {Wf , n = ^{h^)\{H>f.
(6100) (6.101)
Here 4> = 4>+lo a n d 70 is a diagonal matrix, and the internal product () = (f>+jo
j
which plays the role of isotropic space metric and the dagger sign (*) denotes Hermitian conjugate, the isotransformation belongs to the noncompact group U(i,j\. Zakharov and Shabat showed that (6.99) is completely integrable, in the case of C/(l) group for both positive (b > 0) (the U(l,0) model) and negative (b < 0) (the C/(0,1) model) coupling constants. Furthermore, Manakov integrated (6.99) for the U(2) group (the C/(2,0) model, b > 0), and the integrability for the case of C/(l, 1) group was shown by Makhankov. These can be considered as special cases of the general system.
314
Quantum Mechanics in Nonlinear Systems
In the U(1,O) model, Makhankov et al. found that the integrals of the particle number, momentum and energy can be written in the field and the actionangle variables as follows
N = J \4>\2dx = J n(p)dp + ^2 Ns, P =  3 [(
E
2
4 dx
= I (l^l  > )
J
^sNs,
(6.102)
s=l
= / A(P)*+ E f (**  \b2N2) .
The continuous action n{p) and angular variable 0(p) are related to the scattering matrix element. Prom (6.102), we see that the angular variables 9 and vs are cyclic and the action functionals corresponding to them are integrals of motion. In quantum language the continuous variables (wave background) correspond to linear "microparticle" or elementary excitations (phonons, magnons, etc.), with dispersion E = P2 and mass 1/2 (in natural unit system). The nonlinear microscopic particle (solitons) (i.e., the discrete variable) can be interpreted as a special form of bound states with Ns constituent "particles" of mass 1/2 in the nonlinear quantum field. The energy, momentum and mass of a microscopic particle (soliton) are
Es = ±P?±b2Nl
Ps = \vsNs,
Ms=lNs,
(ATS»1),
(6.103)
The first term of Es is the kinetic energy of the microscopic particle (soliton) with mass Ns/2 and the second term is its binding energy. In the 1/(0,1) model, the dynamic equation is of the form iHt + ^
 b (M 2 p)4> = 0.
r a
(6.104)
A term, bpcj>, is introduced into the equation and a corresponding term, —bp\
N = /
(\
J — oo
P = j
J—oo
Q(
(6.105)
where 8 is the phase shift in the solution when the coordinate varies from — oo to +oo. The discrete part of the spectrum corresponding to the hole excitation mode
Nonlinear versus Linear Quantum Mechanics
315
in quantum case was studied by Lieb. In the classical limit, this mode is denoted by a complex kink solution Vie
<j)k = ak tanh[ajt(a;  vkt  x0)] + i — , where
Thus we get
The negative sign in the expression of N corresponds to a particle "deficiency" in the condensate, i.e., the presence of holes in the system. The hole number is given by — Nk = Nh > 0. Therefore, this model can be used to describe the nonlinear excitation of holes or holesolitons in the nonlinear quantum mechanical systems.
6.5
Eigenvalue Problem of the Nonlinear Schrodinger Equation
As mentioned in Chapter 3, the eigenvalue problem of the nonlinear Schrodinger equation (3.2) with the Galilei invariance is determined by the eigenequation of the linear operator L in the Lax system in (3.20)  (3.22). The eigenequation corresponding to the nonlinear Schrodinger equation is found by the linear ZakharovShabat equation (4.42) from Lax equation (3.20), or iil>x> + $V = A03V1
(6.106)
This is an eigenequation for an eigenfunction ip with a corresponding eigenvalue A and a potential $, where,
•=(£)•  ( ;  . ) •  t i t y
<"•»>
Here <j> satisfies (4.40). It evolves with time according to (3.21). However, what are the properties of the eigenvalue problems determined by these relations? This deserves further consideration. As it is known, the eigenequation is invariant under the Galilei transformation. As a matter of fact, if we substitute the following Galilei transformation
{
x = x'  vt', i=t', (j>'(x,i) =
eivx'~iv2t'/2(x',t')
(6.108)
316
Quantum Mechanics in Nonlinear Systems
into (6.107), $ is transformed into
*'(*) = (% e i / 2 J * ( * ' ) ( e 0 ^ J ,
(6109)
where 0 = vx'  v2t' + 0O, and #o is an arbitrary constant. If the eigenfunction ip{x') is transformed as
*&={ 0 ei/aj^)'
( 6  U °)
ifa + $ V = (A  £) orf.
(6.111)
then (6.106) becomes
It is clear that in the reference frame that is moving with velocity v, the eigenvalue is reduced to v/2 compared to that in the rest frame. It shows that the velocity of the microscopic particle (soliton) is given by 23?(A«). When & is constant, i.e., 0 = 60, the eigenvalue is unchanged because v = 0. This implies that the nonlinear Schrodinger equation is invariant under the gauge transformation, 0' = e%e°<j>{x'). Satsuma and Yajima studied the eigenfunction of (6.106) that satisfies the boundary condition tp = 0 at a; » oo. The eigenvalues and the corresponding eigenfunctions were denoted by Ai,A2, ,XN and tpi,ip2, ,I}>N For a given eigenfunction, tpn(x'), equation (6.106) reads i
^ ax
+ $(i')i(i') = U i ( i ' )
1
n = 1,2, •••,7V.
(6.112)
$(x') was expressed in terms of the Pauli's spin matrices a\ and a2,
$(x') = RMaOfci  9fe(a;')]"2.
(6.113)
Multiplying (6.112) by a2 from left and taking the transpose of the resulting equation, we get
where the superscript T denotes transpose. Multiplying the above equation by %i>n from right and (6.111) by tymci fr°m 1 ^ a n < i subtracting one from the other, Satsuma and Yajima obtained the following equation (AB  Am) /
Vm^ndx'
= 0.
Nonlinear versus Linear Quantum Mechanics
317
The boundary conditions, ipn,tpm > 0 as x' > oo, were used in obtaining the above equation. The following orthonormal condition was then derived. /•OO
/ J — oo
Vm°^ndx' = Snm.
(6.114)
Satsuma and Yajima further demonstrated that (6.112) has the following symmetry properties. (I) If <j>{x') satisfies <j>{x') = 4>*(x'), then replacing x' by x' in (6.112) and multiplying it by 02 from left, we can get {
lb ^n(~x'K
+
^
[^"(z')] = Ana3 [<x2i/>n(z')] •
Since cr2ipn(—x') is also an eigenfunction associated with An, its behavior resembles that of ij)n{x') in the asymptotic region, i.e., a^tpni—x') » 0 as a;' > oo, Thus ipn has the following symmetry
or
^n{x')
= S<72^n{x'),
(8 = ±1).
Therefore, if <j)(—x') = —<j>*(x'), then tpn(x') satisfies the symmetry property ipn(x') — (7iV>n(z') with S = ± 1 . This can be easily verified by replacing <TI withCT2in the above derivations. (II) If 4>(x') is a symmetric (or antisymmetric) function of x', i.e., <j){x') = ±^>(a:'), then ?/>^s'(a:') = a\^{—x') is the eigenfunction belonging to the eigenvalue —\*n, and ij>n {x') = CT2^(X') is the eigenfunction belonging to the eigenvalue An. The suffix s {ox a) to the eigenfunction ip'n indicates that (j> is symmetric (or antisymmetric). Since fi(—x') = 4>{x'), replacing x' with —x' in (6.112) and taking complex conjugate, we get
i£; [*!#;(*')] + *(^) ki<(^)] = A>3 [a^^x')] • Compared with (6.112), the above equation implies that — A£ is also an eigenvalue and the associated eigenfunction ip'n (xf) is oi$*(—i'), with an arbitrary constant. For <j)(—x') = —
= ffi [6a2rn(x')] =
5o^{x%
i>i \x') = a2 [ScrMx')} = Sa^Hx'),
318
Quantum Mechanics in Nonlinear Systems
i.e., ipn(x') has the same parity as ipn(x'), while ip'Jta\x') has the opposite one. When (/>(x') = ~(p(x'), and An is pure imaginary (An = A*), the eigenvalues corresponding to the positive and negative parity eigenfunctions degenerate. (III) If 4>{x') is real but not antisymmetric, then the eigenvalue An is pure imaginary, i.e., 3?(An) = 0. Prom (6.112) and its Hermitian conjugate, Satsuma and Yajima found that 9J(An)(na2n> = (nm
(6.115)
with (m<72n> = /
(6.116)
iPln
where [$,<TI] = 2iQ($)<73 was used. From (6.115), we see that Jt(An) vanishes if 4> is real and (n<72n) ^ 0. When <>/ is a real and an antisymmetric function of x', symmetry property (I) gives /•OO
(n\a2\n) = d2 /
J—oo
ip+(x')tTia2crixl)n(x')dx' = (n\a2\n).
Thus (n\a2\n) = 0. (IV) If the initial value takes the form of
rn(x') = idcT3ipn(x'),
where 6 = ±1. Because 5ft(An) = 0, from the complex conjugate of (6.112), one can get ipn(x) a ozipn(x). Substituting (6.117) into the normalization condition (6.114), one then has 8 = ±1. If the eigenvalue of (6.106) is real, i.e., A = £ is real, then
(6118)
{
lt> + ^ = ^^
and the adjoint function of ip, ip = ia2ip*", is also a solution of (6.118) i.e., di>
i— + $ip = fr3i>From this and (6.118), Satsuma and Yajima obtained the following
A(^+V) = ±$++) = ^«>+V0 = ±m)
= 0.
(6.119)
Nonlinear versus Linear Quantum Mechanics
319
Using the above boundary conditions, they found that the solutions of (6.106), xpi(x',£), i>2{x',0) a n d ifa(x',Q satisfy the following relations
Prom ipi = a(^)%p2 + HOV^, we get a = t/>^Vi &nd & = V'Ji/'i, where
^ l(x '' e) = (o) e "^'' as ar' = —oo and
V'2(x',O=(°)e+^',
as x' = oo. As it is pointed out earlier, if real (not antisymmetric) initial value is considered, the microscopic particle does not decay into moving solitons, but forms a bound state of solitons pulsating with the proper frequency. Satsuma and Yajima developed a perturbation approach to investigate the conditions for the solutions to evolve and decay into moving solitons. If the wave function cf> in (6.106) undergoes a small change, i.e.,
An and ipn change as An + AA n and tpn + Aipn, respectively. To the first order in the variation, equation (6.112) becomes rdx7
+
^ ~
Xn(T
^\
A
^ " + ( A * ~ AAn<73)n = 0.
Multiplying the above equation by Vn°2 from left and integrating with respect to x' over (—oo, oo), we get nOO
AAn = i
Vn^A^Vnda:' J — OO /•OO
=/
J~
OO
/OO
^(A^as^ndx'+i
J~
i>lZ{&4>)^ndx'.
OO
If 4> is a real and nonantisymmetric function of x', equation (6.117) holds and AA n = 6(n\^{A<j>)a3\n) + W(nR(A0)n).
(6.120)
Equation (6.120) indicates that if (n3(A)(T3n) ^ 0, the perturbation A
320
Quantum Mechanics in Nonlinear Systems
the solution of the nonlinear Schrodinger equation (4.40) breaks up into moving solitons with velocity 23?(AAn). If tf> is a real and is either a symmetric or an antisymmetric function of x', the above symmetry properties of eigenvalues of the nonlinear Schrodinger equation lead to (n3(A#c')V 3 n) = (n\$s(A<j>{x))a3\n). Therefore, if %(A
where
AA± = 8 fanfloMrsMz' + 2a£)n) Tsin (j)
{n\<j>0{x')e2x'^dldx^\n)\ . (6.121)
iS \cos60(n\<j>o(x' + 2x'0)\n) ± cos (j) where (n\Mx')e2x'o{d/dx)\n)i6Sin
Jcos {^\ =
J— oo
^2
CT2$2^i
d,X '=
(^) {n\az
Vl
02*1^2
dx
\
J—oo
5cos(00)(n\
= I
J—oo
tf{n)Ta2*2tf(n)dx'
pOO
= I
J—oo
^n)Ta2^2'{n)dx'.
Here $(O=*l(z') + $2(*'), and $i(a;') = ai
$ 2 (x ) = [cos(0o)"'i  sm(0o)a2]
The corresponding eigenvalue equation is given by
321
Nonlinear versus Linear Quantum Mechanics
The eigenfunction
CtCz') = ±6 [cos (jfj a2 + sin (  ) CTl] ^±(x'),
S = ±1,
J—oo
when 6Q = 0, <j>(x') is real and symmetric, A An ' is pure imaginary. When 0O = n,
(6.122)
Thus the solution of the nonlinear Schrodinger equation (4.40) decays into paired solitons and each pair consists of solitons with equal amplitude and moving in the opposite directions with the same speed. For arbitrary 6'0, we can see from (6.122) that the solution of (4.40) breaks up into an even number of moving solitons with different speeds and amplitudes. 6.6
Microscopic Causality in Linear and Nonlinear Quantum Mechanics
The microscopic causality is an interesting problem in quantum mechanics, and understanding the difference in microscopic causalities in the linear quantum mechanics and the nonlinear quantum mechanics is important for a better understanding of the essential features of the nonlinear quantum mechanics. The microscopic causality will be discussed in this section. In classical physics, the causal relations, which crystallizes the intuitive notion that a cause must precedes its effects, are often expressed in the Green's function of a theory or by the KramersKronig type of dispersion relations. These forms can be easily generalized to the linear quantum mechanics. In fact, the commutator of two operators or its matrix form contains causality information, and for this reason, it is referred to as an interference function, and the relations are called microcausal. Using solutions of the dynamic equation represented by the creation and annihilation operators, Burt expressed the microscopic causality in terms of probability amplitudes. The microcauslity concerns two processes or events that must be classified into two independent data sets if the events are separated by a spacelike interval. As shown in Fig. 6.3, two such events may appear to have quite different temporal orientations in various coordinate systems. Thus, in the absence of superlumimal signals, the description of the events must be independent. In the canonical quantum theory, the information concerning the interference of the two events is described by
322
Quantum Mechanics in Nonlinear Systems
Fig. 6.3 Minkowski diagram illustrating relativity of temporal order of two events separated by a spacelike interval in three coordinate systems related by Lorentz transformations (see book by Burt).
the commutator of the operators denoting the two events. This commutator gives the interference when the sequence of events is reversed and the difference taken between the two orderings. The representation in terms of probability amplitudes results in an interference function which can be built by the creation and annihilation operators. The sequence of events to be considered consists of the creation and annihilation of microscopic particles from the vacuum at two points in spacetime. In the reversed sequence the points in spacetime are exchanged. The difference between the amplitudes for the two sequences defines the interference function. Using the knowledge of quantum field theory, the amplitude for the creation of a particle with positive energy from the vacuum at x and its subsequent annihilation into the vacuum is represented by
k,p,€
= ^(oi4 + ) 4 + ) t i°)( £ ) 2 w P w ^ 2 )" 1 / 2 e * i "^' s '
(6123)
where V is the volume of the system. Similarly, the sequence for a particle with an negative energy is
Attp(y,i)
= ^(04 ) 4 )t 0)(D 2 o; p a; fc y 2 ) 1 / 2 /e ii •***.
(6.124)
Here At and A^ are the creation and annihilation operators of the particle with wave vector k, respectively, (+) and () indicate the positive and negative energy states, respectively, the field operators 0 of the particles is represented by
where uik is the frequency of the particle, k — (tJk,k), w\ — k2 + m 2 , x — (x, ict), \Ar, At 1 = 6tt,, D is a normalization constant, Ay'elk'x is a creation operator. Thus, the probability amplitude that a positive energy particle with momentum k,
323
Nonlinear versus Linear Quantum Mechanics
created from the vacuum at position x, is found in the intermediate state \e) is
Similarly, the probability amplitude that a negative energy particle annihilated from the vacuum at position x with momentum k is found in the state e) is
Then, the amplitude for annihilation of a negative energy particle from the vacuum at y, followed by the creation (to fill the negative energy state) of a negative energy Thus we get the particle at x (the time sequency is reversed) is p\~\x)pC'{y). representations (6.123) and (6.124). Since the two amplitudes have equal weight, total amplitude is given by the arithmetic average,
A amp (y, x) = \ [4J) p (j/,x) + A[)p(y, x)] .
(6.125)
However, from the definition of the vacuum state, it is required that AJ,~)tO)=O.
(6.126)
Therefore, AAmp(y,x)
(6.127)
= ±A{+}p(y,x).
The amplitude of the reversing sequence can be calculated following the same steps. The total interference amplitude is then given by A>lamp(y, x) = A amp (y, x)  Aamp(x, y) = \ ] T ( 0  4 + ) 4 + ) t  0 ) (e****"*  e**"*) {D2cjpu;kV2)1/2
(6.128) •
For the neutral, spin zero system of particle, using the commutator [/l^^lt] = finkrip and converting the sum into an integral using
the total interference amplitude can be obtained AA (v x\ AA amp (2/,x; 
1 2(27r)3
f — [P«*(*S) _ p»*(Ss)l y w £ ) [e e j .
Cfi loqi (b.129)
If the constant D is set to unity, this becomes the interference function denned in the canonical formalism by Bjorken et al. and Bogoliubov et al. (see book by
324
Quantum Mechanics in Nonlinear Systems
Burt)
AA&mp(y, i) = A(y  x) = ^^
J ^
[>»*>  e^'*"*)] .
(6.130)
This Lorentz invariant function clearly vanishes for spacelike intervals. If (y — x)2 < 0, a Lorentz transformation can always be found which takes us to a coordinate system in which XQ = y0. Replacing k by — k in the second term does not change the integrand. Hence A(y  x) = 0,
(y  x)2 < 0.
(6.131)
Properties of this function were studied by Bjorken et al. and Bogoliubov et al. The statement of microscopic causality in terms of probability amplitudes is equivalent to the usual canonical relation for free fields in the linear quantum mechanical theory. This idea was generalized to the selfinteracting fields in nonlinear quantum mechanics by Burt. As in the establishment of propagator including persistently interacting fields in the nonlinear quantum mechanics, the interference function describing microcausality with interactions is obtained by employing the superposition principle of quantum theory. The algorithm developed in the linear quantum mechanics is utilized for the persistently interacting field operators since the latter can be interpreted as creation or annihilation operators. For the representation of the interference function in (6.128), Burt generalized it to the following form AAamp(y,*r rS = Amp(y,x) perS  A&mp(x, y)*™
= \ E w i p t t o ^ W  4+)(*)^+)(j')ti0>' k,q
(6132)
where
A&mp(y,xr™ = \A{$p(y,xr™ = \ £<04 +) (y)^ + W0>.
(6.133)
Using the general expressions for persistently interacting fields given in section 5.3, the current / is denoted by a series containing only positive powers of the creation or annihilation operators,
2
where <j>n(\i;m ;p) contains all the coupling constant dependence and depends on the parameters m2, p and po, the p and po are indeces of interaction which are related to the form of the selfinteraction current. The interference function can be
325
Nonlinear versus Linear Quantum Mechanics
written as AA amp (y,x)P ers = i J2 ^n(Ai;m 2 ;p)0 s (A i; m 2 ;p)x
(6.134)
k,q,n,s
pn+P
{(01 [u?Hy)] ° [u?>t(i)p°
_ [uf\i)]pn+P0 [u?»(y)]PS+P° o>} •
Using
UJ^tf) =
AWe^iDuV)1'2,
where D is a constant, V is the volume of the system, u = Vk2 + m2, and
with
equation (6.134) becomes AA amp (£,*)P ers = i 5 ] «^n(Ai;m2;p)
a
e
o
_ei{pn+po)ki+i(ps+po)qy]
\
= \YJ{
(6.135)
If we consider events for which the interval (y — x)2 is spacelike, a Lorentz transformation can be performed to the system in which the events are simultaneous. Prom (6.135), Burt found that AAAmp(y • x)Pers = \'£
+ p0)\ x
fc,n
fei{pn+po)k(yx)_ei(pn+po)'k(xy)\
(6.136)
Replacing k by —k in the second term, the total expression vanishes. Since the interference function is Lorentz invariant, the result is independent of the coordinate systems. Thus, Burt obtained that AA amp (y, x)pers = 0,
(x y)2 < 0.
(6.137)
326
Quantum Mechanics in Nonlinear Systems
Therefore, the interference between the creation and annihilation operations of the persistently interacting fields vanishes when the processes are separated by spacelike intervals. The principle of microscopic causality, stated in terms of probability amplitudes, is satisfied by the persistently interacting fields. As is the case for a propagator, the causal function AA amp (y,x) pers typically reduces to a sum of terms of the form AA amp (j/, x)*ers = A(y £) + APers(£  x; A;),
(6.138)
where A(y  x) is the usual free particle interference function obtained by Bjorken and Bogoliubov, and APers(y  x; A,) contains the interaction effects. Thus, for the \(f>z current or (/>4field equation, Burt found that APers(z;Xi) = \J2 k,n=l
Vn + iy.iDajV)2"1
" {c<(2n+i)t« _ ei(2n+i)*.« j
(^) \
m
/
Due to the presence of the factor (DuV)~2n~1, this function is less singular than A(z). The effect of the interaction is to smear the singularity into a smooth function with a tail characterized by the Compton wavelength of the higher mass contributions and by strength proportional to the coupling constant. Therefore, there are only somewhat differences between the representations of the causalities in the nonlinear quantum mechanics and linear quantum mechanics, but their results are basically the same. Bibliography Asano, N., Taniuti, T. and Yajima, N. (1969). J. Math. Phys. 10 2020. Bjoken, J. D. and Drell, S. D. (1964). Relativistic quantum mechanics, McGrawHill, New York. Bjoken, J. D. and Drell, S. D. (1980). Relativistic quantum fields, McGrawHill, New York. Bogoliubov, N. N. and Shirkor, H. (1959). Introduction to the theory of quantized fields, Interscience, New York. Bogoliubov, N. N. (1962). Problems of a dynamical theory in statistical physics, NorthHolland, Amsterdam. Burt, P. B. (1981). Quantum mechanics and nonlinear waves, Harwood Academic Publishers, New York. Carr, J. and Eilbeck, J. C. (1985). Phys. Lett. A 109 201. Chen, X. R., Gou, Q. Q. and Pang, X. F. (1996). Chin. Phys. Lett. 13 660. Chen, X. R., Gou, Q. Q. and Pang, X. F. (1997). Chin. J. Atom. Mol. Phys. 14 393; Chin. J. Chem. Phys. 10 145. Chen, X. R., Gou, Q. Q. and Pang, X. F. (1998). Acta Phys. Sin. 7 329; J. Sichuan University (nature) Sin. 35 362. Chen, X. R., Gou, Q. Q. and Pang, X. F. (1999). Acta Phys. Sin. 8 1313; Chin. J. Chem. Phys. 11 240; Chin. J. Comput. Phys. 16 346; Commun. Theor. Phys. 31 169. Dodd, R. K., Eiloeck, J. C, Gibbon, J. D. and Morris, H. C. (1984). Solitons and nonlinear wave equations, Academic Press, London.
Nonlinear versus Linear Quantum Mechanics
327
Drazin, P. G. and Johnson, R. S. (1989). Solitons, an introduction, Cambridge Univ. Press, Cambridge. Eilbeck, J. C, Lomdahl, P. S. and Scott, A. C. (1984). Phys. Rev. B 30 4703. Eilbeck, J. C, Lomdahl, P. S. and Scott, A. C. (1985). Physica D16 318. Finkelstein, D. (1966). J. Math. Phys. 7 280. Goldstone, J. and Jackiw, R. (1975). Phys. Rev. D 11 1486. Guo, Bailin and Pang Xiaofeng, (1987). Solitons, Chin. Science Press, Beijing. Heisenberg, W. (1974). Across the frontiers, Harper and Row. Kartner, F. X. and Boivin, L. (1996). Phys. Rev. A 53 454. Kleinert, H., SchulteFrohlinde, V. (2001). Critical properties of ^theories, World Scientific, Singapore. Kolk, W. R. (1992). Nonlinear system dynamics, Reinhold. Korepin, V. E., Kulish, P. P. and Faddeev, L. D. (1975). JETP Lett. 21 138. Lai, Y. and Haus, H. A. (1989). Phys. Rev. A 40 844 and 854. Lieb, E. (1963). Phys. Rev. 130 1616. Lieb, E. and Liniger, W. (1963). Phys. Rev. 130 2605. London, R. (1986). The quantum theory of light, 2nd ed., Oxford University Press, Oxford. Majernikov, E. and Pang Xiaofeng (1997). Phys. Lett. A 230 89. Makhankov, V. G. and Fedyanin, V. K. (1984). Phys. Rep. 104 1. Makhankov, V. G., Makhaidiani, N. V. and Pashaev, O. K. (1981). Phys. Lett. A 81 161. Manakov, S. (1974). Sov. Phys. JETP 38 248. Pang, X. F. (1994). Acta Phys. Sin. 43 1987. Pang, X. F. (1995). Chin. J. Phys. Chem. 12 1062. Pang, X. F. and Chen, X. R. (2000). Chin. Phys. 9 108. Pang, X. F. and Chen, X. R. (2001). Commun. Theor. Phys. 35 323; J. Phys. Chem. Solids 62 793. Pang, X. F. and Chen, X. R. (2002). Chin. Phys. Lett. 19 1096; Commun. Theor. Phys. 37 715; Phys. Stat. Sol. (b) 229 1397. Pang, Xiaofeng (1994). Theory of nonlinear quantum mechanics, Chongqing Press, Chongqing. Pang, Xiaofeng (1999). Acta Physica Sin. 8 598; Chin. Phys. Lett. 16 129; Phys. Lett. A 259 466. Pang, Xiaofeng (2000). J. Phys. Chem. Solid 61 701; Phys. Stat. Sol. (b) 217 887. Pang, Xiaofeng (2001). J. Phys. Chem. Solid 62 491. Pang, Xiaofeng (2002). Inter. J. Mod. Phys. B16 4783. Perring, J. K. and Skyrme, T. H. (1962). Nucl. Phys. 31 550. Potter, J. (1970). Quantum mechamics, NorthHolland, Amsterdam. Rubinstein, J. (1970). J. Math. Phys. 11 258. Saji, R. and Konno, H. (1998). J. Phys. Soc. Japan 67 361. Satsuma, J. and Yajima, N. (1974). Prog. Theor. Phys. (Supp) 55 284. Scott, A. C. (1982). Phys. Rep. 217 1. Scott, A. C. (1985). Phil. Trans. Roy. Soc. Lond. A315 423. Scott, A. C. and Eilbeck, J. C. (1986). Chem. Phys. Lett. 132 23; Phys. Lett. A 119 60. Scott, A. C, Bernstein, L. and Eilbeck, J. C. (1989). J. Biol. Phys. 17 1. Scott, A. C , Lomdahl, D. S. and Eilbeck, J. C. (1985). Chem. Phys. Lett. 113 29. Strang, G. (1975). Linear algebra and its applications, Academic Press, New York. Tomboulis, E. and Woo, G. (1976). Nucl. Phys. B 107 221. Zakharov, V. E. and Shabat, A. B. (1972). Sov. Phys.JETP 34 62. Zakharov, V. E. and Shabat, A. B. (1973). Sov. Phys.JETP 37 823.
Chapter 7
Problem Solving in Nonlinear Quantum Mechanics In Chapters 3  6, we have established the fundamental principles and theory of the nonlinear quantum mechanics. The next important step is to discuss methods for solving the dynamic equations. In this Chapter, some common approaches for solving these equations will be discussed.
7.1
Overview of Methods for Solving Nonlinear Quantum Mechanics Problems
From earlier discussions, we know clearly that problems in nonlinear quantum phenomena are very complicated. This complexity is due to the different mechanisms of nonlinear interactions. Therefore, understanding the mechanism and property of nonlinear interaction is essential for establishing correct dynamic equations and for solving these equations. The dynamic equations, (3.2)  (3.5), cannot be solved unless the exact form of the nonlinear interaction, b, is known. Therefore, the first step one should take in solving a nonlinear quantum mechanics problem is to carefully study the nonlinear quantum phenomenon and its properties, look into the mechanism of the nonlinear interaction, and come up with an appropriate model for the nonlinear interaction. Based on these understandings, the Lagrangian, Hamiltonian function, and operators of the system can be established. Finally, the dynamical equations of the microscopic particles can be obtained from the corresponding classical quantities, or Lagrangian, using the EulerLagrange equation or the Hamilton equation, or from the quantum Hamiltonian using the Schrodinger equation or the Heisenberg equation in the second quantization representation in which the Hamiltonian is given in terms of creation and annihilation operators of the microscopic particle. Once the dynamical equation is known, we can solve the equation to obtain its the solution and study the nonlinearphenomena. Related systems can also be studied based on the fundamental dynamic equation. For example, external fields or forces can be considered and included in the dynamical equations. This is a general approach for solving nonlinear quantum mechanics problems. Of course, certain problems can be easily solved using some special techniques. 329
330
Quantum Mechanics in Nonlinear Systems
The commonly used methods for obtaining soliton solutions of nonlinear dynamical equations in nonlinear quantum mechanics are summarized in the following. 7.1.1
Inverse scattering method
The inverse scattering method was first proposed by Gardner, Green, Kruskal and Miura (GGKM) in 1967, for solving the KdV equation which connects a nonlinear dynamic equation to the linear Schrodinger equation
through the GGKM transformation
where <j> satisfies the nonlinear dynamic equation. In 1968, Lax gave a more general formulation of the inverse scattering method. The nonlinear Schrodinger equation was solved by Zakharov and Shabat in 1971 using the inverse scattering method which was discussed in Section 4.3. In 1973, the SineGordon equation was solved by Ablowitz et al.. According to Lax, given a nonlinear equation
4> =
(x,t),
the inverse scattering method for solving the equation consists of the following three steps. (1) Find operators L and B, which depend on the solution <j>, such that iLt = BL LB, as given in (3.20) and (3.21). (2) The scattering operator, L, satisfies its eigenvalue equation, Lip — \
Bdcklund transformation
Given d<j> = Pdx + Qdt, then the integrability of the corresponding firstorder differential equations j>x = P,
(7.1)
Problem Solving in Nonlinear Quantum Mechanics
331
requires Pt = Qx. The Backlund transformation consists of the following three steps. (1) Find P and Q, as functions of 4>. (2) Choose a trial function, <j>0. (3) Use fax = P(fa,(f>0) and fat = Q(fa,
Hirota method
The Hirota method changes the independent variables and rewrites the original dynamic equation into the form of F(Z?™Dtn) = 0, where D is an operator. Analytic solutions are then obtained using perturbation method. Here, (7.2)
The Hirota method is widely used. 7.1.4
Function and variable
transformations
Quite often a complicated equation can be transformed into a simpler or a standard equation which is readily solved. The forms of transformation vary and depend on the problem and equation. The following are some commonly used transformations. 7.1.4.1 Function transformation As an example, we consider the transformation <j> = Jpeie^t\
(7.3)
Using this transformation, the nonlinear Schrodinger equation (3.2) with V{f,t) = 0 = A(
332
7.1.4.2
Quantum Mechanics in Nonlinear Systems
Variable transformation and characteristic line
The following variable transformation is often used. £ = a; — vt,
r] = x + vt.
(7.4)
For example, the onedimensional linear fluid equation,
at
ox
can be written as
^=0,
4> = f(t) = f{xvt)
under this transformation. The equation dx/dt = v or dx/v — dt/1 — d
, then £ = f(x, t) and $ = g(x, t, cj>) are called a selfsimilarity transformation, the solution {x, t) (nonlinear Schrodinger equation) is a condition of compatibility of auxiliary linear equations (3.20) and (3.21), where A is a spectral parameter (generally speaking, complex), and ip in (3.21) is often called a Jost function and is represented by
334
Quantum Mechanics in Nonlinear Systems
For the linear heat conduction equation, cty _ d2<j> where 7 is a constant, its selfsimilarity transformation is,
<7f * = f and its solution is
Under this transformation, the equation becomes .
V
y
4.
2
Z_L
2 d£
/v — n 7
~
For the SineGordon equation
its selfresemblant transformation is C = £j,
<£ = <£,
and the equation becomes
under this transformation. The solution of this equation can be easily obtained. 7.1.4.5
Galilei transformation
Function transformation is a general approach for solving dynamic equations. As an example, we consider the nonlinear Schrodinger equation (3.2) with V(x) = A{4) = 0, and 6 = 1 . If we let
,
X=X its solution is of the form
[2^
V^'
t
and
* = n' /
A2t'\
(78)
<j>(x',t') = ^sech(Ax')exp f  t ^ " J
Since this nonlinear Schrodinger equation is invariant under the Galilei transformation (6.108), its solution can be written as
[
C,,2 _ j 2 w n
ivx' + i±
2 ~  "
^
Problem Solving in Nonlinear Quantum Mechanics
335
Let v = 2£, and A — 2T], then it reduces to the soliton solution (4.56) which was obtained by the inverse scattering method. 7.1.4.6
Travelingwave method
If we can write the solution of an equation of motion in the form of <j)(x, t) = (£)> with £ = x — vt, it is called a travelingwave solution. This method is widely used in wave mechanics and nonlinear quantum mechanics. 7.1.4.7 Perturbation method In reality, due to the existence of boundaries, defects, impurities, dissipation, external fields, etc., a system usually experiences some perturbation. As a matter of fact, when the soliton behavior of microscopic particles in the system is probed, the measurement itself is bound to alter the state of the system to some extent. Thermal excitation at finite temperature is another form of perturbation that no system can avoid. In fact, the appearance of microscopic particles (solitons) itself implies some form of internal excitation has occured. When such perturbations are relatively weak, they may be treated by perturbation methods. The main spirit of soliton perturbation was illustrated in Chapter 3. 7.1.4.8
Variational method
The variational method will be discussed later in this Chapter. 7.1.4.9 Numerical method When analytic solutions cannot be found and perturbation methods are inapplicable, one can always solve the nonlinear equation by numerical methods. Sometimes the numerical solutions may lead to a correct guess of their analytic forms. There are many numerical approaches for solving differential equation such as MonteCarlo, RungeKutta, finite elements etc. With the rapid increase in computing power in the last decades, numerical simulations plays a more and more important role in practical calculations. 7.1.4.10 Experimental simulation Some dynamic equations may be simulated by mechanical models or electric circuits. For example, the SineGordon equation was simulated by a chain of torsional pendula. The overdamped SineGordon equation
336
Quantum Mechanics in Nonlinear Systems
7.2
TravelingWave Methods
7.2.1
Nonlinear Schrodinger equation
For the nonlinear Schrodinger equation given in (3.2) with V(x, t) = A(cf>) = 0, a travelingwave solution can be assumed <j>{x,t) =
<j>{^kx'ut'\
where
£/• = *i  < J.1
/ ,, t Ji = yjWx, t' = . 2
m
In terms of the new variables, equation (3.2) can be written as
^ p + (w _ k>m)  if = o. The soliton solution of this equation can be easily obtained when b > 0 and 7 = w  k2 > 0, * ( M ) = y ^ s e c h [ V 7 ( ^  &)]e'< fa ' wt< '.
(7.10)
On the other hand, when b < 0 and 7 < 0, the solution is
4>(x,t) = y^tanh n^E^fc) j e ^ '  O .
(7 . n)
If we assume the travelingwave solution is of the form <j>(x',t')=4>(OeiQt'+iQx>,
(7.12)
then (3.2) with 6 = 2, V(x) = A(4>) = 0, becomes
0«(O + i(2Q  v)fa  0(0 (n + g 2 ) + 203(o = 0.
(7.13)
If the complex coefficient of 0$(O vanishes, then <5 = u/2. Furthermore, from A = Q2 + fi, we get that ft =  « 2 / 4 + A. Finally from (7.13), we obtain
fa)2 = D + A\ where D is a integral constant. The solution (j) is then obtained by inverting an elliptic integral,
r , *>
=±e
Problem Solving in Nonlinear Quantum Mechanics
337
Let
P{4») = (ft  0 2 )(0 2  ft), we have,
±={K(k)F(4>,k)} = ±$, where F(
Using these and j3\t2 = 0f 2 > t n e n
we nave
0(O=0i{l[(l)sn2(C,*)]}12. W h e n D » 0, i >
4>,(x,t) = VAsech [y/A(x'  vt')] exp ft (~\v2t' + ^vx' + AA] .
(7.14)
The corresponding canonical form of the onesoliton solution has been given in (7.9) and (4.56). 7.2.2
SineGordon equation
Using the travelingwave method in natural unit system,  = x—vt, the SineGordon equation,
(7.15)
can be written as
(vivl)£g+hlsm
= 0.
(7.16)
If v2 > VQ, (7.16) can be written as ^+/? <%2
2
sin 0 = 0,
with P
~
v2vl
(7.17)
338
Quantum Mechanics in Nonlinear Systems
Equation (7.17) resembles the equation of motion of a undamped pendulum, and can be separated into the following two equations
 = .. £*>.*
«,18)
This gives /32 sin <j)
d$ _ Thus we can get
l$2 + p2{lcos4>) = A, where A is a constant of integration. Using 1  cos <j> = 2 sin2 ^ and (7.18) and letting A = 2/32k2, the above equation can be written as
1
(719)
()'"I* "(!)]• If 0 < k2 = A/202 < 1, the solution of (7.19) is given by
sin (£j = ±ksn[f3(£   0 ),
fc].
(7.20)
li k —> 1, equation (7.20) is replaced by
sinf0=tanh[±/3(o)], or
[lsin((A/2)J
\4
4J
Thus the kink solution of the SineGordon equation (7.15) is finally given by 4>± = v + 4 tan" 1 e ± / 3 ( «« o ) . When v2 < VQ, equation (7.19) is replaced by
(3)'«K!M. where oil _ hQ P — ~o
o
vlv2
(7.21)
Problem Solving in Nonlinear Quantum Mechanics
339
The kink solution of the SineGordon equation (7.15) in the case of k' > 1 and k'2 — 1 — k2 becomes
0 = 4 tan" 1 {eT/3'(««°>} .
(7.22)
Using the following dimensionless variables, t' = hot, x' = 5x, 6 = ho/vo, equation (7.15) can be written as
(7.23)
According to the properties of (7.21) and (7.22), the solution of (7.23) is of the form
^ = 4 t a n " 1 [wl
(7>24)
In terms of Y and T, equation (7.23) becomes
(Y2 + T2) {^ + ^fr 1 )  2[(Y,02 + (Tv)2} = T2  Y2.
(7.25)
Differentiating (7.25) with respect to t' and x\ and then dividing the two resulting equations by YYx* and TTf, we can get 1
1 (Tt.t>\,
(Yx.x,\
.
(79R.
where 4q is a constant. Integrating (7.26), the following equation can be obtained ^ 
= 2qY2+ai,
T^ = 2qT2 + a2,
(7.27)
(Tt,)2 = qT* + a2T2 + d2,
(7.28)
and (Yx,)2 = qY4 + aiY2 + du
where a\, a2 and d\, d2 are some integral constants. Inserting (7.27) and (7.28) into (7.25), we can get ai  a2 = 1, and d\ + d2 = 0. If we choose a2 = —a, d\ = d, then ai = 1  a and d2 = d. When q = —1, and d = 0, equation (7.28) becomes Y2, = Y4 + (1  a)Y2,
T2, = T 4  aT2.
(7.29)
If 0 < a < 1, integrating (7.29) and taking the integral constant to be zero, we can get
jl— sech"1 (JLJ) Vl  a
= ±x' = ±Sx,
Vvla/
isin'(^)= ± ( ' = ±M.
340
Quantum Mechanics in Nonlinear Systems
Thus Y = y/l  asech(\/l  a Sx),
— = ±—~ sin(y/ahot). (7.30) yd 1 Substituting (7.30) into (7.24), we get the breather solution of the SineGordon equation (7.15) (
. 7.31
Following the same procedure, we can also obtain the travelingwave solitonsolution of the 04field equation in natural unit system (j>ttvl(j>xx+a'cl>pl^=Q. Since the procedure is the same, we only give the results here,
^ ± ^ t anh[^^ ( f^ , when v2 > ug, a' > 0, 0' > 0 or v2
0, /?' < 0, and
, , /ic7 [ / a'  ^=±y—sech^^^(eeo)j, when v2 > wg, a ' < 0, /3' < 0 or w2 < ug, a ' > 0, /3' > 0.
7.3
Inverse Scattering Method
We now determine the soliton solution of the nonlinear Schrodinger equation (4.40) with an initial pulse using the inverse scattering method mentioned earlier, following the approach of Zakharov and Shabat, in accordance with the Lax operator equation (3.20), where L and B are some linear operators with coefficients depending on the function
GO where L is given in (3.22) for the nonlinear Schrodinger equation (3.2) with V = A = 0. Therefore, equation (3.21) can be rewritten in the form of ZakharovShabat
Problem Solving in Nonlinear Quantum Mechanics
341
equation (4.42), where
b is the nonlinear coefficient in the nonlinear Schrodinger equation (4.40). Here the characteristic value, £ = £' + it). Equation (4.42) can be further written in the form
*!*<*' ~ y *i*< + *i ^ ' 2 + \q\2  tf y ) = 0,
(7.32)
*2 =  [ * ! » ' + ^ ' * i ] .
(733)
According to the inverse scattering method, it is necessary to find the solutions of the system (7.32) which have the following asymptotic behavior for a given function q(x') and real £ = £'
ifV>oo,
l(l)e**', l i m b s ' ) ={
VVJ
,
[ o(O (JJ e  ^ '  6(n ( J J e^'', if x' > oo.
(7.34)
The coefficients a(^') and b(£') are complex and satisfy
\a(a2 + Ha2 = i. The quantity, l/a(^'), determines the transmission coefficient at a;' of a plane wave incident on the potential q(x') except at x' w oo, while the ratio, R(£') = b(£')/a(£') determines its reflection coefficient. The functions in (7.34) continue to be analytic into the upper half (rj > 0) of the complex plane of £. With the inverse scattering method, we can obtain the values of a(£') and &(£') at time t' ^ 0 from their values at t' = 0 using the relations a(U')=a(0,
b(C,t') = b(0e^2t'.
(7.35)
If for certain values £j, a(£j) vanishes in the upper half of the complex £ plane, then the asymptotic behavior of ^(Q,x') is given by
((jW«i>'\ *&,*')
={
V
,nx
x'^oo, (j = l,,N).
(7.36)
For r)j 7^ 0, *l!(Q,x') decreases asymptotically when a;' —> oo and therefore describes bound states corresponding to complex Q given in (4.42) or in (7.32) (7.33). Values of Q and Cj(t'), (j = 1,2, • • • , N), where
c,(O^(C^')[^]^=fe4^',
(7.37)
342
Quantum Mechanics in Nonlinear Systems
and
form a set of "scattering data". Prom these data, Davydov obtained the following auxiliary function
Pi*',?) = ±
z n J
°°
deRtf,t)e*'' + Y,cj(t')ei^'x>.
(7.38)
j=i
Inserting (7.38) into the GelfandLevitanMarchenko (GLM) equation rOO
K(x',y') = F*(x'+y',t')
/
Joo
/OO
dp /
Jx'
dzF*(p + y'1t!)F(p + z^)K(xf,e).
(7.39)
the undetermined function q(x',t') = 2K(x',x')
(7.40)
and the wave function satisfying the nonlinear Schrodinger equation (4.40), with the initial value
(7.41)
Vb
Using the inverse scattering method, Davydov obtained a soliton solution of the nonlinear Schrodinger equation (4.40) with the following initial data
4>(x', t' = 0) = l^be™*1
sech (^pj ,
(7.42)
where 2k = mexrov/h. Here ro is the lattice constant, mex and v are effective mass and velocity of the particle, respectively. To obtain the scattering data, it is sufficient to find the asymptotic solutions of (7.32) and (7.33) with
, 0) = je™*' sech ( ^ ) .
(7.43)
If \x'\ > oo, equation (7.32) becomes *i'  (2ik ±  ) * i + C (C  * =F ^ ) * i = 0, where ~,
dV dx'
(7.44)
Problem Solving in Nonlinear Quantum Mechanics
343
The upper and lower signs in (7.44) correspond to x' > —oo and x' > oo, respectively. It can be shown that (7.44) has the following solution (7.45)
with mi=C + ki~,
m2 = C + k + i. 8
o
lib ^0, (7.45) coincides with the asymptotic function (7.34) with
£=fc,
m = . o
Prom (7.34), we found that as x' »• 00, # 2 (£') = 6(^')e^'*'. However, (7.45) and (7.33) imply that * 2 (oo) = 0. Therefore, for the potential (7.43), b(f') and R(£) must equal to zero. Such potentials are called nonreflective potentials and cannot be observed by scattering of plane waves coming from infinity. In such a case, the spectral data (7.37) include R(t',t') = 0,
Ci =  « + f  ,
d(t') =
coe^',
o
where Co = ib/4. Thus, F(x',t')=cl(t')ei^'. Inserting this expression into (7.39), we get K(x',y') = f(x')e
/(*') = cUt')e^' [l + !6i£ip! e 6*72p . The square of the modulus of the wave function, including q(x',t') in (7.40) is given by
\
\
8 cosh [b(x'  x'o  4/ct')/4]
(7.46)
where
This indicates that given an arbitrary nonzero value of b and the initial pulse (7.42), the microscopic particle propagates with velocity v in the form of a single soliton.
344
Quantum Mechanics in Nonlinear Systems
Davydov further obtained solution of the nonlinear Schrodinger equation (4.40) with a rectangular step initial pulse
{
J _ 6 2ikx'
Vi
if 0X <1 r' < / li0
'
(7.47)
^^
0, if x' < 0 and x' > I. In this case the solution of (4.42) or (7.32)  (7.33) can be written as
*=rj)e«*\
forz'<0,
* = "(0 (I) e~Kx' + KCKCl\ ^ x' > i, while in the region of 0 < x < £, * i = [Ai sinnx' + A2 cos nx']elkx>, * 2 = 7^{[(C + ^)^i + inA2]smnx' Qo
+ [((  K)A2  inA±]
sinnx'}e~ikx',
with
n2 = Q 2 _ ( c _ K ) 2
^0 = ^ 1 ,
(748)
It follows from the continuity of the solutions that 4i(C) =  » — ,
(7.49)
A2 = l,
n 6(0 = ^Qoc^* + c ) i ',
iiK+Ot
a ( 0 = ^ « , k)e
',
where S(C, k)=n cos(nl)  i{k + Q sin(nl). Using (7.49), we can find the reflection coefficient at time t' = 0 R(^') = iQoS1^1
,k)e2i^k+^t.
According to (7.36), the reflection coefficient at time t' is given by R(?,t!) = R(?)eiit"t'.
(7.50)
The parameters of the bound states are obtained from the condition 5(C, k) — 0. This equation has the following solution, £0 = —k + irfo, where 770 is determined from
Ibl 1
&
2
[bi 1
&
~
345
Problem Solving in Nonlinear Quantum Mechanics
If ?7o ^ 0, according to (7.35), (7.37) and (7.48), Davydov obtained that co(0 = «e 4 *o'\
(7.51)
where n2P2lrlo Hit
„
Q
2
" g 0 [niIcos(ni0sin(niZ)]'
/^2
2
"i^o"^
Thus, the auxiliary function F(x', t') defined in (7.38) can be written as
F(x',t') = coit'y^'  ~ r dt'RtfJ)e**\
(7.52)
27r7_oo where Co(i') and R(£',t') have been given in (7.51) and (7.50), respectively. The first term in (7.52) represents the soliton solution, while the second term determines the "tail" accompanying the soliton. For a sufficiently long time £', the integral in (7.52) can be approximated by 2TTJ
s
^' '
4v^r75(0,fc)
I
[ 16*'
V 8/JJ
Then, the "tail" decreases with time according to l/y/¥. Keeping only the first term in (7.52) in the long time limit, obtained K(x\y') and the wave function, by solving (7.39), ^
( a r
_ i2>/2tt,exp{2t[fts' 2(fc 2  q g ) f ] } '°" VSoo8h[2»(x'x' 0 4*f)] '
(
}
where 2%
2rjo
Thus, given the initial excitation in the form of a pulse (7.47), soliton solutions exist if the nonlinearity parameter b is greater than the critical value bCT = TT2 /4£. The amplitude of the soliton increases and its width decreases with increasing b. 7.4
Perturbation Theory Based on the Inverse Scattering Transformation for the Nonlinear Schrodinger Equation
In this section, we establish the perturbation theory for the nonlinear Schrodinger equation based on the inverse scattering method. According to the basic idea of the inverse scattering method described in (3.21), we seek a pair of Lax operators L and B and the corresponding scattering data. We will first build the general theoretical framework. The first step in using the inverse scattering method is to solve directly the scattering problem, i.e., to find the eigenfunctions of the spectral equation (3.21)
346
Quantum Mechanics in Nonlinear Systems
(or scattering data). Since the coefficients of L depend on
•• ^
= ilnbn{t).
(7.54)
To solve the Cauchy problem for the nonlinear Schrodinger equation, we first determine the initial scattering data, S(X,t = 0) and Sn(t = 0), corresponding to an initial condition (x, t = 0) (direct scattering problem). Next we find S(X, t) and Sn(t) from (7.54). And finally reconstruct
(7.55)
where P[] represents the perturbation, x' — yp^mfWx and t' = t/h. The following was obtained by Kivshar et al.,
df
 ;_„
dx
mx',t')n
sct>(x',t')rm
= in(X)S(X,t>) + e£y^P[
(7.56)
can be obtained similarly, (7.57)
where
F[
347
Problem Solving in Nonlinear Quantum Mechanics
Following the approach of Zakharov and Shabat discussed in Chapter 4 and (3.22), the operators L and B corresponding to the unperturbed nonlinear Schrodinger equation (7.55) can now be written as
L=
A
 i £  <S>*{x') &
[  4 U 3 + 2A2 + 4>*4>x.  # ; , ~ [ 4»AV + 2iA0,. + t0,» a . + 2i0 2 0*
e
,
(758)
 4 i A V  2i\ij>*x, + i
(7.59) The Jost functions for real spectral parameter A are defined by the boundary conditions *±(z', A) = eiXcr3X> + 0(1),
when x' >• ±oo,
(7.60)
where 03 is the Pauli matrix. This matrix form of the Jost functions can be expressed as \P+ = (tp1,^), iff = (ip,ip'). Here ip' and
transforms it into
"[SI
The monodromy matrix T relates the two fundamental solutions * + and *_, ¥  0 0 ) = ¥+(!', A)T(A). Kivshar, et ai. gave this matrix in the following form [a*(A) 6(A)1
f7gn
where the two Jost coefficients a(A) and b(X) satisfy o"(A)2 + 6(A)2 = 1.
(7.62)
The vector Jost functions ij}'(x',\), ip'(x',X) and the Jost coefficient a(\) are analytic and continuous in the upper half of the complex Aplane. The zeros, An = £„ + ir)n (n = 1,2, • • •), of the functions a(X) in the upper halfplane give the discrete spectrum of the corresponding linear problem (4.41). The Jost functions ip'(x',\n) and ip'(x',Xn) are linearly dependent, cp'(x',\n) = bnip'(x', Xn). They decay exponentially as x' —> co. a(A) and 6(A) with real A constitute the continuousspectrum scattering data, and the set of complex numbers An and bn
348
Quantum Mechanics in Nonlinear Systems
constitutes the discretespectrum scattering data of the corresponding scattering problem. Their time evolution gives the linear equation (4.41) of the operator B in (7.59), b(X, t') = 6(A, 0)e 4 i A V ,
o(A, t') = a(X, 0),
bn(t') = M0)e 4 a »*' •
Xn(t') = AB(0),
(7.63)
The inverse scattering problem for the operator L in (7.58) now reduces to a system of singular integral equations
+(x,\)e
{1)+L[
{XXn)a'(Xn)
2 m J_0O
n*,Xn)e
+
{l)
+
1
'
X'  X + IO
v
;
2^i (KXm)a'(Xm) A00 i?(A)^(x',A)e^'
ssy_00—>r^—dA'
(7 65)

where a'(Xn)=^
, 5A
/?(A)HM.
a(A)
A=Xn
(7.66)
Finally, (a;', i) is given in terms of the scattering data and the solutions (7.64) and (7.65),
tfV.f) = 2 E ^r^i(*',A n )e iA »*' + 1 /°° ^(AJ^Car'.AJc^W. (7.67) n = 1
a
\*n)
Kl
Joo
The reflectionless potentials (f>(x',t'), for which b(X) = 0, are soliton solutions of the unperturbed nonlinear Schrodinger equation. The refectionless scattering data with the single zero Ai = £' + if] of the function a(X) corresponds to the case of one soliton (4.56) or (7.53). When a perturbation is present as in (7.55), in the lowest order approximation, Kivshar et al. obtained the following from the evolution equations (7.56) and (7.57)
^%P
=e /+°°dx'P[^i(x'>A)V,2(x'>A) r+oo
+e* /
J — oo
ds'P'[M>a(x',A)<M*'lA)>
(768)
Problem Solving in Nonlinear Quantum Mechanics
=4iA26 + e
^%Pm
[+0O<&P[4>]MJ,*)
f+OO
e*
J — oo
ri\
1
^ T = ^TTT / at
r+o°
a \<\n) j  o o
dxT'i^ixWteKxW),
= 4
T
(7.69)
dx'{eP[
^
349
(7.70)
/ + 0 °dx'{ePMg 1 (x',A B ) + e *P»Mg 3 (* l > A B )}. (7.71)
Here Qi(x', A,,) = ^ foOc', A n )^(x', A)  ^ ( x ' , A)^(x', An)]A=An , i = 1,2, (7.72) and a'(Xn) has been denned in (7.66). The general evolution equations for the parameters for the one soliton solution of the nonlinear Schrodinger equation (4.56) are (7.73)
(7.74)
(7.75)
(7.76)
where C' = — 4f'i + x{,, 0 = 4(^'2  T?2)* + 0O Applying these formulae, we can obtain the timeevolutions of the scattering data and the corresponding solutions of the equation (7.55) only if the perturbation P[
eP[
(7.77)
350
Quantum Mechanics in Nonlinear Systems
Ott et al., Karpman et al., and Nozaki et al. obtained the evolution equations for the amplitude r\ and velocity v = —4f' of the soliton d
V
_
(
n
=
_27? ^
4
,
+ rir]2
8 +
,\
_a3T]2j
1
o ,
_ _ a2r?2?;2;
*  y W
(778)
In contrast to the unperturbed case, in which the soliton's amplitude 77 and velocity 4£' may take arbitrary values, the stationary values in the presence of this perturbation are uniquely determined from if = 3ai/4(a 2 +2a 3 ) and £' = 0. However, it is clear that the corresponding soliton is unstable, since as a;' > 00,
] . + %[([ 4«p.#\ df +
(P
[fdzj
[fj
=
9{x'+e)f{x'e)
f(x> + e)g{x'  e)
Gxp(eDx,)g
•f
exp(eDx,)f • f
376
Quantum Mechanics in Nonlinear Systems
where t' = t/h, x' — x^j2m/h2,
we find that g and / must satisfy
(iDt, + Dpg • / _ g (PDl,ffbgg*\
A
P
P
_
) ~
Substituting <j> = g/f into (7.179), we have {iDt, + pD2x,)g • f = 0, pDlf
• f  bgg* = 0,
(7.180)
where / and g satisfy (7.2). The oneenvelopesoliton solution of the nonlinear Schrodinger equation is given by g = Ae^e71,
f = 1 + ae"+r1',
(7.181)
with
and the dispersion relation is i(n + iui) + p{K + ik)2 = 0. Here we see that the decoupling of (7.180) does not result in a simple relation between g and / , and the Hirota form (7.179) is a coupled system (7.180) in g and / . To obtain solution of (7.180), we assume that g = e g x + e 3 g 3 + ••• ,
f = 1 + e2f2 + • • • .
(7.182)
We then find that igw + gix'x = 0, Sixx = 2#101>
ig3t + 93x'x' = {iDf +pDl,)gx • f2, Thus these equations may be solved recursively until an inhomogeneity is found to vanish where upon the term defined by that equation and all subsequent ones may be take to be zero. Substituting (7.181) into (j) = g/f, we obtain the onesoliton solution, as given in (7.53) or (4.56). Continuing in this way, the iVsoliton solution of the nonlinear Schrodinger equation can be constructed. For the SineGordon equation, (7.6) in natural unit system, if we require d(f>/dx —> 0 when \x\ —> oo, we can assume that
^•"^"'[TIM]'
(7 I83)

Problem Solving in Nonlinear Quantum Mechanics
377
where N/2
/(*,*) = EE a ( ii ' i3 "' i ») cIhl+IKa+ "' +lha "' n=0 Nn (JV1)/2
9(z, *) =
E
E
m=0
a
N2m+i
^ 2 • • • ,i2m+i)e"'i+1?>»++f''*»+>.
(7.184)
Here
{
(«) 1,
n = O,l
^ ' ^ " ( P i f c + p ^  ^ f c + nw) 2 '
Inserting (7.183) into (7.6), we get, /&*  2/«Sx + fxxg  (fgtt  2ftgt + fug) = fg, fxxf  Vl  ffxx  (fttf  2/f + fftt) = gXx9  2g
2
x
(7.185)
 (gag ~ 2g\ + ggtt).
Substituting (7.184) into (7.185), we can determine a{ik,i[) and t)i2n, r/j2m+1. We can thus finally obtain the solution of (7.183), and find out the iVsoliton solution of (7.6). To obtain the onesoliton solution, we insert (7.183) into (7.6) to get {DlDi)(f.g)=fg, (DlD2t)(ffgg)=0.
(7.186)
Obviously, g = 0 and / = 1 is a solution. Let /
=
g
l
+ e
2 / ( l )
= egW
+ £
4 / ( 2 )
+
...
+ e 3 g W + .
)
(7.187)
Substituting (7.187) into (7.186) and equating the coefficients of the e terms, we get the following linear equation
g£gV
= g(1)
The solution is given by g^ = exp(fcx — ut + S), where k2 = u2  1, or equivalently,
378
Quantum Mechanics in Nonlinear Systems
For the 04field equation (3.5) with A((t>) = 0 in natural unit system, we can assume cj> = g/f, and insert it into (3.5). Tajiri et al. obtained that {D*Dla)gf = 0, (D2D2x)f.f/3\g\2=0, where f = l + ae"+r>\ P
S = e", 1
2(fi + n*)±(p + n*)' Here j3 is a nonlinear coefficient and a is a linear coefficient in this equation, 77 = Px — £lt±r)0, and P2 — fi2 = —a. Its onesoliton solution can be finally written as
~ p r (*  vt) + %r I sech[Pr(a;  vt + 6r)),
where Pr, fl r , and 7jor are real parts of P , Q and 770,
and w2 > 1. Tajiri et al. generalized the Hirota's method to two and threedimensional systems and to solve dynamic equations of the microscopic particle in these systems. For the threedimensional nonlinear Schrodinger equation, i(f>t> + p(j>x'X' + q'
(7.188)
with 6 > 0 (for p, q' > 0 and r' < 0), we introduce the dependent variable transformation, 9
g{x',y',z\t') f(x',y',z',t')'
where / = /*. Substituting it into (7.188), we have f {iDv + pDl + q'D2y, + r'D2z,)g • f = 0, \(pDl, +q'D2yl +r'D2z,)f • f b\g\2 = 0. Here, the bilinear operators are defined by
DZlD1y,DyD?,a(x',y',z',0 • &(*',v\*\0 d_\K fd_ d_\l f_d d_\m (_d__d_Y = f_d ~ \dx'
dx'J
\dy'
•a(x',y',z',t')b(x,y,z,i)\
dy'J \dz'
dz<)
, . , . , . , . \x'=x,y'=y,z'=z,t'=t
\dt'
di')
Problem Solving in Nonlinear Quantum Mechanics
379
The oneenvelopesoliton solution of the 3D nonlinear Schrodinger equation is given by g = Ae^e",
/ = 1 + ae"+7)*
where 9 = kx' + ly' + mz'  ut', r] = Kx' + Ly' + Mz'  nt1  r?°, bA2 2 ° ~ 2 \p(K' + K*) + q'{L + L*)2 + r'{M + M*)2]' and the dispersion relation is given as follows i(fi + iw) + p(K + ik)2 + q'(L + ilf + r'(M + im)2 = 0. 7.9
Backlund Transformation Method
Applying the Backlund transformation (BT), we can obtain new soliton solutions from a given solution of the nonlinearly dynamic equations in nonlinear quantum mechanics. However, there are many different types of Backlund transformations in nonlinear quantum systems as described in Section 3.3. Therefore, it is important to choose an appropriate Backlund transformation in order to obtain the solutions of a given equation. 7.9.1
AutoBacklund
transformation
method
The autoBacklund transformation method was studied by Rao and Rangwala, and Rogers et al. For a 2 x 2 eigenvalue problem
(7.189)
A = H, with *=(*\
Y=(
X
q{ X
 ^\
f_(A{x,t,X)
B'(x,t,X)\
(7.190)
where ipi and «/>2 are eigenfunctions, while A is an eigenvalue, the compatibility condition ipxt = iptx yields Ytfx = [Y, f] = YT TY.
(7.191)
Equation (7.191) results in the following set of conditions on A, B' and C, Ax=qCrB',
B'x  2XB' = qt  2Aq,
Cx + 2XC = rt + 2Ar.
(7.192)
380
Quantum Mechanics in Nonlinear Systems
If we set
r
\
to)
then the Lax system in (3.20)  (3.22), where B and L are linear operators subject to the compatibility condition Lt + [L, B] = 0, can result in the AblowitzKaupNewellSegur (AKNS) system. Specializations of A, B' and C in the AKNS system lead to nonlinear evolution equations amenable to the inverse scattering method for the solution of privileged initial value problems. For the SineGordon equation (7.6), there are r — q = <j)x/2 and A=lcos
B' = C =^r sin <j).
For the nonlinear Schrodinger equation (4.40) with y/2m/h2x there are r = — q* = <j>*, and A = 2i\2 +i\4>\2,
B' = i<j>x, + 2i\
(7.193) t x', t/H >• t',
C = i
(7.194)
AutoBacklund transformations for the above equations are constructed via the AKNS formalism. The key step in the derivation involves the introduction of F = * i / * 2 , so that the AKNS system (7.189)  (7.194) results in a pair of Riccati equations rx = 2\r + qrr2,
(7.195)
2
rt = 2^r + s'cr . If r = —q in the SineGordon equation, the AKNS system yields ipxx  A V = H>,
(7196)
where tfi = tpi + iip2 arid <)> = — iqx — q2.
The system (7.196) is invariant under the Crumtype transformation rlt = \
r' = \ ,
t
<j>' = ^ + 2(lnrp')xx,
^ ' = ^+2(lnV»*')x«.
Whence
^—« ta (i). + .«.(fe),
(7.197)
381
Problem Solving in Nonlinear Quantum Mechanics
where w and w' are potentials given by q = wx, q' — w'x. Thus, under the transformation (7.197),
T = cot I ^ (w'  to) j .
(7.198)
Substitution of (7.198) in the Riccati equations (7.195) provides a generic Backlund transformation for the present subclass of the AKNS system with r = —q, namely wtw't = 2Asin(w'  w)  (B' + C) cos(u/  w) + B'  C, wx+wx=2\sin(ww'),
x = x,
(7.199)
i=t.
If we set a' = A/2, w = —0/2, substitution of the specialization (7.193) in (7.199) produces the following autoBacklund transformation for the SineGordon equation (7.6) 4fx = 4>x 2a' sin ( ^ p  ) =  B T i O M ^ ' ) ,
0t =  &> ^')'
( 7  20 °)
where x = x', t = t', and a' is a nonzero Backlund parameter. The invariance can be easily proved. In fact, applying the integrability condition
dBTi dt
8BT2 dx ~ '
we can get <j>xt = sin cj>. However, if we write (7.200) in the following form,
4>x = 4>'x ~ 2a'sin ( ^ ) = 2?ri(0\&,0), 4>t = ft + 1 sin {^Y^J = BT^'A't,
dBTj dt
dBT± dx ~ '
we can also get
^ =  ; sin (  ) ,
which yield a soliton solution (6.14), i.e.,
A
A.  i /
[, xxovt]\
(£ = 4tan Wexp ± —
L
L
Ia12 V, « =
V l  v2 J J
,2
1+ "
382
Quantum Mechanics in Nonlinear Systems
For the nonlinear Schrodinger equation with r = — q*, generic autoBacklund transformations were generated by Konno and Wadati via I" and q' which leave the pair of Riccati equations (7.195) invariant. In fact, (7.194) can result in the autoBacklund transformation + l\2, x = x,
<j>x + <S>'x = {4><j>'W^\
(7.201)
i = t,
& + # = »(*.  0 ' X ) V 4 A 2   0 + 0 T + ±{4> +
(7202)
(02  0i 2 )\/4A 2 [02 + 01212 ~ ( 0 i  0 i 2 ) ^ 4 A 2   0 1 + cM 2 = 0, where & = 5A0, (i = 1,2), 4>12 = BXlBX2
Backlund transform of Hirota
The Backlund transformation provides a mean of constructing new solution from known solutions of the dynamic equations. In the context of bilinear equations, the Backlund transformation method was introduced by Hirota to solve nonlinear dynamic equations in nonlinear quantum mechanics based on the D operator in Section 7.8. For the nonlinear Schrodinger equation (7.179) and /2m
,
, •
t
Hirota assumed that n = [(iDt, + D2X,) (g • /)] /' 2  f [{iDt, + Dl) (g' • /')] ,
(7.203)
which vanishes provided that (g, f) and (g1, /') satisfy the first of (7.180). By means of a pair of identities he rewrote (7.203) as
n = iDf,(gf' + fg')ff
 (gf + fg')iDt,f • f'+
22?s.[Z?s(ff • / ' + / • 9')} • / / ' + (5 • / '  / • 9')Dlf
f'DUgf'f
\ [(gf1 + fg1) (gsT  g*g')  (gf  fsf)(gsT + g*g')},
gf) • / / ' + (7204)
where it is also assumed that (g, f) and (g1, /') satisfy the second equation in (7.180). Considering the relations between successive "rungs of the soliton ladder", Hirota obtained Di.(gf
+ fg') =
n(gff91)
383
Problem Solving in Nonlinear Quantum Mechanics
Thus (7.204) decouples to give iDt,(9 f' + f  g 1 )  (D% + X)(g •f'fg')
= 0,
2
+ D\,  (2/z  A)] / • / ' = gg'*.
[iDt, + itiDv
(7.205)
Equation (7.205) constitutes a Backlund transformation between (g, / ) and ((x, t) is given. In such a case we must choose \i = KN — K^,, A = 2{Kjf + K$) + (K% — KH)2, where KN is a constant introduced by integration of the first of (7.182). Applying the Hirota's transformation,
• f = \{f2  p 2 ),
DtDxg • g = \{g2  ?).
The Backlund transformation of the SineGordon equation is Dxf • f = jg' • g,
Dtfg
= ±g'f,
(7.206)
as well as their complex conjugates, where if is a real constant. In order to obtain a superposition formula, we rewrite (7.206) as Dtfigo =  j j ^ S i • /o, Dtf2gi = ^g2fi,
Dtfi • go = ^gifo, Dthgi = ^g2f\,
(7.207)
where f\ and g\ are onesoliton solution with parameter K%, K\ and Ki are the corresponding parameters of soliton solutions. Using the properties of the D operator, Dl(g
• f)hq  gf(D2xh
• q) = Dx{(Dxg
•q)hf
+ gq. (Dxh • / ) } ,
Hirota obtained the following superposition formula for the soliton solution of the SineGordon equation, V/o
=
*2(d//i)gi(gi//i)
(? 2 m
Inserting <\> = 2i \og{g/f) into (7.208) we obtain, e*(**o)/a =
K2K^**l* Kx+Kie1^^)/2'
which gives
tan ^(fo  *,)] = f ^ ^ tan ^(0!  ^)] .
(7.209)
384
Quantum Mechanics in Nonlinear Systems
This is the same as (3.44) with a.\ = K\, ai = K2, i.e., it is the superposition formula of the SineGordon equation. Applying (7.206), or (7.208), or (7.209) we can get a new solution of the SineGordon equation from a known solution. 7.10
Method of Separation of Variables
Lamb proposed a method of separation of the variables to solve nonlinearly dynamic equations. For the SineGordon equation given in (6.13) in natural unit system, Lamb expressed its solution in the form tanfi#M) = g(x)F(t).
(7.210)
Inserting (7.210) into (6.13) results in two first order differential equations as follows gl{x) = C'g\x)+lg2(x) 4
+ p', 2
F?(t) = p'F (t) + {£ l)F (t) + C",
(7.211)
where C", /?' and £ are integral constants. When C" = /?' — 0, equation (7.211) become gx = nyflg,
Ft = nV£  IF,
n = ±1.
The solutions of this set of equations can be easily obtained. Equation (7.210) can now be written as
tan [ ^ ( M )  = 3?exp ln>Tt \x  x0  y'l~ 1 fl  .
(7.212)
Thus there are three kinds of solutions corresponding t o £ > l , 0 < £ < l and I < 0 in (7.212) respectively. (1) In the case of £ > 1, we assume that y/l  l/£ = v, •/£ = 7 = 1/Vl  v2 with 0 < v2 < 1. In such a case, equation (7.212) is replaced by
(7.213)
This is consistent with results given in (7.22). (2) In the case of 0 < £ < 1, we assume that \J£  l/£ = iu, \f£ = 1/Vl + w2 with 0 < u2 < 1. Then (7.212) becomes
"
This represents a stable solitary wave with frequency ui/VT+u2. (3) In the case of £ < 0, assuming yj{£  l)/£ = v, 7 = 1/Vi>2  1, we get 4>{x,t) = 4tan" 1 {cosftOc  vt)]} .
385
Problem Solving in Nonlinear Quantum Mechanics
This solution depicts a periodic wave with a velocity v > 1 and wavelength A = 2TT/7 = 2ny/v2 — 1.
When C" = 0, fi = 1, (7.211) becomes g2x = £g2 + l,
F? = F4 + (£l)F2.
(7.214)
In the case of £ > 1, there are two types of solutions. The first one is given by g(x) = sinh(7z)/v^,
F(t) = s/£lcsch(jvt).
Equation (7.210) can now be written as (7.215)
This solution describes two interacting kinks moving with velocities v and —v, respectively, as described in Section 6.2. Equation (7.214) also has the solution g(x)  iVtcoshes),
F(t)  iV£  1 sech(^vt).
Thus (7.216)
It describes the collision between a kink and an antikink in which they pass through each other, as described in Section 6.2. When C = 0 and /?' = 1 and 0 < £ < 1, equation (7.214) has the following solutions g(x) = i cosh(v^a;)/v^,
F(t) = n / l  £sech(Vl  It).
Thus ,, . . _! [ sin(wt/y/l + w2) 1 /l^ / O1 x v y (x,t) —¥ 0 as x > oo. It has an internal frequency w — w/y/l + w2, spread over a region which is inversely proportional to 1/Vl + w2. The corresponding traveling breather can be obtained by the Lorentz transformation t —> j(t — x/v),
x —> 7(0; — vi),
386
Quantum Mechanics in Nonlinear Systems
where 7 = l / \ / l  v2. The solution can be written as
*(*,t) = 4 tan"1 (tan^sinyx/.) T cos^] ^
cosh [7(x  vt)} smfi'
(7.218)
j
with ^i' = ta,n~1(l/w). The frequency of the pulsation in the moving breather is u/ = 7cos/x' = 710'/Vl + w2. According to dx [<j)l + cj>2 + 2 ( 1  c o s
E = \ f
a n dP =  [
* J00
J00
dx<j>t(j>x,
we can obtain the energy and momentum of the moving breather _ 16 sin u! Ebr= y—'i, VI — v* We then obtain the relation
, „ \&vs\nu! and Pbr = ftm VI — vl
(7.219)
El = Pi + Ml, where Mbr = 16 sin/x' is the mass of a motionless breather. Since the total energy of a free kink or an antikink is equal to 167, the binding energy of the breather is EB = 167(1  s i n fi'). In the case of C" ^ 0 and /?' ^ 0, Lamb and Davydov assumed that
a' = v W ,
9{x)F{t) = g(x)F{t).
Equation (7.211) then becomes g2x = a'g* + lg2 + a1,
F? = oclFi + (£ 1)F2 + a1.
When a' > 0, we can denote
gl=a'(g20g2)(g2gl), where
Thus ^(ar) = 50 cn(u, K),
F(t) = Fo en  ^ 2 * , j ,
Problem Solving in Nonlinear Quantum Mechanics
387
where u = aK
l
go{xXo),
^
______
=
^o = ^ 7 ^  l + V^l) 2 +4a'}, j2
_ i  1 + y/(£  1)2 + 4a' 2^/(1  I) 2 + 4a'
The solution of the SineGordon equation in such a case is
t_ [*(£!_] = <j0F0 en [*^(«  z0),if] en [ ^ t , j \ .
(7.220)
This solution represents "plasma vibrations" of standing periodic waves. If /?' in (7.210) is replaced by /?' and C" 7^ 0, /3' < 0, ^" = y/pO > 0, in accordance with the above method, Lamb and Davydov obtained solution of the SineGordon equation tan i,/>(z,t) I = 30^0 dn(u, K) sn(V, J'), with u=~goyrPJ(xxo), (_ji\2 _
x
<
V= v i,1
<;
^Fot, ^p
~ lz+VCi^)2^1'' 2 _ l  z + y / a  Q g  ^ " . 2 _ t + y/py ' °~ 4/9" 5o 2/?" This solution describes breather oscillations. 7.11
Solving HigherDimensional Equations by Reduction
In the previous section, we solved dynamic equations in onedimensional systems. In this section, we discuss methods for solving higherdimensional, for example, two and threedimensional, dynamic equations in the nonlinear quantum mechanics. We introduce a method of reduction from higher to lower dimensional equations proposed by Hirota and Tajiri. As a matter of fact, for the twodimensional case, the solutions for some similarity variables have been studied by Nakamura et al. They obtained an explosiondecay mode solution by generalizing the similarity type plane wave solution derived by Redekopp. Tajiri investigated the similarity solutions of one and twodimensional cases using Lie's method.
388
Quantum Mechanics in Nonlinear Systems
(I) For the twodimensional nonlinear Schrodinger equation with r' = 0 in (7.188), i.e., i
(7.221)
where p, q, b are constants, Tajiri considered an infinitesimal oneparameter (e) Lie's group in the (x',y',t',
(7.222)
2
i = t' + eT(x',y',t',
fc = 4,t + e[Ut] + O(e2), k* = 4>X'X + e[Ux>x>] + O(e2), 4>vv =
(7.223)
where ,TT ,
[Ut>] =
8U
+
fdU
dT\
8Y
Pt
w {di~w)' '
dX
dT o
x
dY
dX
~ W^'" ~d**tliPx''
W**  ~dv* '" W
Similarly rrr
.
d2U
/
82U
d2X\
d2Y
d 2T
[U*'x>] = ^ + ( ^ 2 ^  ^  — j
 — d2r
w
{W~
d^>) **'" *!*€?*** ~ 2d*d4>ipx"pt'
d2x v
d2Y
3
W " ~W* dY
2 xl
dT
dT
d2r
fdu 2 Vx Pv +
' ~W " odX
dY
 *QT
•QVVVVX'X'
\w~ dY
ndx\
w)*x'x'
dT
2,—lfx'Vx't',
where ,
t
.
12m
.
l2m
Assuming that the 2Dnonlinear Schrodinger equation (7.221) is invariant under the transformations (7.222) and (7.223), we get by comparing terms in first order
389
Problem Solving in Nonlinear Quantum Mechanics of £, i[Ur] + p[Ux>x>] + q'[Uylyl} + b{2\<j>\2U + <j>2U*) = 0.
(7.224)
The solutions of (7.224) gives the infinitesimal elements (X, Y, T, U) while leaving invariant (7.221). For (7.224), Tajiri obtained the following for the infinitesimal elements
X = ax' + kt'x' + 4y # + PJt' + 0i, Y = ay' + kt'y'  ^x' + pSt' + 03,
(7.225)
Q
T = 2at' + kt'2 + 03, {px'2  ^y'2)
U=[iUan(t
+ i 7 z' + i Sy'} 0,
where a, /3, 7, S, K, 9I, 82, 03, and u> are arbitrary constants. Thus, the similarity variables and form are given by solving the characteristic equations, dx' _ dy' _ dt' _ d<j>
{7 Zb)
T ~ Y ~Y ~ u•

The general solution of (7.226) involves three constants, two of them become new independent variables and the third constant plays the role of a new dependent variable. Note that different similarity variables and form can be obtained by integration of (7.226) for different choices for values of the constants a, 0, 7, S, K, 01,02,03 and win (7.225). Case (1), a, 7, S, K, 0 I , 02, 03 and w ^ 0 and /? = 0. From the integrals of dx'/X = dt'/T, dy'JY = dy/T and dt'jT = d<j>/U, Tajiri obtained the new independent and dependent variables as follows ._
(0i«p7a)t + {61apyd3)
x'
V\0W\
(«*3aViGi
V
_ (02*  q'Sa)t + (02a  q'693)
V\QW\
(K03a2)y/)Q\
'
(7.227)
and + = JL=e""<*'»''''>«;(&i 7 ) >
Q{t') = Kt'2 + 2at' + 03,
(7.228)
390
Quantum Mechanics in Nonlinear Systems
where
+
If
1
\f ibryt' + Oi) + 6(q'6t' + 02) 1
2/ T i l [ / o7W\ dt\dt 2 +± [ (f ^ + £ ^ V dt>+^ [(f tEpiA *pJ \J QV\Q\ J
dt.
WJ \J Qy/\o\ )
+£[([ n
Kt' + afe
4 {j
, V2\\
+
7)l'
and in the above, ax is a step function in Q, i.e., ai
f 1 ~ \ 1
for Q > 0, for Q < 0.
Inserting (7.228) into (7.221), we have pwu + q'wm + b\w\2w  IA ($ + ?L\ + B\ W = 0,
(7.229)
with A=J(K93OJ),
B
" {[» i s s b i ) ] [ f + f + ^ + " " *  2O<791+Ms) ]} •
Following the same procedure as in case (1), we can get the similarity variables and forms for other cases. The results are summarized in the following. Case (2) 9 i / 0 and 82 ^ 0, the similarity variables and forms are £ = x' — 6y', T — t\
(7.230)
Case (3) S ^ 0 (or 7 ^ 0), the similarity variables and forms are £ = x' (or y'), T = t', <j> = expfay2/(4g't')]w(£,T) Or exp[ix'2/(4pt')]w($,T). The differential equation is iwT + —w + pwtf + b\w\2w = 0, AT
or iwT + — w + q'wtf + b\w\2w = 0. /iT
391
Problem Solving in Nonlinear Quantum Mechanics
Case (2) above is the reduction transformations for essentially onedimensional propagation and symmetry. The reduction of case (3) was first found by Nakamura. As an example of case (1), we give the reductions of the 2Dnonlinear Schrodinger equation to the nonlinear KleinGordon equation or the <£4equation in the following. If K = 1, w ^ O , a = £ = 7 = (5 = 0 i = 0 2 = 03 = O, £ = x'/t', IJ = y'/t', then
+
P) ]«•'>•
*=^K$ $
(7 23i)

and pwtf + q'w^ LJW + b\w\2w  0. But if w = 1, 0i ^ 0, 02 ^ 0, 0 3 ^ 0, a = /3 = 7 = <5 = K = 0, £ = a:'  (0i/0 2 )*', f = 2/'  (02/*3)f, then
and p w ^ + 4'wqq — mw + b\w\2w = 0, where 1 03
0 4g'0
6{ 4p0f
We note that the above equations are the nonlinear KleinGordon equation or the 0 4 equation when pq' < 0. The solutions of the IDnonlinear Schrodinger equation and nonlinear KleinGordon equation can also be transformed into the solutions of the 2Dnonlinear Schrodinger equation through the similarity transformations. Substituting the soliton solutions of the IDnonlinear Schrodinger equation and the nonlinear KleinGordon equation into the transformations, we get the well known soliton solutions extended in the (x,y) plane. The solutions obtained by substituting the soliton solutions of the nonlinear KleinGordon equation into the transformation (7.231) are the selfsimilar solitonlike solutions,
^4 e x p K^ + ^^)] x \ ~r sech. / x —  c\ —V b V p + gcj \f t' fu / —w (x1 WTtanhJ c Vo y 2 (p + g'cj) \ t' which was first obtained by Nakamura.
(7232)

— Co su
, /
v' \ » _^0 t' /
for —^ > 0, p + q'c\ p+ ui for < 0, p + q'c{ p+
—=• > 0, q'c\ r 2 > 0, q'cf
392
Quantum Mechanics in Nonlinear Systems
(II) Tajiri considered further the solutions of the 3Dnonlinear Schrodinger equation (7.188) using the same reduction method. Assuming that the coefficients p, q' and r' are not all the same sign, for example, p, q' > 0 and r' < 0, and using the transformation,
X = j=(x'elt'),
Y = j=(y'02t),
Z==(z'93f),
z' = zyj^,
<j> = exp {i \^{x'  Otf) + ^,{y' W) + ^z'~ W) + 9^\ } W y ' Z )> (7.233) Tajiri obtained Uzz  Uxx  UYY + CU b\U\2U = 0,
(7.234)
which is formally the 2Dnonlinear KleinGordon equation, where 6\, 82, 63, and 6^ are arbitrary constants, and 4p
4g' 4r'
Making transformation once more
a ^
a ax/g7
a^rq'
y^/p
ay/q1
ay/rq' J
(7.235)
U = exp {i [ ± ^ 4 (^  M + \ (a + £) ^1} G^' we get
iGT ± f ^/^T^G« ±  ^ L =  G  2 G = 0,
(7.236)
which is formally the IDnonlinear Schrodinger equation, where a and a are arbitrary constants. It is well known that the IDnonlinear Schrodinger equation can be reduced to the following equation ^
= 2F3 + CF,
(7.237)
393
Problem Solving in Nonlinear Quantum Mechanics
by the similarity transformation. Using results in (7.233)  (7.235), equation (7.188) with r < 0 can be reduced to (7.237) by the following transformation,
'•(^^•^(Va)']} 2 ^' f[l
1 ,
,
y/l + a2 ,
6>2
(Oi
\ZlTo7. \ ,] 2 ]
,_( *2_,)1/>(.?a^)I/1^{
V
o
I L2P
/
2
9
(7.238)
1/
X
cw'g 3 t'
V y/P
p (x'erf
a
^4 ^
av^
a(y'e2t')\
7
)
where /9 is an arbitrary constant. We note here that (7.236) with 6 > 0 is transformed into the equation with b < 0 by the substitution r > —r and £ > i£. The 2Dnonlinear Schrodinger equation (7.221) can be easily reduced to the IDnonlinear Schrodinger equation by the similarity transformation Z = x'9y',
r = t',
x>x> + ifJ'\4>\2(t>  "'V The solitonlike solution was found to be
4>{x',t) =
A{x'yv^iut'\
where A(x') and 0(x') are some unknown functions which are determined by the original equation. It was found that in general the microscopic particles in such a case are unstable except in a few special cases and that there are some regions in the parameter space where soliton solutions are stable. Various profiles of microscopic particles described by the following nonlinear Schrodinger equation,
i
(8187)
were studied numerically by Moussa et al.. They found that solitonlike solution of this equation is of the form
453
Microscopic Particles in Different Nonlinear Systems
where rf = ^ [ 5 ( 2 £  P) ± \/25(2e/0 2 480cd)], A and 0' are arbitrary constants. The results reveal various shapes of the solitary solutions in such a case.
8.12 Interaction of Microscopic Particles and Its Radiation Effect in Perturbed Systems with Different Dispersions We first consider collision between microscopic particles in a perturbed system which was investigated by Malomed. The microscopic particles are described by the following nonlinear Schrodinger equation with a nonlinear damping i f c +
(8.188)
where R{4>) is a perturbation, e is real small number. Malomed considered R(4>) = (\
(8189)
and
m
= iP r midx, •K
Joo
X'
X
(8.190)
Here P stands for the principle value of the integral. R{(j>) given in (8.189) specifies the dispersion effect, and that in (8.190) describes a nonlinear Landau damping effect. Obviously, when e = 0, the soliton solution of (8.188) is given by (8.19). When e ^ 0 the microscopic particle experiences an acceleration induced by the nonlinear damping. In the lowest order approximation, the equations of motion for the amplitude 77 and velocity v of the microscopic particle are r/t = 0, and vt = S'er]n, with 5' = 128/13 and n = 4 in the case of (8.189), or 6' «a 7.443 and n = 3 in the case of (8.190). The accelerated motion of the microscopic particle can generate radiation and selfdamping. However, these effects are very weak, and depend on the amplitude of the microscopic particle. Consider the collision between two microscopic particles and collisioninduced changes in their amplitudes. Such changes are useful in understanding of energy transfer from the slower microscopic particle to the faster one. As it is known, the sum of amplitudes is conserved because the number of quanta bound in the soliton is Nso\ = 4r) which is conserved for the soliton (8.19). We now examine the changes in the amplitude and velocity of the microscopic particle in the system described by (8.188), using the perturbation theory. Assuming that the relative velocity v of the colliding microscopic particles (solitons) is large compared to their amplitude, Malomed considered the lowest order
454
Quantum Mechanics in Nonlinear Systems
approximation and expressed the full wave field during the collision as a linear superposition of the unperturbed solutions (8.19), # M ) = (&oi)i + (<M 2 .
(8.191)
Inserting (8.191) into the righthand side of (8.188), both (8.189) and (8.190) become polynomials in {<j>so\)i and (soi)2 The number of quanta is conserved in the system described by (8.188), i.e., rOO
\<j>\2dx' = constant.
N = J00
It can be shown that d
d
r+o°
(TVso,)! = T. / a i
" ' J00
(
r+00
dx!4>itf)Pi(x') + c.c,
(8.192)
J00
where Pi =e(&oi)i(&oi)2[(&,i)2]*'
(8.193)
Further analysis shows that evolution of the first soliton is dominated by the term of the polynomial given in (8.193) in the case of (8.189). Indeed, inserting the polynomial (8.193) into (8.192), one notes that the dominating term must contain one power of (0soi)2 and one power of (<^>gOi)2 (or their derivatives), lest the integral on the righthand side of (8.192) will be exponentially small (the integrand will contain a rapidly oscillating exponent), and that either (0Soi)2 or (<^>*ol)2 must be expressed by its derivative, which gives an additional large multiplier w v. Using r) = TVsoi/4 and (8.19), (8.193) and (8.192) in the following I
J(1)??i=
/+OO
iL^
J
(iVsoi)i
'
we get <5(1)f?i,2 = 4«7i??2Sgn(i>1,2  v2,i).
(8.194)
The superscript (1) in (8.194) indicates result obtained in the first order of e. Evidently, (8.194) satisfies the conservation law S^rji + 5^r)2  0. For (8.190), the result is similar SWVl,2 = 80JiffeK2  W2.1)"1'
( 8  195 )
Malomed considered an ensemble of solitons with different initial amplitudes and velocities. Solitons with the largest initial amplitude will acquire the largest velocity according to vt< = 8'erf1, and from (8.194) and (8.195), their amplitudes will be further increased due to collisions with the slower solitons. Malomed estimated that if the mean distance between the microscopic particles (solitons) is L and a characteristic value of the initial amplitudes is 770, then the time T during which the
Microscopic Particles in Different Nonlinear Systems
455
microscopic particles (solitons) will separate into "large" and "small" ones is about for (8.189) and T » L/(en0) for (8.190). T « y/Lhl/t Using perturbed theory and numerical method, Malomed also investigated collision of microscopic particles described by (8.188). The collisioninduced changes in the amplitudes of both microscopic particles and their radiation losses in the collision process were obtained for the case of relative velocity of the microscopic particles being much larger than their amplitudes. The collision between two microscopic particles of amplitudes 771 and rj2 and the relative velocity v(v2 ^ vh V2) results in a radiation loss which depends on the changes S^7]n (n = 1,2) in the amplitudes of the colliding microscopic particles, i.e., S{2)Vi,2={
f 4^1,2^,1, 16s2 2
I
2~I?l,2l?2 1.
for . f
(8.189), ,fi1Qn.
° r (8190)
(8.196)
However, the changes in the amplitudes of the microscopic particles produced by the adiabatic (nonradiative) exchange should also be considered, which are first order effects in e and are given by (8.194) and (8.195). Therefore, total changes in the amplitudes of the microscopic particles should be the sum of the above two effects. Contribution of the collision to the change in the velocities of the solitons can be calculated. However, this effect, unlike the energy exchange and the radiative losses, is not of principal importance since the velocities of the solitons change continuously between collisions. Malomed also investigated radiative losses in collision processes of microscopic particles obeying the following nonlinear Schrodinger equation, with highorder (third order) dispersions i
+ ie2\4>\2
(8.197)
where e\, £2 and £3 are coefficients of the thirdorder linear and nonlinear dispersions, respectively, which are some real and small perturbation effects. Malomed calculated the radiative losses using perturbation theory based on the inverse scattering method. In his treatment of the collision process, he also assumed that the relative velocity of the two microscopic particles is much larger than their amplitudes, i.e., v = \v\ — i>2 ^$> T}i,ri2. Hence, what we discussed in the above is still applicable here. In the lowest order approximation, Malomed expressed the solution of (8.197) in the form of
(8.198)
71—1
where the correction 8
456
Quantum Mechanics in Nonlinear Systems
It is easily seen that the lowestorder approximate solution to (8.199) is given by keeping only the first term on either side of the equation, and the fully modified solitonic wave form can be expressed as
*% =4$+64$ =*%*»,
(8.200)
where v — vi — v2 is the relative velocity of the colliding microscopic particles, and 8 =*^ta,nh[2r)2(x'
 vt')].
The collision results in the following change in the phase of the first microscopic particle A0i = 6{x'  vt' = co)  0(x'  vt' = +co) = 16—.
(8.201)
As discussed in Chapter 4, the expression of the collisioninduced phase shift in the unperturbed nonlinear Schrodinger equation is (8.202)
Thus (8.201) is nothing but the lowestorder term in the expansion of (8.202) in powers of IJ" 1 . Furthermore, the collision between the microscopic particles gives rise to a shift in the center of mass of the first soliton, and a corresponding expansion which starts from the term v~2. According to the inverse scattering method, the radiation part of the wave field is described by the spectral amplitudes (reflection coefficient) B(X) and the spectral parameter 2A (radiation wave number). The basic ingredient of the perturbation theory is the general perturbationinduced evolution equation for B(X), which is given by
^
= a(A)e4i*V f~dx' {[^>{X,x')fP*(x')+ [^2>{\,x')fP(x')),
(8.203)
where P(z') denotes an emissiongenerating perturbing term on the righthand side of (8.197), and ^ l i 2 )(A, x') are components of the Jost function, which are related to (j){x') by the ZakharovShabat equations (4.42) or (3.22). The quiescent (v  0) onesoliton solution is given by (8.19). The transmission coefficient in (8.203) is a(A) = (A — ir))/(\ + ir)), and
^ ( A , * ' ) = ^  [ A + ^tanh(2^')], ^ ( A , x') = ^eiXx'+4^t'sech(2r1x').
(8.204)
Microscopic Particles in Different Nonlinear Systems
457
Since it is assumed that \v\ — D2I ^ ^1,^2 in the lowest order approximation, a correction corresponding to the modified soli ton wave form given by (8.200) can be found. Substituting (8.200) into the ZakharovShabat equation (4.42), the correction to ip(A, x1) can be found to be
8i> = id I ? 2) j .
(8.205)
Inserting (8.200), (8.204) and (8.205) into (8.203) with the perturbing term iei
/,N
Bfin(A) =
f°° JtdB(X)
L ^M
=
l
.167^772A 17T?2 + 3A 2
15,
^+A»
,_ / nX \
S6Ch
(M) •
(8 2 6)
 °
Equation (8.197) (with ei = €3 = 0) conserves the number of quanta, the momentum and the energy, r+00
N= /
J—00 r+00
<&'#:,,
p=i j—00
Thus the spectral densities of the radiation parts of these three conserved quantities are N'(X) = ^B f i n (A) 2 , P'(A) =  ^  B f i n ( A )  2 , 4A2
—\BRn(\)\2,
e'M =
so that the net number of quanta, momentum, and energy carried by the radiation are given by /"+00
iV rad = /
dAJV'(A),
J—00 r+00
Prad = /
d\P'{\), />+oo
EraA = / dXE'(X). J—oo
(8.207)
458
Quantum Mechanics in Nonlinear Systems
The same quantities for the unperturbed soliton are Psol=2r}v,
NSOI=4T],
Esoi =
—ri3+r)v2.
Substituting (8.206) into (8.207) and integrating the spectral density, the total number of quanta emitted by the first microscopic particle (soliton) can be obtained and is given by
„(!) _ fienrhmeiV
JV
rad  1
^
I
, »?W)
,
.
(H.ZUii)
where T
f+o°
, rz(17 + 3^ 2 )l 2
^firzs
Using the conservation of the net number of quanta and (8.208), the collisioninduced change in the amplitude of the microscopic particle (soliton) can be obtained as 8m = \N&.
(8209)
For the second soliton, changes in the amplitudes of the microscopic particles are given by (8.208) and (8.209) with 771 and 772 replaced by their transposes. Using the momentum and energy conservations we can find the collisioninduced changes in the velocities of the microscopic particles through the balance equations for the momentum and energy in the centerofmass reference frame defined by rjivi + Tj2V2 = 0. In fact, from (8.206) and the assumption of \v\ — t;2 > 171,772, o n e c a n conclude that the spectral wave packets emitted by the two microscopic particles are centered at Ai = ui/4 and A2 = t^/4, respectively, and the widths of the packets are given by SXn « 77,, (n = 1,2). According to (8.207), this implies that, in the firstorder approximation, the emitted radiation carries zero momentum, and ST]I/ST]2 = 771/772 in such a case. Based on these, Malomed obtained changes in the velocities of the particles 6v\ and Sv2 as 771^1+772^2 = 0.
(8.210)
Substituting (8.207), (8.210) and 771^1 = 772K2 into the energybalance equation, Malomed determined Svi, viSvi = —^(771 + 772  771772)^771. For the effects of the other two perturbing terms involving €2 and 63 in (8.197), on the emission of radiation and on the amplitudes and velocities of the solitons, a tedious calculation cannot be avoided. As mentioned above, the particular cases of ei = 0, €2 = 2t3 and £3 = 0, £2 = 6ei can be integrated exactly. The solitonsoliton collisions are elastic in these cases. Thus one may infer that the contributions to
Microscopic Particles in Different Nonlinear Systems
459
B(X) due to the second and third terms, respectively, in (8.197), or the first [see (8.206)] and second, may cancel each other. If this is true, all the above results are applicable to the general case, (8.197), if the parameter ex in (8.206) and (8.208) is replaced by 1
1
Ceff = Cl  7 e 2 + o€3' D 6
Besley et al. used multiscale perturbation theory in conjunction with the inverse scattering method to study the interaction of many microscopic particles (solitons) obeying the following nonlinear Schrodinger equation which has a small correction to the nonlinear potential, i