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Stephen J. Gustafson Israel Michael Sigal
Mathematical Concepts of Quantum Mechanics
Springer
Stephen J. Gustafson University of British Columbia Department of Mathematics Vancouver, BC V6T 1Z2 Canada email:
[email protected] Israel Michael Sigal University of Toronto Department of Mathematics Toronto, ON MSG 3C3 Canada email:
[email protected]math.toronto.edu
CataloginginPublication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): 81S, 47A, 46Nso
ISBN 3540441603 SpringerVerlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from SpringerVerlag. Violations are liable for prosecution under the German Copyright Law. SpringerVerlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de ©
SpringerVerlag Berlin Heidelberg 2003
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: design & production GmbH, Heidelberg Typeset by the authors using a Springer ID'EX macro package Printed on acidfree paper 46/3142db 5 43 2 1 0
Preface
The first fifteen chapters of these lectures (omitting four to six chapters each year) cover a one term course taken by a mixed group of senior undergraduate and junior graduate students specializing either in mathematics or physics. Typically, the mathematics students have some background in advanced analysis, while the physics students have had introductory quantum mechanics. To satisfy such a disparate audience, we decided to select material which is interesting from the viewpoint of modern theoretical physics, and which illustrates an interplay of ideas from various fields of mathematics such as operator theory, probability, differential equations, and differential geometry. Given our time constraint, we have often pursued mathematical content at the expense of rigor. However, wherever we have sacrificed the latter, we have tried to explain whether the result is an established fact, or, mathematically speaking, a conjecture, and in the former case, how a given argument can be made rigorous. The present book retains these features. Prerequisites for this book are introductory real analysis (notions of vector space, scalar product, norm, convergence, Fourier transform) and complex analysis, the theory of Lebesgue integration, and elementary differential equations. These topics are typically covered by the third year in mathematics departments. The first and third topics are also familiar to physics undergraduates. Those unfamiliar with Lebesgue integration can think about Lebesgue integrals as if they were Riemann integrals. This said, the pace of the book is not a leisurely one and requires, at least for beginners, some amount of work. Even in dealing with mathematics students we have found it useful, if not necessary, to review basic mathematical notions such as the spectrum of an operator, and the Gateaux or variational derivative, which we needed for the course. Moreover, to make the book relatively selfcontained, we recall and sometimes discuss the basic notions mentioned above. As a result, the text is interspersed with mathematical supplements which occupy in total about a third of the material. A mathematically sophisticated reader can skim through them, or skip them altogether, and concentrate on physical applications. On the other hand, readers familiar with the physical content of quantum mechanics, and who would like to enhance their mathematics,
VI
Preface
could concentrate on those detours and consider the physics chapters as an application of the mathematics in a familiar setting. Though we tried to increase the complexity of the material gradually, we were not always successful, and first in Chapter 8, and then in Chapter 13, there is a leap in the level of sophistication required from the reader. This book consists of fifteen main chapters and one supplementary chapter, Chapter 16. The latter chapter is more technical than the preceding material. We did not include many standard topics which are wellcovered elsewhere. These topics are referenced in Chapter 17, where we also give some comments on the literature and further reading. Acknowledgment: The authors are grateful to W. Hunziker, Yu. Ovchinnikov, and especially J. Frohlich and V. Buslaev for useful discussions, and to J. Feldman, G.M. Graf, 1. Herbst, L. Jonsson, E. Lieb, B. Simon and F. Ting for reading parts of the manuscript and making useful remarks. The second author acknowledges his debt to his many collaborators, and especially to V. Bach, J. Frohlich, Yu. Ovchinnikov, and A. Soffer.
Vancouver /Toronto, Sept. 2002
Stephen Gustafson Israel Michael Sigal
Table of Contents
1
Physical Background ..................................... 1.1 The DoubleSlit Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 State Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Schrodinger Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Mathematical Supplement: Operators on Hilbert Spaces. . . . .
1
1 4 4 5 6
2
Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13 2.1 Conservation of Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13 2.2 Existence of Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14 2.3 The Free Propagator.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18 2.4 Mathematical Supplement: Operator Adjoints. . . . . . . . . . . . .. 19 2.5 Mathematical Supplement: the Fourier Transform. . . . . . . . .. 22 2.5.1 Definition of the Fourier Transform. . . . . . . . . . . . . . . .. 22 2.5.2 Properties of the Fourier Transform. . . . . . . . . . . . . . . .. 22 2.5.3 Functions of the Derivative. . . . . . . . . . . . . . . . . . . . . . .. 24
3
Observables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.1 Mean Values and the Momentum Operator. . . .. . ... . . .. . .. 3.2 Observables............................................ 3.3 The Heisenberg Representation. . . . . . . . . . . . . . . . . . . . . . . . . .. 3.4 Quantization........................................... 3.5 Pseudo differential Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . ..
4
The 4.1 4.2 4.3
5
Spectral Theory. . .. . . .. . . ... . . . . . .. .. . .. . . . . .. . . . . . . .. . .. 5.1 The Spectrum of an Operator. . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.2 Functions of Operators and the Spectral Mapping Theorem.. 5.3 Applications to Schrodinger Operators. . . . . . . . . . . . . . . . . . .. 5.4 Spectrum and Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.5 Variational Characterization of Eigenvalues . . . . . . . . . . . . . . ..
25 25 26 27 27 29
Uncertainty Principle. . . . .. .. . . . . . . . .... .... .. .. . . ... 31 The Heisenberg Uncertainty Principle. . . . . . . . . . . . . . . . . . . .. 31 A Refined Uncertainty Principle. . . . . . . . . . . . . . . . . . . . . . . . .. 32 Application: Stability of Hydrogen. . . . . . . . . . . . . . . . . . . . . . .. 33
35 35 39 41 46 48
VIII
Table of Contents 5.6 5.7
Number of Bound States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53 Mathematical Supplement: Integral Operators. . . . . . . . . . . . .. 58
6
Scattering States ......................................... 6.1 Shortrange Interactions: p, > 1. . . . . . . . . . . . . . . . . . . . . . . . . .. 6.2 Longrange Interactions: p, :s; 1 . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.3 Existence of Wave Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . ..
61 62 64 65
7
Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.1 The Infinite Well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.2 The Torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.3 A Potential Step ....................................... 7.4 The Square Well ....................................... 7.5 The Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.6 A Particle on a Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.7 The Hydrogen Atom.. . ... .. . . .. . . .. . .... . . . . . . . . .. . . . .. 7.8 A Particle in an External EM Field. . . . . . . . . . . . . . . . . . . . . ..
67 67 68 68 69 71 74 74 77
8
Manyparticle Systems. .. . ... .. .. . . .. . .. . . . . . . . . .. . . . . ... 8.1 Quantization of a Manyparticle System. . . . . . . . . . . . . . . . . .. 8.2 Separation of the Centreofmass Motion .................. 8.3 Breakups............................................. 8.4 The HVZ Theorem ............................. , . . .. . .. 8.5 Intra vs. Intercluster Motion. . . . . . . . . . . . . . . . . . . . . . . . . . .. 8.6 Existence of Bound States for Atoms and Molecules. . . . . . . .. 8.7 Scattering States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8.8 Mathematical Supplement: Tensor Products ...............
79 79 83 85 86 88 90 91 93
9
Density Matrices ...................... . . . . . . . . . . . . . . . . . .. 9.1 Introduction........................................... 9.2 States and Dynamics. . .. . .. . . . .. . . .. .... . . . . . . . .. .. . . ... 9.3 Open Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.4 The Thermodynamic Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.5 Equilibrium States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.6 The T > 0 Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.7 Example: a System of Harmonic Oscillators ................ 9.8 A Particle Coupled to a Reservoir ........................ 9.9 Quantum Systems ...................................... 9.10 Problems .............................................. 9.11 Hilbert Space Approach ................................. 9.12 Appendix: the Ideal Bose Gas ............................ 9.13 Appendix: BoseEinstein Condensation .................... 9.14 Mathematical Supplement: the Trace, and Trace Class Operators .............................................
95 95 95 96 98 99 99 100 102 102 103 103 104 109 112
Table of Contents
10 The 10.1 10.2 10.3
IX
Feynman Path Integral .............................. The Feynman Path Integral. ............................. Generalizations of the Path Integral ...................... Mathematical Supplement: the Trotter Product Formula ....
115 115 118 119
Quasiclassical Analysis ................................... 11.1 Quasiclassical Asymptotics of the Propagator .............. 11.2 Quasiclassical Asymptotics of Green's Function ............ 11.2.1 Appendix ....................................... 11.3 BohrSommerfeld Semiclassical Quantization .............. 11.4 Quasiclassical Asymptotics for the Ground State Energy .... 11.5 Mathematical Supplement: Operator Determinants .........
121 122 125 127 128 130 132
12 Mathematical Supplement: the Calculus of Variations .... 12.1 Functionals ............................................ 12.2 The First Variation and Critical Points .................... 12.3 Constrained Variational Problems ........................ 12.4 The Second Variation ................................... 12.5 Conjugate Points and Jacobi Fields ....................... 12.6 The Action of the Critical Path .......................... 12.7 Appendix: Connection to Geodesics .......................
135 135 137 141 142 143 147 150
13 Resonances ............................................... 13.1 Tunneling and Resonances ............................... 13.2 The Free Resonance Energy .............................. 13.3 Instantons ............................................. 13.4 Positive Temperatures .................................. 13.5 Preexponential Factor for the Bounce .................... 13.6 Contribution of the Zeromode ........................... 13.7 BohrSommerfeld Quantization for Resonances .............
153 153 155 157 159 161 162 163
14 Introduction to Quantum Field Theory ................... 14.1 The Place of QFT ...................................... 14.1.1 Physical Theories ................................. 14.1.2 The Principle of Minimal Action ................... 14.2 KleinGordon Theory as a Hamiltonian System ............. 14.2.1 The Legendre Transform .......................... 14.2.2 Hamiltonians .................................... 14.2.3 Poisson Brackets ................................. 14.2.4 Hamilton's Equations ............................. 14.3 Maxwell's Equations as a Hamiltonian System ............. 14.4 Quantization of the KleinGordon and Maxwell Equations ... 14.4.1 The Quantization Procedure ....................... 14.4.2 Creation and Annihilation Operators ............... 14.4.3 Wick Ordering ...................................
167 167 168 168 169 169 170 170 171 173 174 175 179 181
11
X
Table of Contents 14.4.4 Quantizing Maxwell's Equations .................... Fock Space ............................................ Generalized Free Theory ................................ Interactions ............................................ Quadratic Approximation ............................... 14.8.1 Further Discussion ................................
182 183 186 186 189 195
15 Quantum Electrodynamics of Nonrelativistic Particles: the Theory of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 The Hamiltonian ....................................... 15.2 Perturbation Setup .................................... 15.3 Results ................................................ 15.4 Mathematical Supplements .............................. 15.4.1 Spectral Projections .............................. 15.4.2 Projectingout Procedure ..........................
197 197 200 202 205 205 206
14.5 14.6 14.7 14.8
16 Supplement: Renormalization Group ..................... 16.1 The Decimation Map ................................... 16.2 Relative Bounds ........................................ 16.3 Elimination of Particle and High Photon Energy Degrees of Freedom ............................................... 16.4 Generalized Normal Form of Operators on Fock Space ...... 16.5 The Hamiltonian Ho(c, z) ............................... 16.6 A Banach Space of Operators ............................ 16.7 Rescaling .............................................. 16.8 The Renormalization Map ............................... 16.9 Linearized Flow ........................................ 16.lOCentralstable Manifold for RG and Spectra of Hamiltonians. 16.11Appendix .............................................
207 207 210 211 216 218 221 223 225 226 230 234
17 Comments on Missing Topics, Literature, and Further Reading .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Index ......................................................... 245
1 Physical Background
In this introductory chapter, we present a very brief overview of the basic structure of quantum mechanics, and touch on the physical motivation for the theory. A detailed mathematical discussion of quantum mechanics is the focus of the subsequent chapters.
1.1 The DoubleSlit Experiment Suppose a stream of electrons is fired at a shield in which two narrow slits have been cut (see Fig. 1.1.) On the other side of the shield is a detector screen. ~ shield ~
1
~ electron ~ gun ~
~
1)'s 1I
screen
~
Fig. 1.1. Experimental setup. Each electron that passes through the shield hits the detector screen at some point, and these points of contact are recorded. Pictured in Fig. 1.2 and Fig. 1.3 are the intensity distributions observed on the screen when either of the slits is blocked.
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
2
1 Physical Background
7
t
PI (brightness)
7 7 7 7
T Fig.1.2. First slit blocked.
7 7
1
7 7 7 Fig.1.3. Second slit blocked. When both slits are open, the observed intensity distribution is shown in Fig. 1.4.
1.1 The DoubleSlit Experiment
3
1
I
T Fig. 1.4. Both slits open. Remarkably, this is not the sum of the previous two distributions; i.e.,
P
I PI + P2 . We make some observations based on this experiment.
1. We cannot predict exactly where a given electron will hit the screen, we can only determine the distribution of locations. 2. The intensity pattern (called an interference pattern) we observe when both slits are open is similar to the pattern we see when a wave propagates through the slits: the intensity observed when waves EI and E2 (the waves here are represented by complex numbers encoding the amplitude and phase) originating at each slit are combined is proportional to lEI + E212 I IEI12 + IE212 (see Fig. 1.5).
Fig.1.5. Wave interference. We can draw some conclusions based on these observations.
4
1 Physical Background
1. Matter behaves in a random way. 2. Matter exhibits wavelike properties. In other words, the behaviour of individual electrons is intrinsically random, and this randomness propagates according to laws of wave mechanics. These observations form a central part of the paradigm shift introduced by the theory of quantum mechanics.
1.2 Wave Functions In quantum mechanics, the state of a particle is described by a complexvalued function of position and time, 1/J(x, t), x E ]R3, t E lR. This is called a wave function (or state vector). Here]Rd denotes ddimensional Euclidean space, ]R = ]Rl, and a vector x E ]Rd can be written in coordinates as x = (Xl, ... ,Xd) with Xj E lR. In light of the above discussion, the wave function should have the following properties. 1. [1/J(', t) [2 is the probability distribution for the particle's position. That is, the probability that a particle is in the region Q C ]R3 at time t is [1/J(x, t)[2dx. Thus we require the normalization In~3 [1/J(x, t)[2dx = 1. 2. 1/J satisfies some sort of wave equation.
In
For example, in the doubleslit experiment, if 1/Jl gives the state beyond the shield with the first slit closed, and 1/J2 gives the state beyond the shield with the second slit closed, then 1/J = 1/Jl + 1/J2 describes the state with both slits open. The interference pattern observed in the latter case reflects the fact that [1/J[2 =I [1/Jl[2 + [1/J2[2.
1.3 State Space The space of all possible states of the particle at a given time is called the state space. For us, the state space of a particle will usually be the squareintegrable functions:
(we can impose the normalization condition as needed). This is a vector space, and has an innerproduct given by
(1/J, ¢):=
r 7[;(x)¢(x)dx
JIR3
(in fact, it is a "Hilbert space"  see Section 1.5)
1.4 The Schrodinger Equation
5
1.4 The Schrodinger Equation We now give a motivation for the equation which governs the evolution of a particle's wave function. This is the celebrated Schrodinger equation. Our equation should satisfy certain physically sensible properties. 1. The state 'I)!(', to) at time t = to should determine the state 'I)!(', t) for all later times t > to (causality). 2. If 'I)! and ¢ are evolutions of states, then o.'l)! + (3¢ (a, (3 constants) should also describe the evolution of a state (the superposition principle). 3. In "everyday situations," quantum mechanics should be close to the classical mechanics we are used to (the correspondence principle). The first requirement means that 'I)! should satisfy an equation which is firstorder in time, namely (1.1) for some operator A, acting on the state space. The second requirement implies that A must be a linear operator. We use the third requirement in order to find the correct form of A. We first recall that one of the fundamental equations of classical mechanics is firstorder in time. It is the HamiltonJacobi equation, f) f)t
S
=
h(x, V x S )
(1.2)
where h(x, k) = ~ + V(x) is the classical Hamiltonian junction, V is the potential, m is the mass, and S(x, t) is the classical action. This equation, in turn, is similar to the eikonal equation,
which is a highfrequency approximation of the wave equation for u = ae i ¢. We make an analogy between the passage from the wave equation to the eikonal equation (that is, from wave optics to geometric optics) and the passage from quantum mechanics to classical mechanics. Thus, we require that Equation (1.1) (whose form we are seeking) has solutions of the form 'I)! (x, t)
= a(x, t)eiS(x,t)/li
where n is some very small constant, with S satisfying equation (1.2). Assuming a, S, and their derivatives are of order one in n, then to the leading order in n, 'I)! satisfies the equation
n
2 f) in~'I)!(x, t) = i1x'l)!(x, t)
ut
2m
+ V(x)'I)!(x, t).
(1.3)
6
1 Physical Background
This equation is of the desired form (1.1). In fact it is the correct equation, and is called the Schrodinger equation. It can be written as (1.4) where the linear operator H, given by
IH1jJ := ~L11jJ + V1jJ I is called a Schrodinger operator. The operator L1 = ~~=1 8J is the Laplacian (in spatial dimension 3), and the function (and multiplication operator) V is the potential. The small constant fi is Planck's constant; it is one of the fundamental constants in nature. For the record, its value is roughly fi ~ 6.6255
X
10 27 erg sec.
Example 1.1 Here are just a few examples of potentials.
1. 2. 3. 4. 5.
Free motion: V == o. A wall: V == 0 on one side, V == 00 on the other (meaning 1jJ == 0 here). The doubleslit experiment: V == 00 on the shield, and V == 0 elsewhere. The Coulomb potential: V(x) = a/lxl (describes a hydrogen atom). The harmonic oscillator: V(x) = m~2IxI2.
We will analyze some of these examples, and others, in Chapter 7.
1.5 Mathematical Supplement: Operators on Hilbert Spaces We have seen in this chapter that the space of quantummechanical states of a system is a vector space with an innerproduct (in fact a Hilbert space). We saw also that an operator (a Schrodinger operator) on this space enters the basic equation (the Schrodinger equation) governing the evolution of states. In fact, the theory of operators on a Hilbert space provides the basic mathematical framework of quantum mechanics. This section describes some aspects of operator theory that are essential to a study of quantum mechanics. Let H be a (complex) vector space. We assume H is endowed with an innerproduct, (., J. Recall that this means the map
(, ):HxH4C satisfies the properties
1.5 Mathematical Supplement: Operators on Hilbert Spaces
7
1. linearity (in the second argument):
(v, aw
+ (3z)
a(v, w)
=
2. conjugate symmetry: (w, v) = (v, w) 3. positive definiteness: (v, v) > 0 for v
=I=
+ (3(v, z)
0
for any v, w, z E H and a, (3 E C. It follows that the map given by
Ilvll
:=
I . I : H +
[0,(0)
(v, v)1/2
is a norm on H. Recall that this means that for any v, w E H and a E C, 1.
2. 3.
lIavll = lalllvil Ilv + wll : : : Ilvll + Ilwll Ilvll > 0 for v =I= o.
(the triangle inequality)
If H is complete in this norm (that is, all Cauchy sequences converge), then H is called a Hilbert space. Our main example of a Hilbert space is the L2space
with the innerproduct
(7/J, ¢):=
r {;¢.
JlR d
Here, and in what follows, we use the notation f for flRd f for flR d f(x)dx. Another important example of a Hilbert space is the Sobolev space of order n, n = 1,2, ... :
Here a is a multiindex: a = (a1, ... , ad), aj a nonnegative integer, and lal := 2::=1 aj. The expression 87/J denotes the partial derivative 8::; ... 8:::7/J of order lal. In other words, Hn(JRd) is the space of functions all of whose derivatives up to order n lie in L 2 (JRd). The innerproduct that makes Hn(JR d) into a Hilbert space is
(7/J, ¢) Hn
:=
L
(87/J,8¢)
1I:Sn
where (.,.) is the L2 innerproduct defined above. We recall here two frequently used facts about Hilbert spaces (see, eg., [Fo] or [RSI] for proofs). Proposition 1.2 (CauchySchwarz inequality) For v, w E H, a Hilbert space,
I(v, w)1 : : : Ilvllllwll·
8
1 Physical Background
Recall that a set {V n } C 1i, n = 1, 2, ... is called orthonormal if II Vn I = 1 for all nand (vn' V m ! = for n f m. It is a complete orthonormal set (or basis) if the collection of finite linear combinations of the v n ' s is dense in 1i. Recall that for a subset D c 1i to be dense in 1i means that given any v E 1i and f > 0, there exists wED such that Ilv  wll < f.
°
Proposition 1.3 (Parseval relation) Suppose {vn } C 1i is a complete orthonormal set. Then for any w E 1i,
n
In this chapter, the main objects of study are linear operators on 1i (often just called operators), which are maps, A, from 1i to itself, satisfying the linearity property A( o:v + (Jw) = o:Av + (JAw for v, w E 1i, 0:, (J E 00
by (2.12). Hence (eiA)*e iA = 1. Similarly, one can show that (e iA )* = e iA . Then applying the above argument to e iA gives eiA(e iA )* = (eiA)*e iA = 1. Thus eiA is unitary. 0 The following simple example illustrates Theorem 2.9. Example 2.11 If ¢ : lRd
;
lR is continuous, then the bounded operator
2.2 Existence of Dynamics
17
is easily checked to be unitary on L 2(JRd) (just note that U* is multiplication by e i .t/fi7jJ, eiH>.S/fi¢)
= Jim (7jJ, eiH>.(Hs)/fi¢;) = (7jJ, U(t + s)¢) / 1/4 is symmetric on the domain (~3 \ {O}) (the infinitely differentiable functions supported away from the origin), but does not have a unique selfadjoint extension (see [RSH]). It is usually much easier to show that a given operator is symmetric than to show that it is selfadjoint, since the latter question involves additional domain considerations.
Co
Problem 2.18 Referring to our earlier list of examples of operators on L2(~d), show that the following operators are symmetric.
1. 2. 3. 4. 5.
The The The The The
multiplication operator V (if V(x) is realvalued). differentiation operator, Pj (hint: integrate by parts). Laplacian, .1. Schrodinger operator, H (again if V is real). integral operator J( (if K(y,x) = K(x,y)).
In fact, Pj and .1 are actually selfadjoint operators on L2(~d), as is V if it is a "nice enough" function (we will be more specific in Section 5.3). To be more precise, there exist domains for these operators, containing CO(~d), for which they are selfadjoint (and, moreover, they are the unique selfadjoint extensions from Co(~d)). For example, D(.1) = H2(~d). The subtleties surrounding domains and the question of selfadjointness are absent for bounded operators, as Lemma 1.6 demonstrates. Lemma 2.19 If A is bounded and symmetric, it is selfadjoint.
Proof. Since A is bounded, by Lemma 1.6 we may assume D(A)
= Ji.
Since
1(1f!, A¢) 1::; 111f!IIIIAIIII¢II, for 1f!, ¢ E Ji, we have D(A*) selfadjoint. 0
= Ji, and so D(A) = D(A*). Hence A is
We will, however, have to deal with unbounded operators. As an example, consider the following: Problem 2.20 Show that the operators Pj := ifWj and T := unbounded on L2(~d).
!!:n .1 are
For what follows, we need the following important result: Theorem 2.21 If A is a selfadjoint operator, then for any z E C with Im(z) =I 0, the operator A  zl has a bounded inverse, and this inverse satisfies (2.19)
2.4 Mathematical Supplement: Operator Adjoints
21
Proof. Suppose Im(z) i= O. Then we compute, for 'I/J E D(A),
II(A  zl)'l/J11 2 = ((A  zl)'I/J, (A  zl)'I/J) = IIA'l/J112  2Re(z)('I/J, A'I/J) + Iz1211'I/J112
(2.20)
so
II(A  zl)'l/J11 2 2: IIA'l/J112  2IRe(z)III'l/JIIIIA'l/J11 + Iz1211'I/J112 = (1IA'l/JIIIRe(z)III'l/JII)2 + IIm(z)1211'I/J112. Hence
(2.21 )
II(A  zl)'l/JII 2: IIm(z)III'l/JII· This shows that the operator A  zl has an inverse (A  Zl)l : Ran(A  zl)
+
1i
(here Ran(A) := {'I/J E 1i I ::J¢ E D(A), 'I/J = A¢} denotes the range of A) and this inverse satisfies (2.19) for any ¢ E Ran(Azl) (just take ¢:= (Azl)'I/J in (2.21)). Moreover, if ¢ .l Ran(A  zl), then (¢, (A  zl)'I/J) = 0 for all 'I/J E D(A). Hence by Definition 2.14, ¢ E D(A*) = D(A). Therefore 0= (¢, (A  zl)'I/J) = ((A  Zl)¢, 'I/J)
for all 'I/J E D(A). Since D(A) is dense, this implies (A  Zl)¢ = 0 and therefore, by (2.21), ¢ = O. Hence Ran(A  zl) is dense. Consequently, by Lemma 1.6, the operator (A  z1)l can be extended to a bounded operator on the entire space 1i, with the estimate (2.19) valid for all ¢ E 1i. 0 We conclude this section with a useful definition. Definition 2.22 A selfadjoint operator A is called positive (denoted A > 0) if ('I/J,A'I/J) > 0
for all 'I/J E D(A), 'I/J i= O. Similarly, we may define nonnegative, negative, and nonpositive operators. Problem 2.23 Show that the operator d on L 2(JRd) is positive (we can take D(d) = H 2 (JRd)). Hint: integrate by parts (equivalently, use the divergence theorem) assuming that 'I/J E D{3 := {'I/J E C 2 (JR d) I I8a 'I/J (x) I ~ Ca (1 + Ixl){3Va, lal ~ 2} for some f3 > d/2, Then use the fact that D{3 is dense in H 2(JRd) to extend the inequality to all 'I/J E H 2(JRd). If A is a positive operator, then we can define the operator e A in a way similar to our definition of eiA above. We take e A := lim).+oo e A ", where A). = (A + >..)1 >"A, a family of bounded operators.
22
2 Dynamics
2.5 Mathematical Supplement: the Fourier Transform The Fourier transform is a useful tool in many areas of mathematics and physics. The purpose of the present chapter is to review the properties of the Fourier transform, and to discuss the important role it plays in quantum mechanics. 2.5.1 Definition of the Fourier Transform The Fourier transform is a map, F, which sends a function 'ljJ : JRd another function {j; : JRd + 0). Hmt: try d = 1 first  complete the square in the exponent and move the contour of integration in the complex plane. 2. e2hxoAlx f+ fid/2(detA)1/2 e!k.Ak (A a positive d x d matrix). Hint: diagonalize and use the previous result. 1. e ~
f+
v'b/1l. 2 Ixl
3. ~e Ixl f+ (lkl 2 + b)l (b > 0, d = 3). Hint: use spherical coordinates. Alternatively, see Problem 2.26 below. In the first example, if Re(a) > 0 then the function on the left is in Ll(JRd), and the Fourier transform is welldefined. However, we can extend this result to Re(a) = 0, in which case the integral is convergent, but not absolutely convergent. 2.5.2 Properties of the Fourier Transform The utility of the Fourier transform derives from the following properties. 1. The Plancherel theorem: F is a unitary map from L 2(JRd) to itself (note
that initially the Fourier transform is defined only for integrable (Ll (JRd)) functions  the statement here is that the Fourier transform extends from L1(JRd) n L2(JR d ) to a unitary map on L 2(JRd)).
2.5 Mathematical Supplement: the Fourier Transform
23
2. The inversion formula: the adjoint F* of F on L 2(JR.d) is given by the map 'ljJ f4 1f where
1f(x) := (27fn)d/2
r eiX.k/I'i'ljJ(k)dk
JJfII.d
(and by the Plancherel theorem, this is also the inverse, F 1 ). For the next four statements, suppose 'ljJ, ¢ E CO"(JR.d).
in~(k) = k(/;(k). x'ljJ(k) = in\lk(/;(k). ¢'ljJ = (27fn) d/2¢ * (/;. ~ = (27fn)d/2¢(/;.
3. 4. 5. 6.
Here
(f
* g)(x):=
r
f(y)g(x  y)dy JJfII.d is the convolution of f and g. The last four properties can be loosely summarized by saying that the Fourier transform exchanges differentiation and coordinate multiplication, and products and convolutions.
Proof. The proof of Property 1 is somewhat technical and we just sketch it here (see, eg, [Fo] for details). In particular, we will show that Ilfll = lIill. Suppose f E CO", and let C e be the cube of side length 2/ E centred at the origin. Choose E small enough so that the support of f is contained in C e • One can show that {Ek := (E/2)d/2 eik.x/1'i IkE En7fZd } is an orthonormal basis of the Hilbert space of functions in L2(Ce ) satisfying periodic boundary conditions. Thus by the Parseval equation (Proposition 1.3),
as
E
+
O.
Problem 2.25 Show that {Ek} is an orthonormal set. We will prove Property 3, and we leave the proofs of the other properties as exercises. Integrating by parts, we have
in\l'ljJ(k)
= (27fn)d/2
J J
= k· (27fn)d/2 D
eix.k/I'i. (in\l)'ljJ(x)dx eiX.k/I'i'ljJ(x)dx = k(/;(k).
24
2 Dynamics
Problem 2.26
1. Show that for b > 0 and d
= 3, under Fl,
(hint: use spherical coordinates, then contour deformation and residue theory). 2. Show that under F 1 ,
Here 5 is the Dirac delta function  not really a function, but a distribution  characterized by the property f(x)5(xa)dx = f(a). The exponential function on the right hand side is called a plane wave.
J
2.5.3 Functions of the Derivative
As an application, we show how the Fourier transform can be used to define functions of the derivative operator. Recall our notation p := iti'v. Motivated by Property 3 ofthe Fourier transform, we define an operator g(p) (for "sufficiently nice" functions g) on L2 (JRd) as follows. Definition 2.27 ~(k) := g(k)(/;(k) or, equivalently, g(p)'I/J := (27rn)d/2 g * 'I/J.
Let us look at a few examples. Example 2.28 1. If g(k) = k, then by Property 3 of the Fourier transform, the above definition gives us back g(p) = p (so at least our definition makes some sense).
2. Now suppose g(k)
= Ik1 2 . Then ~(k) = IkI 2 (/;.
Problem 2.29 Show that
n2 iJ.'I/J = IkI 2 (/;.
Thus we have Ipl2 = n2 iJ.. Extending this example, we can define g(p) when 9 is a polynomial "with our bare hands" . It is easy to see that this definition coincides with the one above. 3. Let g(k) = (2~ Ikl 2 + A)I, A > 0, and d = 3. Then due to Problem 2.26, we have
((Ho
+ A)
1
'I/J)(x)
=
1m
 v"2ffi>: Ixyl IX  YI 'I/J(y)dy.
melt
2 ~2 7rn
IR3
(2.22)
3 Observables
Observables are the quantities that can be experimentally measured in a given physical framework. In this chapter, we discuss the observables of quantum mechanics, as well as the notion of "quantizing" a classical theory.
3.1 Mean Values and the Momentum Operator We recall that in quantum mechanics, the state of a particle at time t is described by a wave function 7/J(x, t). The probability distribution for the position, x, of the particle, is 17/J(·, t)12. Thus the mean value of the position at time t is given by J xl7/J(x, t)j2dx (note that this is a vector in JR3). If we define the coordinate multiplication operator
Xj : 7/J(x)
ft
Xj7/J(x)
then the mean value of the lh component of the coordinate x in the state 7/J is (7/J, Xj7/J). Using the fact that 7/J(x, t) obeys the Schrodinger equation i~ = H7/J, we compute
d
.
dt (7/J,Xj7/J) = (7/J,Xj7/J)
.
+ (7/J,Xj7/J) =
1
(iIi H 7/J,Xj7/J)
1
+ (7/J,Xj iIi H 7/J)
i i i
= (7/J, Yi Hx j7/J)  (7/J,XjYi H 7/J)
=
(7/J, Yi[H,xj]7/J)
(where recall [A, B] := AB  BA is the commutator of A and B). Since H = K Ll + V ' and 2m Ll(x7/J) = xLl7/J + 2V7/J, we find leading to
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
26
3 Observables
As before, we denote the operator in\! j by Pj. As well, we denote the mean value (7jJ, A7jJ) of an operator A in the state 7jJ by (A),p. Then the above becomes
Im~ (Xj),p
= (pj),p
I
(3.1)
which is reminiscent of the definition of the classical momentum. We call the operator p the momentum operator. In fact, Pj is a selfadjoint operator on £2(JR3). (As usual, the precise statement is that there is a domain on which Pj is selfadjoint. Here the domain is just D(pj) = {7jJ E £2 (JR d) I a~ 7jJ E J
£2(JRd).)
Using the Fourier transform, we compute the mean value of the momentum operator
This, and similar computations, show that 10(k)12 is the probability distribution for the particle momentum.
3.2 Observables Definition 3.1 An observable is a selfadjoint operator on the state space £2(JR3). We have already seen a few examples of observables, including the position operators, Xj, the momentum operators, Pj and the Hamiltonian operator, H
11,2
1
2m
2m
= L1 + V = lpl2 + V
(here IpI2 = 2:~=1 p;). Another example (with an obvious classical analogue) is furnished by the angular momentum operators, £j = (x x pk The reader is invited to derive the following equation for the evolution of the mean value of an observable.
Problem 3.2 Check that for any observable, A, we have
We would like to use this result on the momentum operator. Simple computations give [L1,p] = 0 and [V,p] = in\!V, so that *[H,p] = \!V and hence
(3.2)
3.4 Quantization
27
This is a quantum mechanical meanvalue version of Newton's equation of classical mechanics. Or, if we include Equation (3.1), we have a quantum analogue of the classical Hamilton equations. Since obviously [H, H] = 0, we also have d/dt(H),p = 0, which is the meanvalue version of the conservation of energy.
3.3 The Heisenberg Representation The framework outlined up to this point is called the Schrodinger representation of quantum mechanics. Chronologically, quantum mechanics was first formulated in the Heisenberg representation, which we now describe. For an observable A, define
A(t)
:= eitH / 1i Ae itH / Ii .
Let 7/J be the solution of Schrodinger's equation with initial condition 7/Jo: 7/J = e itH / li 7/Jo. Since e itH / 1i is unitary, we have (by a simple computation which is left as an exercise)
(A),p
(A(t)),po·
(3.3)
Problem 3.3 1. Prove equation (3.3). 2. Prove that
d
i
dtA(t) = /i[H,A(t)]. This last equation is called the Heisenberg equation for the time evolution of the observable A. In particular, taking x and p for A, we obtain the quantum analogue of the Hamilton equations of classical mechanics:
and
mi:(t) = p(t)
(3.4)
p(t) = V'V(x(t)).
(3.5)
In the Heisenberg representation, then, the state is fixed (at 7/Jo), and the observables evolve according to the Heisenberg equation. Of course, the Schrodinger and Heisenberg representations are completely equivalent (by a unitary transformation).
3.4 Quantization We now describe a procedure for passing from classical mechanics to quantum mechanics, called quantization. We start with the Hamiltonian formulation of classical mechanics, where the basic objects are as follows:
28
3 Observables
1. The phase space (or state space) lR~ x lR%
equipped with the Poisson bracket, {, .}. The latter is defined on pairs of functions on lR~ x lR% by
~ (Of og
{j, g} = ~
ok·
j=l
J
of Og)
axJ  ax·J ok·J
.
2. The Hamiltonian, h(x, k), a real function on lR~ x lR% (which gives the energy of the classical system). Then the classical dynamics of the system is given by Hamilton's equations: {h,x}
j; =
k=
{h,k}.
The corresponding fundamental objects in quantum mechanics are the following: 1. The state space L2(lR~), and the commutator,
acting on L2(lR~). 2. A Schrodinger operator, H state space L2(lR3 ).
=
*[.,'J,
of two operators
h(x,p) (the Hamiltonian) acting on the
= Ikl2/2m + V(x), the corresponding quantum Hamiltonian is the Schrodinger operator
If the classical Hamiltonian function is h(x, k)
H
Ipl2
n2
m
2m
= h(x,p) = 2 + V(x) = .1 + V(x).
Recall that the dynamics of the quantum system can be described by the Heisenberg equations .
i
x = n[H,x]
.
p
i
= n[H,p]
(which are the same as (3.4) and (3.5)). We note that for classical mechanics
and for quantum mechanics
These are called the canonical commutation relations. The functions x and k are called canonical variables of classical mechanics, and the operators x and p, canonical operators of quantum mechanics. The following table provides a summary of the classical mechanical 0 b jects and their quantized counterparts:
3.5 Pseudodifferential Operators
QM
CM
Object state space evolution of state observable
29
lR 3x x lR 3k and L2(lR~) and Poisson bracket commutator path in phase space selfadjoint operator real function on state space on state space result of measuring deterministic probabilistic observable object determining Hamiltonian Hamiltonian (Schrodinger) dynamics function operator x and k operators canonical coordinates x (mult.) and p
3.5 Pseudodifferential Operators The correspondence between classical observables and quantum observables is a subtle one. For canonical variables, the variable x is mapped to the operator of multiplication by x, and the variable k is mapped to the operator p = ifi'il x . Continuing in this way, we should have a function f(x) mapping to an operator of multiplication by f(x), and a function g(k) mapping to an operator g (p). However, the following simple case shows the ambiguity of this correspondence. The function X· k = k· x could, for example, be mapped into any of the following distinct operators:
x·p,
p·x,
1
2"(x. p + p .x).
In this case, the ambiguity is resolved by requiring that the resulting operator be selfadjoint. But in the general situation
f(x, k)
4
f(x,p),
showing that this condition resolves the ambiguity and that it can be implemented in terms of a mathematical formula requires some work. In mathematics, operators obtained by a certain quantization rule from functions f(x, k) satisfying certain estimates are called pseudodifferential operators. Differential operators with smooth coefficients, as well as certain integral and singular integral operators are examples of pseudo differential operators.
4 The Uncertainty Principle
One of the fundamental implications of quantum theory is the uncertainty principle  that is, the fact that certain pairs of physical quantities cannot be measured simultaneously with arbitrary accuracy. In this chapter, we establish precise mathematical statements of the uncertainty principle.
4.1 The Heisenberg Uncertainty Principle We consider a particle in a state 'IjJ and think of the observables x and p as random variables with probability distributions i'IjJi 2 and i¢i 2 respectively. Recall that the means of Xj and Pj in the state 'IjJ (E D(xj) nD(pj)) are (Xj),p and (Pj),p, respectively. The dispersion of Xj in the state 'IjJ is
and the dispersion of Pj is
Theorem 4.1 (The Heisenberg uncertainty principle) For any state 'IjJ E D(xj) n D(pj), ,1x .,1p.
n
> . 2
J 
J
Proof The basic ingredient is the commutation relation i
n[P,x]=1
*
(this is a matrix equation, meaning [pj, Xk] = Ojk). For notational simplicity, we assume (x),p = (p),p = o. Note that for two selfadjoint operators A, B, and 'IjJ E D(A) n D(B), (i[A, B]),p = 2Im(A'IjJ, B'IjJ).
So assuming 'IjJ is normalized
(ii'IjJii =
1), and 'IjJ
E
D(xj)
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
(4.1)
n D(pj), we obtain
32
4 The Uncertainty Principle
1
i
2
2
= ('Ij;,'Ij;) = ('Ij;, n[pj,Xj]'Ij;) = n1m(Pj'lj;,Xj'lj;) ::::: nl(Pj'lj;,xj'lj;)1 2
::::: n11pj'lj;llllxj'lj;ll
=
2
n C,1Pj)(.1Xj).
This does it. 0
4.2 A Refined Uncertainty Principle The following result is related to the Heisenberg uncertainty principle.
Theorem 4.2 (Refined uncertainty principle) On L2(JR3),
.1 ~
1
41x1 2 '
Recall that for operators A, B, we write A ~ 0 if ('Ij;, A'Ij;) ~ 0 for all 'Ij; E D(A), and we write A ~ B if AB ~ O. So by the above statement, we mean ('Ij;, (.1  41;1 2 )'Ij;) ~ 0 for 'Ij; in an appropriate dense subspace of D( .1). We will prove it for 'Ij; E C8"(JRd ).
Proof. We will ignore domain questions, leaving these as an exercise for the reader. Compute d
2:>[lxl 1pjlxl 1,xj]
=
Ildlxl 2
j=l
(d = space dimension = 3). Hence, using (4.1) again, d
ndlllxl 1'1j;11 2 = 2 L 1m(lxl 1pj Ixl 1'1j;, Xj'lj;) j=l
and therefore, using
we obtain
d
ned 
2)lllxl '1j;11 = 21m L(Pj'lj;, 1:12 'Ij;). 1
2
J1
Now the CauchySchwarz inequality implies d
1t;(Pj'lj;, ,:12 '1j;)1 : : :
('Ij;, IpI2'1j;)1/211Ixl1'1j;11
(prove this!), which together with the previous equality gives
4.3 Application: Stability of Hydrogen
11ld 
33
2111Ixl 17jJ11 2:::; 2(7jJ, IpI27jJ)1/211Ixl17jJ11.
Squaring this, and observing that 112(7jJ, d7jJ) yields (for d ;::: 3)
Illxl 17jJ11 2 =
(7jJ, Ixl 27jJ) and (7jJ, IpI27jJ)
=
which, for d = 3, implies the desired result. 0
4.3 Application: Stability of Hydrogen In classical mechanics, the hydrogen atom is unstable: as the electron orbits the nucleus (proton), it radiates away energy and falls onto the nucleus. The demonstration that this is not so in quantum mechanics was one of the first triumphs of the theory. The statement that the hydrogen atom is stable is expressed mathematically by the property that the Hamiltonian, H (and therefore the energy), is bounded from below. The hydrogen atom is described by the Schrodinger operator
112 e2 H=d
2m
Ixl
on L2(~3) (m and e are the mass and charge of the electron respectively). The refined uncertainty principle gives
The right hand side reaches its minimum at
Ixl 1= 4me 2/112
and so
2me 4 H >  11 2 ' Thus, the energy of the hydrogen atom is bounded from below, and the electron does not collapse onto the nucleus.
5 Spectral Theory
Our next task is to classify the orbits (i.e. solutions) of the Schrodinger equation
in fJ'l/J = H'l/J fJt
with given initial condition
'l/Jlt=o = 'l/Jo according to their behaviour in spacetime. Naturally, we want to distinguish between states which are localized for all time, and those whose essential support moves off to infinity. Such a classification is made with the help of a very important notion  the spectrum of an operator. We begin by describing the general theory, and then we proceed to applications.
5.1 The Spectrum of an Operator Definition 5.1 The spectrum of an operator A on a Hilbert space H is the subset of C given by cr(A) :=
p. E C I
A  A is not invertible (has no bounded inverse)}
(here and below, A  A denotes A  AI). The complement of the spectrum of A in C is called the resolvent set of A: p(A) := C\cr(A). For A E p(A), the operator (A  A)I, called the resolvent of A, is welldefined. The usual reasons that A  A is not invertible are
= 0 has a nonzero solution, 'l/J E H. Then A is called an eigenvalue of A, and 'l/J a corresponding eigenvector. 2. (A  A)'l/J = 0 "almost" has a nonzero solution. More precisely, we say {'l/Jn} cHis a Weyl sequence for A and A if a) II'l/Jn II = 1 for all n b) II(A  A)'l/Jnll> 0 as n > 00 c) 'l/Jn > 0 weakly as n > 00 (this means (cp, 'l/Jn) > 0 for all cp E H). This is what we mean by (A  A)'l/J = 0 "almost" having a nonzero solution. 1. (A  A)'l/J
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
36
5 Spectral Theory
Definition 5.2 The point spectrum of an operator A is iTp(A) = {A
Eel
A is an isolated eigenvalue of A with finite multiplicity}
(isolated meaning some neighbourhood of A is disjoint from the rest of iT(A)). Here the multiplicity of an eigenvalue A is the dimension of the subspace
Null(A  A)
:=
{v
E 1{
I
(A  A)V
=
O}.
Problem 5.3 1. Show Null(A  A) is a vector space. 2. Show that if A = A*, eigenvectors of A corresponding to different eigenvalues are orthogonal. Definition 5.4 The continuous spectrum of an operator A is iTc(A)
= {A
I there is a Weyl sequence for A and A}.
Remark 5.5 Some authors use the terms "discrete spectrum" and "essential spectrum" rather than (respectively) "point spectrum" and "continuous spectrum'. The following result says that when A is selfadjoint, the disjoint sets iTp(A) and iTc(A) comprise the whole spectrum (see, eg., [HS] for a proof):
Theorem 5.6 (Weyl) If A = A*, then the spectrum of A is the union of the point spectrum of A and the continuous spectrum of A:
Problem 5.7 Show that if U : 1{ 7 1{ is unitary, then iT(U* AU) = iT(A), iTp(U* AU) = iTp(A), and iTc(U* AU) = iTc(A). Example 5.8 This example determines the spectrum of the operators Xj and Pj on L2(~d). We show that 1. iT(Pj)=iTc(Pj)=~ 2. iT(Xj) = iTc(Xj) = ~ Proof of the second fact: assume, for simplicity, d = 1. For any A E ~, we find a Weyl sequence for x and A. This sequence is such that its square approximates the deltafunction 6>. (x) = 6(A  x) which solves the equation
(xA)6>. = 0 exactly. Such a sequence is sketched in Fig. 5.1.
5.1 The Spectrum of an Operator
37
Fig.5.l. Weyl sequence for x, A. How do we construct such a sequence 'l/Jn? Let ¢ be a fixed nonnegative function supported on [1,1], and such that
We compress this function, increasing its height, and shift the result to A:
Then
and
as n  00. Thus A E 17(x), at least. Now we show that 'l/Jn  0 weakly. Indeed, for any f E L2(JR d),
which  0 as n  00 by a wellknown result of analysis. Thus, A E 17c (x). It is easy to convince yourself that x has no eigenvalues. Now we prove the first fact. We seek a Weyl sequence, {'l/Jn}, for p and A. Using properties of the Fourier transform, we have
Take for 'l/J~ the Weyl sequence constructed above:
38
for
5 Spectral Theory
¢ supported on
[1,1], and
J 1¢1 2 = 1. So we have II'l/Jnll = II'!f7nll = 1 and
and so 'l/Jn is a Weyl sequence for p and A. Thus o'(p) us see how 'l/Jn looks. We have
= O'c(p) = R Now let
Suppose, for example, that ¢ == 1 for Ixi ::; 1/2. Then 'l/Jn looks like a plane wave (with amplitude n 1 / 2 and wave vector A), cut off near 00 by ¢(x/n) (I'l/Jnl is sketched in Fig. 5.2).
nl2
nl2
Fig.5.2. Weyl sequence for p, A. We remark that the fact O'(pj) = O'c(pj) = JR also follows directly from the fact O'(xj) = O'c(xj) = JR, together with Problem 5.7 and properties of the Fourier transform. The following exercise asks for the spectrum of two of our other favourite operators. Problem 5.9 Prove
1. For V : JRd + C continuous, the spectrum of the corresponding multiplication operator on L 2(JRd) is O'(V) = range(V). 2. On L2(JR d), 0'(..:1) = O'c(..:1) = [0,(0). We observe that selfadjoint operators have real spectrum. Theorem 5.10 If A
= A*,
then O'(A) C R
Proof This follows immediately from Theorem 2.21. D
Proposition 5.11 Let A be a selfadjoint operator on 'H. If A is an accumulation point of O'(A), then A E O'c(A).
5.2 Functions of Operators and the Spectral Mapping Theorem
39
Proof. Suppose Aj E O"(A), Aj t A as j t 00. Then we can find '¢j E O"(A) with lI'¢jll = 1 satisfying II(A  Aj)'¢jll :::; l/j (by the Weyl criterion, Theorem 5.6). Since
as j t 00, we see A E O"(A) (either '¢j t 0 weakly, in which case A E O"c(A), or else one can show that A is an eigenvalue of A). Since A is an accumulation point of O"(A), it is not isolated, and so A tj. O"p(A). Hence A E O"c(A). D If the operator A is a Schrodinger operator on L2 (Jl~d), we can strengthen Theorem 5.6 by showing that it is enough to consider Weyl sequences whose support moves off to infinity.
Definition 5.12 Let A be an operator on L 2(JR.d). A sequence {'¢n} C L 2(JR.d) is called a spreading sequence for A and A if 1. II'¢n II = 1 for all n 2. for any bounded set B C JR.d, sUPP('¢n) n B = 0 for n sufficiently large 3. II(A  A)'¢nll t 0 as n t 00.
Problem 5.13 Show that a spreading sequence for A, A is also a Weyl sequence for A, A. Theorem 5.14 If H =  ;~.::1 + V is a Schrodinger operator, with real potential V(x) which is continuous and bounded from below, then 0" c
(H)
=
{A I there is a spreading sequence for H and A}.
We skip the proof of this theorem (see [HS]).
5.2 Functions of Operators and the Spectral Mapping Theorem Our goal in this ~ection is to define functions f(A) of a selfadjoint operator A. We do this in the special case where A is bounded, and f is a function analytic in a neighbourhood of dA).
Problem 5.15 Let A be a bounded selfadjoint operator. Show O"(A) C [IIAII, IIAII]· Hint: use that if Izl > IIAII, then II(A  z)ull ~ (lzlIIAII)llull· Suppose f(A) is analytic in a complex disk ofradius R, {A Eel IAI < R}, with R > IIAII. SO f has a power series expansion, f(A) = E~=o anA n , which converges for IAI < R. Define the operator f(A) by the convergent series
L
(Xl
f(A)
:=
n=O
anAn.
(5.1)
40
5 Spectral Theory
We have already encountered an example of this definition: the exponential e A discussed in Section 2.2. As another example, consider the function j()...) = ()...  Z)l, for Izl > IIAII, which is analytic is a disk of radius R, with IIAII < R < Izl, and has powerseries expansion
j()...)
1
1
= 
zl).../z
1
00
.
""().../z)J. z~
= 
The corresponding operator defined by (5.1) is the resolvent (5.2) and the series in (5.2) is called a Neumann series. Of course, the resolvent is defined for any z in the resolvent set p(A) = lC\lT(A). In fact, (A  Z)l is an analytic (operatorvalued) function of z E p(A). To see this, we start with the relation
(A  z)l
=
(A  ZO)l  (zo  z)(A  zo)l(A  Z)l
for z, Zo E p(A), which the reader is invited to verify (this relation is called the first resolvent identity). Thus
If Iz  zol < (II (A  zo)lll)l, the first inverse on the right hand side can be expanded in a Neumann series, yielding 00
(A  z)l
=
L(z  zo)j (A  zO)jl. j=O
Thus (A  z)l is analytic in a neighbourhood of any Zo E p(A). The following useful result relates eigenvalues and eigenfunctions of A to those of j(A). Proposition 5.16 (Spectral mapping theorem) Let A be a bounded operator, and j a function analytic on a disk of radius> ItAII. If A¢ = )...¢, then j(A)¢ = j()...)¢. Proof. If {aj} are the coefficients of the power series for j, then
o
00
00
j=o
j=o
5.3 Applications to Schrodinger Operators
41
We conclude with a brief discussion of alternatives to, and extensions of, the above definition. Suppose J is analytic in a complex neighbourhood of a(A). We can replace definition (5.1) by the contour integral (called a Riesz integra0 J(A) :=
~ 1 J(z)(A 27ft
Ir
z)ldz
(5.3)
where r is a contour in C encircling a(A). The integral here can be understood in the following sense: for any 1/;, ¢ E 'H, 1. (1/;, J(A)¢) := 2 7ft
1 J(Z)(1/;, (A z)l¢)dz Ir
(knowledge of (1/;, J(A)¢) for all 1/;, ¢ determines the operator J(A) uniquely).
Problem 5.17 Show that the definition (5.3) agrees with (5.1) when J(A) is analytic on {IAI < R} with R > IIAII. Hint: by the Cauchy theorem, and analyticity of the resolvent, the contour r can be replaced by {Izl = R o}, IIAII < Ro < R. On this contour, (Az)l can be expressed as the Neumann series (5.2). A similar formula can be used for unbounded operators A to define certain functions J(A) (see [HS]). If A is an unbounded selfadjoint operator, and J is a continuous, bounded function, the bounded operator J(A) can still be defined. One example of this is the definition of e iA in Section 2.2. The definition of J(i'V) using the Fourier transform (Section 2.5.3) provides another example. The reader is referred to [RSI] for the general theory.
5.3 Applications to Schrodinger Operators In this section we address two central issues. Firstly, we want to know for which potentials, V, we can conclude that the Schrodinger operator H =  2'::..1 + V is selfadjoint. As we saw in Chapter 2, only when H is selfadjoint do we know that the quantum dynamics exist. Secondly, we wish to describe the spectrum, a(H), of H. As we shall see in the next section, knowledge of a(H) gives us important information about the nature of the solutions of Schrodinger's equation (and hence about the evolution of physical states). Our first result covers confining potentials  that is, potentials which increase to infinity with x.
Theorem 5.18 Let V(x) be a continuous function on]Rd satisfying V(x) 2: 0, and V(x) > 00 as Ixl > 00. Then 1. H
= ..1 + V is selfadjoint on L 2(JR3)
2. a(H) consists of isolated eigenvalues
{An}~=l
with An
> 00
as n
> 00.
42
5 Spectral Theory
Remark 5.19 Here and elsewhere, the precise meaning of the statement "the operator H is selfadjoint on L2(JR d)" is as follows: there is a domain D(H), with Co(JR d) c D(H) C L2(JRd), for which H is selfadjoint, and H (with domain D(H)) is the unique selfadjoint extension of .:1+ V(x), which is originally defined on Co(JRd). The exact form of D(H) depends on V. If V is bounded (a case we consider shortly), then D(H) = D(.:1) = H 2(JRd). Proof The proof of selfadjointness is fairly technical, and can be found in [HS], for example. To prove the second part, suppose ..\ is in the continuous spectrum of H, and let {'l/Jn} be a corresponding spreading sequence. Then as n 7 00,
o ~ ('l/Jn, (H 
..\)'l/Jn)
=
= ~
('l/Jn, .:1'I/Jn)
+ ('l/Jn, V'l/Jn)
J1V''l/JnI + J 2
inf
yE supp( 1/Jn)
VI'l/JnI 2
V(y) ..\
 ..\
..\
7
00
(because {'l/Jn} is spreading), which is a contradiction. Thus the continuous spectrum is empty (or consists of a point at (0). Further, there must be eigenvalues tending to +00. Indeed, let ..\l, ... ,..\n be the first n eigenvalues of the operator H (counting multiplicities), and let 'l/J1, ... ,'l/Jn be corresponding normalized eigenfunctions. In the proof of Theorem 5.28, we show that inf
{1/JE7t:k nD (H)
I 111/J11=1}
('I/J, H'I/J)
=
inf{u(H)\{..\l' ... ' ..\n}},
(5.4)
where Hn := span { 'l/J1, ... , 'l/Jn}. Since the right hand side is < inf uc(H) = 00, we conclude that (5.4) is the smallest eigenvalue of H which is ~ ..\n. Hence it is ..\n+1 (..\n+! = ..\n is allowed, but remember that the eigenfunctions for ..\n and ..\n+1 are orthogonal). By repeating this argument, and using the fact that the eigenvalues "\1, ..\2, ... have finite multiplicity, we see that H has an infinite number of eigenvalues. Since H has no continuous spectrum, these eigenvalues cannot accumulate at a finite number. Hence ..\j 7 00 as j 7 00.
o
Our next theorem covers the case when the potential tends to zero at infinity. Theorem 5.20 Let V : lR d Then
7
lR be continuous, with V(x)
70
as Ixl
700.
1. H = .:1 + V is selfadjoint on L 2(JRd) 2. uc(H) = [0,(0) (so H can have only negative isolated eigenvalues, possi
bly accumulating at 0). Proof. For a proof that H is selfadjoint (with domain D(H) = D(.:1) = H2 (JR d)), we refer the reader to [HS], for example. We prove the second part.
We have, by the triangle inequality,
5.3 Applications to Schrodinger Operators
43
Suppose {'ljln} is a spreading sequence. Then the term IIV'IjIn I goes to zero as n + 00 because V goes to zero at infinity and {'ljln} is spreading. So A is in the continuous spectrum of H iff A E O"c( ,1). We have (see Problem 5.9) O"c( ,1) = [0,00), and consequently O"c(H) = [0,00). D Next we consider a Schrodinger operator on a bounded domain, with Dirichlet boundary conditions. Theorem 5.21 Let A be a cube in ~d, and V a continuous function on A. Then the Schrodinger operator H = ,1+ V, acting on the space L2(A) with Dirichlet boundary conditions, has purely point spectrum, accumulating at
00. To be precise, the operator "H on L2(A) with Dirichlet boundary conditions" should be understood as the unique selfadjoint extension of H from Co(A). Proof Suppose A = [0, L]d. Consider the normalized eigenfunctions of the operator ,1 on L2(A) with Dirichlet boundary conditions:
(see Section 7.1), so that (5.5) Now we recall that the eigenfunctions ¢k, k 'IjI
=
L
E
L(Z+)d, form a basis for L2(A):
(¢k, 'IjI)¢k
kEicz+)d
for any 'IjI E L2(A) (this is a special case of a general phenomenon valid for selfadjoint operators). We show now that the operator H has no continuous spectrum. Assume on the contrary that A E O"c(H), and let Un be a corresponding Weyl sequence; i.e. Ilunll = 1, Un + weakly, and II(H A)unll + 0. By the triangle inequality
°
I (H since
A)Un I 2: II (,1  A)unll 
IIVunll 2: II (,1 
A)Un II
Ilunll = 1. Writing Un = La~¢k k
where a~ = (¢k, un), and using (5.5), we compute
(,1  A)U n =
L(lkl2 k
A)a~¢k

sup IVI
(5.6)
44
5 Spectral Theory
and so by the Parseval relation (Proposition 1.3) II( ,1 .\)unI1 2 = ~)lkI2  .\)2Ia~12.
(5.7)
k
The Parseval relation also gives 1
=
IIun l1 2 =
L
la~12.
(5.8)
k
Now choose K such that Ikl 2 .\ ~ V2(sup IVI +1) for Ikl ~ K. Then by (5.7) and (5.8), 11(L1.\)un I1 2 ~ 2(suplVl+1)2
L
la~12 = 2(suplVl+1)2(1
Ikl;:O:K Since
Un '>
0 weakly, a~
= ((Pk, un)
L
la~12).
Ikl 00, for each k. Choose N sufficiently large that la~1 : : : (2K#)1/2 for k with Ikl < K and n ~ N, where
Then for n ~ N, Llkl 0, there is an R such that infl t Ixl::::R To see this, note that if H'4"o
1
Ixl2:R
1'4"12 ~
1 E.
(5.9)
= )..'4"0, then e i';;t '4"0 = e i~t '4"0, and so
1'4"12 =
1
Ixl2:R
1'4"01 2 > 0
as R > 00. Such a '4" is called a bound state, as it remains essentially localized in space for all time. A proof of (5.9) in the general case is given at the end of this section. On the other hand, if
'4"0
E
{span of eigenfunctions of H} ~
then for all R,
(5.10)
5.4 Spectrum and Evolution
as t
f (t)
7 7
47
Strictly speaking, this convergence is in the sense of ergodic mean: 0 in ergodic mean as t 7 00 means
00.
T1 Jro
T
f(t)dt
7
0
as T 7 00. This result is called the Ruelle theorem (see, eg, [CFKS, HS] for a proof). Such a state, 'l/J, is called a scattering state, as it eventually leaves any fixed ball in space. We conclude that the classification of the spectrum into point and continuous parts corresponds to a classification of the dynamics into localized (bound) states and locallydecaying (scattering) states. Finally, we prove equation (5.9) in the general case. If
'l/Jo
E {
span of eigenfunctions of H},
then 'l/Jo can be written as 'l/Jo = I: j aj'l/Jj where aj E "j'l/Jj. We will assume that the above sum has only a finite number of terms, say 'l/Jo = I:f=l aj'l/Jj. Otherwise an additional continuity argument is required below. The solution 'l/J = eiHt/n'l/Jo can be written as
'l/J =
L eiAjt/naj'l/Jj N
j=l
and therefore
Applying the CauchySchwarz inequality to the sum on the right hand side, we obtain
(5.11) The first factor on the right is just
(5.12) To estimate the second factor on the right hand side, for any c > 0, we choose R such that for all j, k,
48
5 Spectral Theory
(5.13) Combining equations (5.11)(5.13) yields
and so (5.9) follows.
5.5 Variational Characterization of Eigenvalues We consider, for the moment, a selfadjoint operator H, acting on a Hilbert space H. By applying variational techniques to the "energy" functional ('l/J, H'l/J), we will derive an important characterization of eigenvalues of H. We begin with some useful characterizations of operators in terms of their spectra. Theorem 5.25 Let H be a selfadjoint operator with u(H) H 2': a (i.e. (u, Hu) 2': allul1 2 for all u E D(H)).
c [a, 00).
Then
°
Proof. Without loss of generality, we can assume a = (otherwise we can consider H  al instead of H). First we suppose H is bounded (we will pass to the unbounded case later). If b > IIHII, then the operator (H + b)l is positive, as follows from the Neumann series
(H
+ b)l =
b 1 (1
+ b 1 H)l
'2) _b00
=
b 1
1 H)n
j=O
(see (5.2)). For all >. > 0, the operator (H + >.)1 is bounded, selfadjoint, and differentiable in >. (in fact it is analytic  see Section 5.2). Compute
a
Hence for any
°
. (H + >.)1 c
< b,
=
(H + >.)2 < 0,
5.5 Variational Characterization of Eigenvalues
49
and therefore (H +C)I > (H +b)I > O. Now any U E D(H) can be written in the form u = (H + C)Iv for some v E 1i (show this). Since
(u, (H + c)u)
=
(v, (H
+ C)IV) > 0
for any c > 0, we conclude that (u, Hu) 2: 0 for all u E D(H), as claimed. In order to pass to unbounded operators, we proceed as follows. Let c > 0 and A := (H + c)I, a bounded operator since c rf'. (J(H). For any'\ =I 0, we have (5.14)
Hence for ,\ > 0, the operator A + ,\ is invertible, and so (J(A) C [0,00) (in fact, one can see from (5.14) that A + ,\ is also invertible if ,\ < cI, and so (J(A) C [0, c I]). By the proof above, A := (H + C)I 2: O. Repeating the argument at the end of this proof, we find that H + c 2: O. Since the latter is true for any c > 0, we conclude that H 2: o. D. Theorem 5.26 Let S('lj;) := ('lj;, H'lj;) for 'lj; E D(H) with 11'lj;11 = 1. Then inf (J(H) = inf S. Moreover, ,\ := inf (J(H) is an eigenvalue of H iff there is a minimizer for S('lj;) among'lj; E D(H) with the constraint II 'lj; II = 1.
Proof In the proof of Theorem 2.21, take z E lR satisfying z < inf S to conclude that z rf'. (J(H). Therefore inf S ::::: inf (J(H). Now let ,\ := inf (J(H). By Theorem 5.25, ('lj;, H'lj;) 2: '\11'lj;112 for any'lj; E D(H). Hence inf S 2: ,\ = inf (J(H), and therefore inf S = inf CJ(H) as required. Now if ,\ = inf (J(H) is an eigenvalue of H, with normalized eigenvector 'lj;o, then S( 'lj;o) = ('lj;o, H'lj;o) = ,\ = inf S, and therefore 'lj;o is a minimizer of S. On the other hand, if'lj;o is a minimizer of S, among 'lj; E D(H), 11'lj;11 = 1, then it satisfies the EulerLagrange equation (see Section 12.3)
S'('lj;o)
=
2'\'lj;o
for some ,\. Since S'('lj;) = 2H'lj; (see Section 12.2), this means that 'lj;o is an eigenvector of H with eigenvalue ,\. Moreover,
Since S('lj;o) = inf S = inf (J(H), we conclude that ,\ = inf (J(H) is an eigenvalue of H (with eigenvector 'lj;o). 0 This result leads us to the Ritz variational principle: for any'lj; E D(H),
('lj;,H'lj;) 2:'\ = inf(J(H) and equality holds iff H'lj; = '\'lj;.
50
5 Spectral Theory In particular, if we can find a'ljJ (called a test function) with 11'ljJ11
= 1 and
('ljJ, H'ljJ) < inf O"c(H), then we know that H has at least one eigenvalue below its continuous spectrum. Indeed, by Theorem 5.26, inf O"(H)
= inf S < ('ljJ, H'ljJ) < inf O"c(H),
so A = inf O"(H) must be an eigenvalue of H. Example 5.27 As an example, consider the Schrodinger operator describing the Hydrogen atom n,2 e2 H=.d2m Ixl acting on the Hilbert space L 2(JR3) (see Section 4.3). Take the (normalized) test function 'ljJ(x) = Vft3/7felllxl for some ft > to be specified later and compute (passing to spherical coordinates)
°
To compute the above integrals, note
and
1
1
d earrdr =  eardr. da 0 Since fooo eardr = aI, we find fooo e ar r 2 dr = 2a 3 , and fooo earrdr = a 2. Substituting these expressions with a = 2ft into the formula for ('ljJ, H'ljJ), we obtain 00
00
o
!i2
('ljJ, H'ljJ) = 2mft2  e2ft. The right hand side has a minimum at ft
=
me 2/!i2, which is equal to
Since O"c(H) = [0, (0) (according to Theorem 5.20, which can be extended to cover singular potentials like the Coulomb potential  see the remark in Section 5.6) we conclude that H has negative eigenvalues, and the lowest negative eigenvalue, AI, satisfies the estimate
5.5 Variational Characterization of Eigenvalues
51
me4 21t
Al < . 2

This should be compared with the lower bound
found in Section 4.3. The variational principle above can be extended to higher eigenvalues. Theorem 5.28 (Minmax principle) The operator H has at least n eigenvalues (counting multiplicities) less than inf uc(H) iff An < inf uc(H), where the number An is given by
An =
inf max ('IjJ, H'IjJ). I dimX=n} {.pEX 111.p11=1}
{XCD(H)
(5.15)
In this case, the nth eigenvalue (labeled in nondecreasing order) is exactly
An. Sketch of proof. We prove only the "if" part of the theorem. The easier "only if" part is left as an exercise. We proceed by induction. For n = 1, the statement coincides with that of Theorem 5.26. Now assume that the "if" statement holds for n ::; m  1, and we will prove it for n = m. By the induction assumption, the operator H has at least m  1 eigenvalues, AI, ... ,Aml (counting multiplicities), all < infuc(H). We show that H has at least m eigenvalues. Let Vm  l denote the subspace spanned by the (nc:irmalized) eigenvectors 'ljJ1, ... , 'ljJml, corresponding to AI, ... , AmI' Then 1. the subspace Vm 
l , and its orthogonal complement, V~_l' are invariant under the operator H, and 2. the spectrum of the restriction, HlvL ,of H to the invariant subspace =1
V~_l is U(H)\{Al,"" AmI}.
Problem 5.29 Prove statements 1 and 2. Now apply Theorem 5.26 to the operator HlvL
=1
inf
{.pEV';;_l nD (H)
111.p11=1}
('IjJ, H'IjJ)
=
to obtain
inf{ u(H)\ {AI, ... ,AmI}}.
(5.16)
On the other hand, let X be any ndimensional subspace of D(H). There exists ¢ E X such that ¢ ~ Vm  l , and II¢II = 1. We have,
(¢, H ¢)
~
inf
{.pEv';;_lnD(H)
and therefore Am defined by (5.15) obeys
111.p11=1}
('IjJ, H'IjJ)
52
5 Spectral Theory Am~
Since Am
inf
{,pEv';;_lnD(H)
11;111I>11=1}
(1/;,H1/;).
(5.17)
< inf aAH) by assumption, and due to (5.16), we have (5.18)
is the mth eigenvalue of H. Moreover, Equations (5.16)(5.18) imply that
Am
~
A;".
Now we show that Am ::; A;". Let 1/;m be a normalized eigenvector corresponding to A;", and let Vm = span{ 1/;1, ... ,1/;m}. Then
Hence Am = A;". Thus we have shown that H has at least m eigenvalues (counting multiplicities) < inf CYc(H) , and these eigenvalues are given by (5.15). 0 This theorem implies that if we find n independent (normalized) vectors (PI, ... ,11=1}
{XCD(H)
A proof of this theorem is similar to the proof above. The following useful statement is a simple consequence of the minmax principle. Suppose A and B are selfadjoint operators with A ::; B. Denote the jth eigenvalue of A below its continuous spectrum (if it exists) by Aj(A) (and similarly for B). Suppose also that the eigenvectors of B corresponding to the eigenvalues A1(B), ... , Aj(B) lie in D(A). Then Aj(A) ::; Aj(B). To see this, let Vj denote the span of the first j eigenvectors of B, and observe that max{,pEVj 1111I>11=1} (1/;, B1/;) = Aj(B). Since Vj c D(A), (5.15) gives A (A) < J

< 
max (.1, A·I,) 1111I>1I=1} ,/" '/'
{1I>EVn
max (.1, B·I,) = A (B). 1111I>11=1} ,/" '/' J
{1I>EVn
5.6 Number of Bound States
53
5.6 Number of Bound States Let H
V(x)
=  :;: ~ + V (x) + 00
as
Ixl
+ 00.
be a Schrodinger operator acting on L2 (JR3). Assume In this section, we address the questions
 Does H have any bound states?  If so, how many bound states does it have? We begin with a very simple example. Suppose that V(x) ~ O. Clearly 0, and so H certainly has no negative eigenvalues. On the other hand, <Jc(H) = [0(0) by Theorem 5.20. So H has no isolated eigenvalues. The next example is physically clear, but mathematically more subtle: H
~
If
X· \7V(x) ::::: 0, i.e. V(x) is repulsive, then H has no eigenvalues.
Let us assume here that V is twice differentiable, and that V(x), together with X· \7V(x), vanish as Ixl + 00. The proof of the above statement is based on the following virial relation: define the selfadjoint operator 1
A := "2(x, p
+ p. x).
This operator is the generator of a oneparameter group of unitary transformations called dilations:
for () E R Let us show formally (i.e. ignoring domain issues) that if'¢ is an eigenfunction of H, then (,¢, i[H, A]'¢) = O. (5.19) Letting A be the eigenvalue corresponding to ,¢, and using (4.1), we have
(,¢, i[H, A]'¢) = (,¢, i[H  A, A]'¢) = 2Im((H  A),¢, A'¢). Since (H  A)'¢ = 0, (5.19) follows. On the other hand, a simple computation (left as an exercise) yields i[H, A]
11,2
= ..)l(x,y) 2m
=
m 2nn2 1x
 yl
/2"'1>11
e Y ~
xy
1
(see Equation (2.22) of Section 2.5.3), we find that the integral kernel for
K(>..) is
5.6 Number of Bound States
57
(see the mathematical supplement, Section 5.7). Using the equation
IIKII:s;
(
r
J'iiI!.3 X 'iil!.3
1/2
IK(x, yWdxdy )
(again, see Section 5.7), we obtain
Since exp( J2rr;Y·llx  yl) ~ 0 as .A ~ 00, Equation (5.27) follows. Now we show that the relations (5.26) and (5.27) imply (5.25). By (5.27), for all .A sufficiently negative, all of the eigenvalues of K(.A) are less than 1. By (5.26), the eigenvalues, vm(.A), of K(.A) increase monotonically with .A. Hence if vm(.A m ) = 1 for some .Am < 0, then vm(O) > vm(.A m ) = 1. Similarly, if vm(O) > 1, then there is a .Am < 0 such that vm(.A m ) = 1. In other words, there is a onetoone correspondence between the eigenvalues vm(O) of K(O) which are greater than 1, and the points .Am at which some eigenvalue vm(.A) crosses 1 (see Fig. 5.3).
Fig. 5.3. Eigenvalues of K(.A). Thus (5.25) follows. The relations (5.24) and (5.25) imply
#{.A < 0
I
.A EV H} = #{v > 1
I v EV K(O)}.
(5.28)
The quantity on the left hand side of (5.28) is what we would like to estimate, while the quantity on the right hand side is what we can estimate. Indeed, we have
58
5 Spectral Theory
L
#{v> 1 I v EV K(O)} =
1:::;
: :;
v;'
)
v m > 1,v",EV K(O)
v m>l,vm EV K(O) 1/2
L
(
1/2
L
(
v;'
)
=
(tr(K(0)2)) 1/2
vmEVK(O)
(5.29) (see Section 9.14 for the definition of tr, the trace). On the other hand (see Section 5.7),
tr(K(0)2) = / IK(O)(x, y)1 2dxdy = (2:n2) 2 /
~~x2~j;) dxdy
Collecting equations (5.28) (5.30) and recalling that V(x) obtain
#{,\ < 0 I ,\ EV H} :::;
~ 27rn2
U(x), we
( / JV(x)V(y)1 dXdY) 1/2
Ix 
Yl2
Under our assumption (5.23) on the potential V(x), with a
/
=
(5.30)
JV(x)V(y)1 d d X Y < Ixy 12
> 2,
00,
so that the number of negative eigenvalues of the operator H is finite. This is the fact we set out to prove. The argument used above (Equation (5.28) in particular) is called the BirmanSchwinger principle, and the operator K('\) is the BirmanSchwinger operator.
5.7 Mathematical Supplement: Integral Operators Let K be an integral operator on L 2 (lR,d):
(K¢)(x):= where K : lR,d x lR,d include
+
r K(x, y)¢(y)dy
JlRd
C is the integral kernel of the operator K. Examples
1. K = g( ih,\!) for which the kernel is
K(x, y) = (27rn)d/2g(x  y) (see Section 2.5.3).
(5.31 )
5.7 Mathematical Supplement: Integral Operators
2. K
=
59
V (multiplication operator) for which the kernel is
K(x, y) = V(x)J(x  y). The following statement identifies the kernel of the composition of integral operators. The proof is left for the reader. Proposition 5.33 If K1 and K2 are integral operators (with kernels K1 and K 2), then the integral kernel of K := K1K2 is
Problem 5.34 Prove this.
Recall (see Problem 2.18) that if the kernel K(x, y) satisfies K(y, x) = K(x, y), then the corresponding integral operator K is symmetric. Proposition 5.35 Let K be an integral operator with kernel, K(x, y), satisfying K(x,y) = K(y,x), and which lies in L2 of the product space: K (x, y) E L2 (JRd XJRd). Then K is a bounded, selfadjoint operator on L2 (JRd) , and (5.32) Proof To show that the operator K is bounded, we estimate by the CauchySchwarz inequality,
IJ K(x, Y)U(Y)dyl ::; (J IK(x, YWdY) (J lu(y)12dY) 1/2 1/2
This implies
IIKuI12::;
J
IK(x, y)1 2dxdy
J
lu(y)1 2dy
which in turn yields (5.32). Since K is bounded and symmetric, it is selfadjoint by Lemma 2.19. 0
6 Scattering States
In this chapter we study scattering states in a little more detail. As we saw in Section 5.4, scattering states are solutions of the timedependent Schrodinger equation
in a7/J at
= H 7/J
(6.1)
with initial condition orthogonal to all eigenfunctions of H: (6.2) where Hb := span{ eigenfunctions of H} is the subspace of bound states of H. We will have to make a more precise assumption on the potential V(x) entering the Schrodinger operator H =  ; : L1 + V (x): we assume, for simplicity, that la~V(x)1 ::::: 0(1 + Ixl)JL1a l (6.3) for lal ::::: 2 and for some J.L > 0 (and 0 a constant). The notation needs a little explanation: a is a multiindex a = (aI, a2, a3) with each aj a nonnegative integer, lal = L~=l aj, and 3
aax = II aaj
Xj'
j=l
The question we want to address is what is the asymptotic behaviour of the solution 7/J = e iHt / li 7/Jo of (6.1)(6.2) as t + 00. First observe that (5.10) shows that 7/J moves away from any bounded region of space as t + 00. Hence, since V(x) + 0 as (6.4) Ixl + 00, we expect that the influence ofthe potential V(x) diminishes as t + 00. Thus the following question arises: does the evolution 7/J approach a free evolution, say cP = eiHot/licpo, where Ho =  ;: L1, for some CPo E L2(JR.3), as t + 007 Put differently, given 7/Jo E is there CPo E L 2(JR.3) such that
Ht,
(6.5)
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
62
6 Scattering States
as t + oo? This (or a modification of this question discussed below) is the problem of asymptotic completeness. It is the central problem of mathematical scattering theory. It conjectures that all the possible free motions, together with the bound state motions, form a complete set of possibilities for the asymptotic behaviour of solutions of timedependent Schrodinger equations. There are two principal cases depending on the decay rate, /L, of the potential V(x) (more precisely, the decay rate, /L + 1, of the force \i'V(x)), and it is only in the first of these cases that the asymptotic completeness property formulated above holds. We now discuss these cases in turn.
6.1 Shortrange Interactions: J.L
>
1
In this case, asymptotic completeness can be proved under condition (6.3). We can reformulate the asymptotic completeness property by defining the wave operator, n+: n+ ¢:= lim eiHt/lieiHot/Ii¢. t+oo
(6.6)
Below we will show that under the condition (6.3) with /L > 1, the limit exists for any ¢ E L 2(JEt3). Further, the operator n+ is an isometry: (6.7) Indeed, (6.6) implies
Iln+ ¢II = t+oo lim IleiHt/lieiHot/Ii¢11 = II¢II since the operators eiHt / 1i and eiHot/1i are isometries. The existence of the wave operator n+ means that given a free evolution eiHot/li¢o, there is a full evolution eiHt/Ii'I/Jo such that (6.5) holds. To see this, note that since eiHt / 1i is an isometry, (6.5) can be rewritten as
as t
+ 00,
which is equivalent to the relation 'l/Jo
=
n+¢o.
(6.8)
Thus the existence of the wave operator n+ is equivalent to the existence of scattering states  i.e. states eiHt/Ii'I/Jo for which (6.5) holds for some ¢o. We have Ran(n+) c Indeed, for any ¢o E L 2(JEt3) and 9 E Jib such that Hg = )..g, we have
Hr.
(g, n+¢o)
= lim (g, eiHt/lieiHot/li¢o) = lim (eiHt/lig, eiHot/li¢o) t~oo
t+oo
(6.9)
6.1 Shortrange Interactions:
JL
>1
63
There is a general theorem implying that the right hand side here is zero, but we will prove it directly. Recalling expression (2.15) for the action of the free evolution operator eiHot/li, we find
and therefore, if ¢o and 9 are integrable functions, i.e.,
r l¢o(x)ldx
 Qt' ¢> as the integral of its derivative: Qt¢> _ Qt' ¢> = .!!:....Qs¢>ds. ds Using the relation d~eiHs = iH e iHs , and similarly for eiHos (see Chapter 2), we find
rt
it'
as H  Ho
= V. Since IleiHsl1 = 1, and using the inequality I
J ¢>(s)dsll
::;
J II ¢>( s) II ds (which can be proved by writing the integral as a limit of Riemann
sums and using the triangle inequality  see [Fo]), we have
(6.12)
66
6 Scattering States
We estimate the integrand as follows:
IlVeiHos¢112 =
r lV(x) (eiHOS¢) (x)1 2dx (eiHOS¢) (X'W r lV(x)1 2dx, E1R JlR
JlR 3
:::: sup I x/
yielding
IlVeiHos¢11 ::::
3
3
11V11£2 sup l(eiHos¢)(x')I· x'
Finally, recalling the bound (see (2.16))
from Section 2.5.3, we obtain
and so
1
00
IlVeiHos¢llds::::
(const)IIVIIL211¢11£1·
This shows that the right hand side in (6.12) vanishes as t', t + 00. In other words, for any sequence tj + 00, {ntj ¢} is a Cauchy sequence, and so {nt¢} converges as t + 00. Convergence for t + 00 is proved in the same way. 0
7 Special Cases
In this chapter we will solve the Schri:idinger eigenvalue equation in a few special cases (i.e., for a few particular potentials) which not only illustrate some of the general arguments presented above, but in fact form a basis for our intuition about quantum behaviour.
7.1 The Infinite Well Let W be the box [0, Lj3 C ~3. We take V(x) = { 0 x
E W ()()X~W
as our potential. This means we take '¢ == 0 outside W, and that we impose Dirichlet boundary conditions
(7.1)
'¢i&w = 0
on the wave function inside W. It is a simple matter to solve the eigenvalue equation
(7.2) in W with the boundary condition (7.1), using the method of separation of variables. Doing so, we obtain eigenvalues (energy levels) >02
En
=
2
3
7r """" 2 2mL2 ~nj It
)=1
with corresponding eigenfunctions (bound states) 3
'¢n(x) = IIsin(7rn j Xj)
.
)=1
L
for each integer triple n = (nl' n2, n3), nj 2:: 1. We see that the eigenvalue En occurs with degeneracy equal to #{(ml, m2, m3)i L m; = L nj}. We remark that the groundstate (lowest) energy
E(1,I,I) =
~~Z~ is nondegenerate.
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
68
7 Special Cases
Problem 7.1 Determine the spectrum of the operator x and the eigenvalues of the operator p on L2(W) with zero boundary conditions.
7.2 The Torus Now we consider a particle on a torus T = ]R3 I'll}. This corresponds to taking V == 0 in the cube W, but this time imposing periodic boundary conditions. That is, we solve the eigenvalue equation (7.2) with boundary conditions 1jJ(x)IXj=o fJ1jJ I fJxk IXj=o
= 1jJ(X)lxj=L
= fJ1jJ I fJxk IXj=L
for all j, k. This leads to (separation of variables again) eigenfunctions 1jJn(x)
= II 3
j=l
j Xj)} { S . (2nn lll
cos
(2n!;:jXj) L
with eigenvalues En
=
v L n; m 2n 2 li 2
3
1
for nj 2: O. The ground state energy, E(o,o,o) = 0 is nondegenerate, with eigenfunction 1jJ(O,O,O) == 1. The spacing between energy levels is greater than for the infinite well, and the degeneracy is higher. Problem 7.2 Determine the spectrum of x, and the eigenvalues of p, on L2(W) with periodic boundary conditions. Challenge: show that p has no
continuous spectrum (hint  use the Fourier transform for periodic functions).
7.3 A Potential Step Here we examine the onedimensional potential V(x)
=
{~
x>O x:S;O
with Vo > O. Consider the "eigenvalue" problem _!f.1jJIf + V1jJ 2m
= E1jJ.
(7.3)
We put the term eigenvalue in quotation marks because we will allow solutions, 1jJ, which are not L2functions. Solving this eigenvalue problem separately in the two different regions gives us a general solution of the form
7.4 The Square Well
'Ij;
= {
69
(n2k5 =E) x 0
There are no bound states (L2 solutions), but we can say something about the scattering states. Suppose 0 < E < Vo, and take kl = iK where K = v(2m/!i2)(Vo  E) > O. Then for a bounded solution, we require D = O. Imposing the condition that 'Ij; be continuously differentiable at 0, that is
leads to the equations
iko(A  B) = KG.
A+B=G,
After some manipulation, we find that
B A
ko  iK ko + iK'
or making the dependence on the energy E explicit,
B A Similarly, if E
v'E  iJVo v'E + iJVo 
E E'
> Vo, we obtain B A
and
R(E) :=
v'EJEVo v'E+JEVo'
1~ 12 = ~2 [2E 
Vo  2VEVE  Vo]2
(R(E) is called the reflection coefficient). In particular, if R(E)
~ 1
EvoVo
«
1, then
4J E  Vo Vo
and almost all of the wave is reflected. This is in spite of the fact that the energy of the particle lies above the barrier. In classical mechanics, the particle would pass over the barrier.
7.4 The Square Well Now we consider a potential well of finite depth Vo, and width a (see Fig. 7.1).
70
7 Special Cases V(x)
aJ2
aJ2 x
Fig. 7.1. The finite well. The determination of the point and continuous spectra is straightforward, and is left as an exercise. Problem 7.3 Show
1. CTc(H) 2.
= [0,00)
CTp(H) c ( Vo, 0)
3. the equations for the eigenvalues E, for  Vo ::; E ::; 0, are ktan(ak/2) = K
kcot(ak/2) = K
where
We study the scattering states  i.e. solutions of (7.3) for E > 0 which converge to plane waves as x + ±oo  in more detail. Consider the situation where a plane wave 'l/Jinc(X) = Aeikx is incoming from the left. Then on the left of the well, 'l/J is a superposition of incoming and reflected plane waves:
'l/J = Aeikx + AB(E)e ikx
x < a/2
while on the right of the well, 'l/J is an outgoing (transmitted) plane wave:
'l/J = AC(E)eik(xa)
x > a/2
Problem 7.4 Show that
C(E)
=
_
[cos(ka)  i
p + k2 2kk
where and
_
sin(ka)r 1
k =
v
2mE 11,2 •
7.5 The Harmonic Oscillator
71
This implies
(T(E) is the transmission coefficient) which is sketched in Fig. 7.2.
0+Fig. 7.2. Transmission coefficient. We see that at the energies satisfying sin(ka) n 2 rr2fi2
E
=  Vo + 2ma 2 > 0,
= 0,
i.e.
n = 1,2, ...
T(E) has maxima (T(E) = 1) which are called resonances. The corresponding values of E are the resonance energies. We remark that for large n these are approximately equal to the energy levels of the infinite well of the same width.
7.5 The Harmonic Oscillator The harmonic oscillator Hamiltonian is H
fi2
1
2m
2
2
2
= .::1+ mw Ixl .
By Theorem 5.18, a(H) consists of isolated eigenvalues, increasing to infinity. We will solve the eigenvalue problem explicitly for this operator. For simplicity, we consider the onedimensional case. First, to remove all the constants we rescale: x f+ AX so that p f+ where p = ifi(djdx). Setting
A=
a
tP
gives H
f+
fiwHnew where
72
7 Special Cases
with pnew = i(d/dx). The number). gives the length scale in the original problem (the scale on which the eigenfunctions of the Hamiltonian Hare concentrated) . We derive a representation of the operator H which facilitates its spectral analysis. It also prepares us for a similar technique we will encounter in the more complex situation of quantum field theory (see Chapter 14). We introduce the creation and annihilation operators
The commutation relation [a, a*] is easily verified. The Hamiltonian a* as follows:
=1
Hnew
H new
(7.4)
can be rewritten in terms of a and 1 2
= a*a + .
We say that this expression is in nO'T'mal form because a* appears to the left of a. The particle number operator, N:= a*a satisfies the relations Na=a(N1) Na*
= a*(N + 1)
(7.5) (7.6)
(use (7.4) to check these). Clearly,
Theorem 7.5 We have 1. N
~
0
2. IJ(N) = Z+ (the nonnegative integers), each eigenvalue having multiplicity 1. Proof.
1. This is easy, because a* is the adjoint of a, so
for any 'Ij;.
7.5 The Harmonic Oscillator
73
2. Therefore, N'IjJ = 0 iff a'IjJ = O. Note that the function
'ljJo(x) := ce x2 / 2 (c a constant) is the unique family of solutions of a'IjJ
=
1
..j2(x + d/dx)'IjJ
= 0,
and hence of N'IjJ = O. Thus 'ljJo (normalized by setting c := (27r)1/4) is the ground state. The commutation relation (7.6) implies
Na*'ljJo
= a*'ljJo
and in general Thus
¢n := (a*)n'IjJo is an eigenfunction of N with eigenvalue n.
Problem 7.6 Show that II¢n11 2 = n!. Hint: write II¢n11 2 = ('ljJo, an (a*)n'IjJo). then push the a's through the a*'s (including the necessary commutators) until they hit 'ljJo and annihilate it. So
'ljJn:=
~(a*)n'IjJo
vn!
is a normalized eigenfunction of N with eigenvalue n. We now show that these are the only eigenfunctions. It follows from the commutation relations that if 'IjJ is any eigenfunction of N with eigenvalue A > 0, then
(7.7) If we choose m so that A  m But this implies
< 0 we contradict N
~
0 unless am'IjJ
= O.
aj'IjJ = c'IjJo for some integer j (c a constant), so by (7.7), A = j. Applying (a*)j to this equation, and using the commutation relations, we can show 'IjJ = c'IjJj (c another constant), so we are done. D
Corollary 7.7
cr(Hnew) = {n + 1/21 n = 0,1,2, ... } with eigenfunctions 'ljJn
= (l/Vnl)(a*)n'IjJo.
Finally, then, the spectrum of the original harmonic oscillator Hamiltonian is cr(H) = {1lw(n + 1/2) 1n = 0,1,2, ... } with eigenfunctions obtained by rescaling the 'ljJn's.
74
7 Special Cases
7.6 A Particle on a Sphere We consider a particle moving on the unit sphere §2 = {x E ~3 I Ixl = I} in ~3. Its Hamiltonian is ;~Lln, the LaplaceBeltrami operator on §2, given in spherical coordinates (B, rjJ), by
The eigenfunctions of Ll n are the wellknown spherical harmonics, (7.8) where l = 0,1, ... ; k E {l, l + 1, ... ,l  1, l}; Czk is a constant; and the Legendre function FZk can be written as (7.9) The spherical harmonics satisfy
LlnY/ = l(l Problem 7.8 bit tedious).
+ l)Y/.
(7.10)
Check (7.10) using (7.8) and (7.9) (unfortunately, this is a
yt,
It turns out that the spherical harmonics, comprise an orthonormal basis of the Hilbert space L2(§2; dD) of L2 functions on the sphere, with the measure dD = sin2(B)dBdrjJ (see, eg, [LL]). We make a few remarks about the connection to angular momentum. The quantummechanical angular momentum L = (L1' L 2, L 3 ) is the selfadjoint (vectorvalued) operator L =x xp where p = ifi,\! as usual. We define also the squared magnitude of the angular momentum, L2 = Li+L§+L§. The following facts are easily checked: 1. L 2Yzk = fi 2l(l + l)Yi k 2. L 3Yzk = fikYik.
Thus we see that the spherical harmonics are simultaneous eigenfunctions of the angular momentum operators L3 and L2.
7.7 The Hydrogen Atom A hydrogen atom consists of a proton and an electron, interacting via a Coulomb force law. Let us make the simplifying assumption that the nucleus
7.7 The Hydrogen Atom
75
(the proton) is infinitely heavy, and so does not move. Placing the nucleus at the origin, we have the electron moving under the influence of the external potential V(x) = e 2 flxl, where e is the charge of the proton, and e that of the electron. The appropriate Schrodinger operator is therefore
lt2 H = .1  e 2 flxl 2m acting on the Hilbert space L 2(JR3). In Chapter 8 we will see how to reduce the problem of the more realistic hydrogen atom  when the nucleus has a finite mass (a twobody problem)  to the problem studied here. As usual, we want to study the spectrum of H. The first step is to invoke Theorem 5.20. As remarked in Section 5.6, Theorem 5.20 can be extended to cover the Coulomb potential V(x) = _e 2 flxl, which is singular at the origin (see, eg, [CFKS]). We may conclude that H is selfadjoint, with continuous spectrum equal to the halfline [0,(0). Our goal, then, is to find the boundstates (eigenfunctions) and boundstate energies (eigenvalues). It is a remarkable fact that we can find these explicitly. Indeed, aside from the infinite well, the only multidimensional potentials for which the Schrodinger eigenvalue problem can be solved explicitly are the harmonic oscillator and the Coulomb potential. Because the Coulomb potential is radiallysymmetric (depends only on r = lxI), it is natural to work in spherical coordinates (r, (), ¢), where Xl
= rsin(()) cos(¢),
a :::; () < 7r, a :::; ¢ < 27r.
X2
= rsin(()) sin(¢),
X3
= rcos(()),
In spherical coordinates, the Laplacian becomes
where .1r is the "radial Laplacian" given by
.1r
[)2
=~ ur
2 [)
+;:)
r
Ur
(.1 r depends only on the radial variable), and .1 n is the LaplaceBeltrami operator on §2, introduced in Section 7.6. To solve the eigenvalue problem, we seek eigenfunctions of H in the separatedvariables form 1jJ(r, (), ¢)
= R(r)Y/((), ¢)
where Yzk is a spherical harmonic. Plugging this into the eigenvalue equation H 1jJ = E1jJ, we obtain
(7.11 )
76
7 Special Cases
The solutions of the ODE (7.11) are wellstudied (see, eg, [LL]). Without going into details, we remark that one can show (by powerseries methods) that (7.11) has squareintegrable solutions only for
n:=
e Fm nV2if E {l+1,l+2, ... }. 2
The corresponding eigenfunctions,
are of the form
Rnl
where p = 2::;,~2 r, and Fnl is a polynomial. In full, then, the solutions of the eigenvalue problem H'Ij; = E'Ij; are
where
l=0,1,2, ... ;
kE{l,~l+l,
... ,l};
nE{l+1,l+2, ... };
and the eigenfunctions are (7.12) So we see that the Hydrogen atom has an infinite number of bound states below the continuous spectrum (which starts at zero), which accumulate at zero (this result was obtained in Section 5.6 by a general technique, without solving the eigenvalue problem). The ground state energy, attained when l = k = 0, n = 1, is El = me 4 /21l? An easy count finds the degeneracy of the energy level En to be nl
2)2l + 1) = n 2 . l=O
Finally, we note that the expression (7.12) is in agreement with the empirical formula ("Balmer series") 1
1
l1E = R(  ).
n}
n;
Here 1 ::; nf < ni are integers labeling the final and initial states ofthe atom in a radiation process, R is a constant, and l1E is the difference of the two energy levels. This formula predates quantum mechanics, and was based on measurements of absorption and emission spectra.
7.8 A Particle in an External EM Field
77
7.8 A Particle in an External EM Field Now we extend our quantization procedure to the case of a charged particle in an external electromagnetic field. Of course, if the external field is purely electric, then it is a potential field, and fits within the framework we have considered already (as we saw in Section 7.7). Suppose, then, that a magnetic field B, and an electric field, E, are present (and are timeindependent: B, E : ]R3 f> ]R3). We know from the theory of electromagnetism (Maxwell's equations) that these fields can be expressed in terms of the vector potential, A ]R3 f> ]R3, and the scalar potential, ifJ : ]R3 f> ]R via
E=V'ifJ
B
=
curlA
(we are using units in which the speed of light, c, is equal to one). According to our general quantization procedure, we write the classical Hamiltonian function for a particle of charge e subject to the fields E and B, 1 2m
hex, k) = (k  eA(x))2
+ eifJ(x)
and then replace the classical canonical variables x and k with the quantum canonical operators x and p. The resulting Schrodinger operator is 1 2m
H(A, ifJ) = (p  eA?
+ eifJ
acting on L 2 (]R3). We remark that the selfadjointness of H(A, ifJ) can be established by using Kato's inequality (see [CFKS]). An important feature of the operator H(A, ifJ) is its gauge invariance. We recall that in the theory of electromagnetism, the vector potential A is not uniquely determined by the magnetic field B. In fact, if we add the gradient of any function X to A (a gauge transformation), we obtain the same magnetic field B:
curl(A + V'X) = curlA = B.
Gauge invariance of the quantum Hamiltonian H(A, ifJ) is reflected in the relation H(A + V'X,ifJ) = eieX/IiH(A,ifJ)eiex/li. (7.13)
Problem 7.9 Check that equation (7.13) holds. Thus if A and A differ by a gradient vectorfield, then the operators H(A, ifJ) and H(A, ifJ) are unitarily equivalent via the unitary map
on L 2 (]R3). Thus the two Hamiltonians are physically equivalent. Of course, this is to be expected as A and A correspond to the same magnetic field.
78
7 Special Cases
One can impose restrictions (called gauge conditions) on the vector potential A in order to remove some, or all, of the freedom involved in the choice of A. A common choice is divA = 0, known as the Coulomb gauge. By an appropriate gauge transformation, the Coulomb gauge can always be achieved. We now consider an important special case  a constant magnetic field with no electric field present. A possible choice for A is 1
2B x x.
A(x) =
(7.14)
Another possibility, supposing B to be directed along the (0,0, b)) is A(x) = b( X2, 0, 0).
X3
axis (B
=
(7.15)
Problem 7.10 Check that both (7.14) and (7.15) yield the magnetic field B, and that the two are gaugeequivalent. Using the second choice for A, the appropriate Schrodinger operator is 1
H(A) = [(PI 2m
2 + ebx 2) 2 +2 P2 + P3l·
To analyze H(A), we apply the Fourier transform to only the first and third variables (XI,3 f+ k l ,3). This results in the unitarily equivalent operator 1 2 mw 2 H = 2mP2 + 2(X2
1
2
1
2
+ eb kl ) + 2m k3
where w = eb/m and kl' k3 act as multiplication operators. We remark that acts as a harmonic oscillator in the variable X2, and as a multiplication operator in kl and k 3 • In the following problem you are asked to determine the spectrum of this operator.
if
Problem 7.11 1. Show that the energy levels of
iI
(called Landau levels) are given by 1
a2 2m
(n+)11w+2
where n = 0,1,2, ... and a eigenfunctions are
E
JR. Show that the corresponding generalized
where ¢n is the nth eigenfunction of the harmonic oscillator. 2. Analyze the same problem in two dimensions, with the magnetic field perpendicular to the plane.
8 Manyparticle Systems
In this chapter, we extend the concepts developed in the previous chapters to manyparticle systems. Specifically, we consider a physical system consisting of N particles of masses m1, ... , mN which interact pairwise via the potentials Vij(Xi  Xj), where Xj is the position of the jth particle. Examples of such systems include atoms or molecules  i.e., systems consisting of electrons and nuclei interacting via Coulomb forces.
8.1 Quantization of a Manyparticle System According to our general framework, we begin with the classical Hamiltonian formulation of an Nbody system. The appropriate phasespace is lR~N x lR~N where x = (X1, ... ,XN) are the particle coordinates, and k = (k1, ... ,kN) are the particle momenta. The Hamiltonian function is 1
N
h(x, k) = "
 kJ2
~2m·
+ V(x)
J
j=1
where V is the total potential of the system, given in this case by
(8.1) The quantization proceeds in the standard way by replacing lR~N x lR~N with L2(lR~N), x with the multiplication operator x, k with p = ifi'Y x, and h(x, k) with the operator HN = h(x,p). Explicitly, the Schrodinger operator HN is HN
=
N
1 2m PJ
j=1
J
L
+ V(x)
(8.2)
acting on L 2 (lR 3N ). Here Pj := ihV xj • Example 8.1 Consider a molecule with N electrons of mass m and charge e, and M nuclei of masses mj and charges Zje, j = 1, ... , M. In this case, the Schrodinger operator, Hmoz, is
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
80
8 Manyparticle Systems (8.3)
acting on £2(JR3(N+Ml). Here x = (Xl, ... , XN) are the electron coordinates, Y = (YI, ... YM) are the nucleus coordinates, Pj = ifi\l Xj is the momentum of the jth electron, qj = ifi\lYj is the momentum of the jth nucleus, and
is the sum of Coulomb interaction potentials between the electrons (the first term on the r.h.s.), between the electrons and the nuclei (the second term), and between the nuclei (the third term). For a neutral molecule, we have M
LZj
=
N.
j=l
If M = 1, the resulting system is called an atom, or Zatom (Z = ZI). Since nuclei are much heavier than electrons, in the leading approximation one can suppose that the nuclei are frozen at their positions. One then considers, instead of (8.3), the Schr6dinger operator 1
HBO
= 2m
N
LP7 + V(x, y) l
on £2(JR3N), the positions Y E JR3M of the nuclei appearing as parameters. This is called the BornOppenheimer approximation. It plays a fundamental role in quantum chemistry, where most computations are done with the operator HBO. The eigenvalues of the operator HBO are functions of the coordinates, y, of the nuclei. Minimizing the lowest eigenvalue  the ground state energy  with respect to Y gives the equilibrium positions of the nuclei, i.e. the shape of the molecule. One can show that if the pair potentials Vij satisfy the condition
(Le. each Vij can be represented as the sum of an £2 function and an £= function) then the operator HN is selfadjoint (see [CFKS, HuS]). Observe that the Coulomb potentials Vij(Y) = eiejlylI satisfy this condition:
lyl I = lylIe 1yl
+ lylI(1 e 1yl ),
for example, with lylIe 1yl E £2(JR3) and lylI(1  e 1yl ) E £=(JR3).
8.1 Quantization of a Manyparticle System
81
However, the spectral analysis of the operator HN for N ~ 3 is much more delicate than that of the onebody Hamiltonians we have considered so far. (We will show shortly that for N = 2, the operator HN is reduced to a onebody Hamiltonian.) We are faced with the following issues:  separation of the centerofmass motion  complicated behaviour ofthe potential V(x) at infinity in the configuration space ]R3N  identical particle symmetries. In the subsequent sections, we will deal with the first two issues in some detail. In the remainder of this section, we comment on all three of them. In what follows, we shall assume "Vij(x) . 0 as Ixi . 00, though for many considerations this condition is not required. Separation of the centreoJmass motion. The Schrodinger operator (8.2) commutes with the operator of total translation of the system
and one can show that, as a result, its spectrum is purely continuous. So in order to obtain interesting spectral information about our system, we have to remove this translational invariance ("break" it). One way of doing this is by fixing the centre of mass of the system at, say, the origin: N
Lmjxj =
o.
j=l
We will describe a general mathematical procedure for fixing the centre of mass below, but first will show how to do it in the case of two particles (N = 2). In this case, we change the particle variables as follows: Xl,
X2
Y
f*
=
Xl 
X2,
Z
=
mlXl
+ m2x2
(8.4)
Here y is the coordinate of the relative position of the two particles, and z is the coordinate of their centre of mass. Using this change of variables in the twoparticle Schrodinger operator H2
acting on
L2(]R6),
=
1
2
2PI ml
1
2
+ 2m2 P2 + V(XI
 X2)
we arrive easily at the operator H2
1
2
1
2
= 2p,P + 2M P + V(y)
where P = ifiVy, P = ifiV z, p, = ;:~m~2 (the reduced mass), and M = ml + m2 (the total mass). In fact, it can be shown that H2 and H2 are
82
8 Manyparticle Systems
unitarily equivalent, with the equivalence given by a unitary realization of the change of coordinates (8.4). The point now is that one can separate variables in the operator H2 . In formal language, this means that H2 can be written in the form
H2 = H on L 2(JR6)
= L2(JR~) @ L2(JR~)
@
1
+ 1 @ HCM
where 1
H = _p2 + V(y) 2p 1
HCM = 2M P
2
acts on L2(JR~) (see Section 8.8 for a description of the tensor product, @). Clearly Hand HCM are the Schrodinger operators of the relative motion of the particles, and of their centre of mass motion, respectively. It is equally clear that of interest for us is H, and not HCM. Note that H has the form of a oneparticle Schrodinger operator with external potential V(y). All the analysis we developed for such operators is applicable now to H.
Behaviour of V(x) at infinity. The second issue mentioned above arises from the geometry of the potential (8.1). The point is that V(x) does not vanish as x * 00 in certain directions, namely in those directions where Xi = Xj for at least one pair i =1= j (we assume here that Vij(O) =1= 0). This property is responsible for most of the peculiarities of manybody behaviour. In particular, the spectral analysis of Chapter 5 does not work in the manybody case, and must be modified in significant ways by taking into account the geometry of manybody systems. Identical particles. Manyparticle systems display a remarkable new feature of quantum physics. Unlike in classical physics, identical particles (i.e., particles with the same masses and charges, or, more generally, which interact in the same way) are indistinguishable in quantum physics. This means that all probability distributions which can be extracted from an Nparticle wave function 'I/J( Xl, ... , XN ), should be symmetric with respect to permutations of the coordinates of identical particles. This is equivalent to the property that 'I/J( Xl, ... , XN) is invariant under such a permutation modulo a change of sign. Assume for simplicity that all N particles are identical. Then the invariance property of 'I/J(XI, ... ,XN) formulated above means that 'I/J(Xl, ... ,XN) belongs to a representation of the symmetric group SN (the group of permutations of N indices) corresponding to a Young tableau with at most two columns (see, eg., [LL] for an explanation of Young tableaux). The shape of the Young tableau is determined by the spin of the particles involved. Onecolumn Young tableaux  i.e. purely symmetric 'I/J(XI, ... , XN)  correspond to
8.2 Separation of the Centreofmass Motion
83
particles with integer spins, or Bosons, while twocolumn tableaus correspond to particles with halfinteger spins, or Fermions. This relation between the symmetry properties of wave functions and the spin of particles, is known as the relation between spin and statistics. We will not go into this topic here, and refer the interested reader to any of the standard books on quantum mechanics given in the references.
8.2 Separation of the Centreofmass Motion This section is devoted to a description of a general method for separating the centreofmass motion of a manybody system. After applying this method, one is left with a manybody system whose centreofmass is fixed at the origin. We begin by equipping the Nbody configuration space ]R3N with the innerproduct N
(x, Y) := L
(8.5)
mixi . Yi
i=l
where ml, ... , mN are the masses of the N particles, and Xi . Yi is the usual dotproduct in ]R3. Next, we introduce the orthogonal subspaces
X:= {x
I
E ]R3N
N
Lmixi
= O}
(8.6)
1::/ i,j}
(8.7)
i=l
and Xl. := {x E ]R3N
I
Xi = Xj
of ]R3N. Recall that orthogonality (X ..l Xl.) means (x, y) = 0 for any x EX, Y E Xl.. To see that X ..l Xl., suppose x E X, and Y E Xl.. Since Yj = Yl for all j, we have n
(x,
y) = L
n
miXi . Yi
y=l
= Yl . L
mixi
=0
y=l
by the definition of X. We recall here the definition of the direct sum of subspaces. Definition 8.2 If Vl , V2 are orthogonal subspaces of a vector space, V, with an inner product, then Vi EB V2 denotes the subspace
Problem 8.3 Show that ]R3N
= X EB Xl..
(8.8)
84
8 Manyparticle Systems
X is the configuration space of internal motion of the Nparticle system, and X..L is the configuration space of the centreofmass motion of this system. The relation (8.8) implies that (8.9) (see Section 8.8 for an explanation of the tensor product, 18l). Let Ll denote the Laplacian on ]R3N in the metric determined by (8.5), i.e.
1
N
Ll = '" Ll x J·. Lm. j=l
J
Under the decomposition (8.9), the Laplacian decomposes as (8.10) where Ix and IXL are the identity operators on L2(X) and L2(X..L) respectively (see again Section 8.8). Problem 8.4 Show that LlxL 1 M
"N uj=l mjxj.
= ~LlXCM where M = 2:f=l mj and XCM =
Let 7rx be the orthogonal projection operator from (7rXX)i
= Xi
1 
]R3N
to X. Explicitly,
N
N
2: j =l mj
L mjxj.
(8.11 )
j=l
Problem 8.5 Show that (8.11) is the orthogonal projection operator from ]R3N to X. Find the orthogonal projection operator, 7rXL, from ]R3N to X..L.
Equation (8.11) implies that (7rXX)i  (7rxx)j
=
Xi  Xj.
Hence the manybody potential (8.1) satisfies V(X)
=
V(7rxx).
(8.12)
Equations (8.10) and (8.12) imply that the operator HN given in (8.2) can be decomposed as (8.13) where H
!i 2
= 2Llx + V(x)
is the Hamiltonian of the internal motion of the system, and
8.3 Breakups
85
is the Hamiltonian of the motion of its centreofmass. Equation (8.13) is the centreofmass separation formula, and H is called the Hamiltonian in the centreofmass frame. It is a selfadjoint operator under the assumptions on the potentials mentioned above. It is the main object of study in manybody theory.
8.3 Breakups Here we describe the kinematics of the breakup of an Nbody system into noninteracting systems. First we introduce the notion of a cluster decomposition a={C1 ... C s } for some s ::; N. The C j are nonempty, disjoint subsets of the set {1, ... ,N}, whose union yields the whole set: s
UC
j
= {l, ... ,N}.
j=l
The subsets C j are called clusters. An example of a cluster decomposition for N = 3 is a = {(12), (3)}. The number of clusters, s, in the decomposition a will be denoted by #(a). There is only one cluster decomposition with #(a) = 1, and one with #(a) = N. In the first case, the decomposition consists of a single cluster f!
= {(I. .. N)},
and in the second case the clusters are single particles:
a = {(l) ... (N)}. In the first case the system is not broken up at all, while in the second case it is broken into the smallest possible fragments. To each cluster decomposition, a, we associate the intercluster potential Ia(x):=
L
Vij(Xi  Xj),
(ij)\la
where the notation (ij) r:t. a signifies that the indices i and j belong to different clusters in the decomposition a. Similarly, we associate to a the intracluster potential Va(X):=
L
(ij)Ca
Vij(Xi  Xj)
86
8 Manyparticle Systems
where (ij) c a signifies that i and j belong to the same cluster in a. Thus Ia(x) (resp. Va(x)) is the sum of the potentials between particles from different (resp. the same) clusters of a. The Hamiltonian of a decoupled system (in the total centreofmass frame) corresponding to a cluster decomposition, a, is
acting on L2(X). For a = {C1 , ... , Cs}, the Hamiltonian Ha describes s noninteracting subsystems C 1 , ... , Cs. For s = 1, Ha = H, and for s = N, Ha =  ~2 Ll x . The operators Ha are also selfadjoint. If s > 1, then the system commutes with relative translations of the clusters C 1 ... Cs: for hEX satisfying
hi = h j
if
(ij) c a.
'Ij;(X1
+ h1, ... ,XN + hN)
Here
Th : 'Ij;(Xl, ... ,XN)
+
for h = (h 1, ... , hN). As a result, one can show that the spectrum of Ha is purely continuous. This is due to the fact that the clusters in a move freely. One can separate the centreofmass motions of the clusters C 1 ... C s in a and establish a decomposition for Ha similar to that for HN (equation (8.13)); we will do this later.
8.4 The HVZ Theorem In this section we formulate and prove the key theorem in the mathematical theory of Nbody systems  the HVZ theorem. The letters here are the initials ofW. Hunziker, C. van Winter, and G.M. Zhislin. This theorem identifies the location of the continuous spectrum of the manybody Hamiltonian, H. Theorem 8.6 (HVZ Theorem) We have
uc(H) = [17, (0), where
17:= min E a , #(a»l
with
Note that 17 is the minimal energy needed to break the system into independent parts.
8.4 The HVZ Theorem
87
Proof. We begin by showing that a(Ha) C a(H) for #(a) > 1. Suppose a(Ha). Then for any c > 0, there is 'lfJ E L2(X) with 11'lfJ11 = 1, such that II(Ha  .\)'lfJ11 < c. Let hEX satisfy hi = h j if (ij) C a, and hi =I h j otherwise. For s > 0, let Tsh be the operator of coordinate translation by sh. Note that Tsh is an isometry. As remarked in the previous section, Tsh commutes with H a , and so .\ E
On the other hand, HTsh'lfJ ~ HaTsh'lfJ as s ~ 00, because the translation Tsh separates the clusters in a as s ~ 00. So for s sufficiently large, II (H '\)Tsh'lfJll < c. Since c > 0 is arbitrary, and IITsh'lfJll = 11'lfJ11 = 1, we see .\ E a(H). So we have shown a(Ha) C a(H). As we remarked earlier, Ha has purely continuous spectrum. In other words, a(Ha) = [Ea, (0). Thus we have shown [17, (0) C a(H). It remains to prove that a c (H) C [17, (0). To do this, we introduce a "partition of unity", i.e., a family {ja} of smooth functions on X, indexed by all cluster decompositions, a, with #(a) > 1, such that
L
j~(x) == 1.
(8.14)
#(a»l
We can use {ja} to decompose H into pieces which are localized in the supports of the ja, plus an error term: (8.15) Indeed, summing the identity
over a, and using (8.14), we obtain
Computing [ja, [ja, H]] = 11,21\7 jal 2 finishes the proof of of (8.15). Now we construct an appropriate partition of unity, namely one satisfying (8.16) for some c
> o. Indeed, the sets Sa
:=
{x E X I Ixl
=
1; IXj  xkl
>0
\:j
(jk)
ct a}
88
8 Manyparticle Systems
form an open cover of the unit sphere, S, of X (i.e., Sa are open sets and S c Ua Sa). For each a, let Xa be a smooth function supported in Sa, and equal to 1 in a slightly smaller set (such that these smaller sets still cover S). Then the functions ja := Xa/(I: X;)1/2 form a partition of unity on S, with supp(ja) C Sa. In fact, (8.16) holds (with Ixi = 1), because supp(ja) is compact. We extend ja(x) to all of X by setting ja(x) := ja(x/lxl) for Ixl > 1, and for Ixl < 1, choosing any smooth extension of ja(x) which preserves (8.15). Thus the partition {ja} satisfies
for
ja(.)..x) = ja(x)
Ixl:::: 1,'>" :::: 1
(8.17)
as well as (8.16). By (8.17),
lV'ja(x) 12 By (8.16),
*
0
Ixl * 00.
as
ja(H  Ha)ja = ja1aja
*
0
Ixl * 00.
as
Returning to (8.15), we conclude that
H =
L
jaHaja
+K
#(a»l
where K is multiplication operator vanishing at infinity. An argument similar to the proof of the second part Theorem 5.20 shows that
Since H :::: (inf O"(H))l for any selfadjoint operator, H, we see
('I/J,jaHaja'I/J) = (ja'I/J, Hja'I/J) :::: 17allja'I/J112 ::::
17all'I/J112
for any 'I/J E X (note 17a :S 0). Thus jaHaja :::: 17a, and therefore I:a jaHaja :::: 17, yielding O"c(I:ajaHaja) C [17,(0), and consequently O"c(H) C [17,(0). This completes the proof of the HVZ theorem. D
8.5 Intra vs. Intercluster Motion As was mentioned in Section 8.3, the Hamiltonians Ha describing the system broken up into noninteracting clusters have purely continuous spectra. This is due to the fact that the clusters in the decomposition move freely. To understand the finer structure of manybody spectra, we have to separate the centreofmass motion of the clusters, as we did with the centreofmass
8.5 Intra vs. Intercluster Motion
89
motion of the entire system. Proceeding as in Section 8.2, we define the subspaces X a := {x E X I mjxj = 0 V i}
L
JEGi
and
Xa
:=
{x
E
X I Xi
=
Xj if (ij) c a}.
Problem 8.7 Show that these subspaces are mutually orthogonal (xa L Xa) and span X: (8.18)
xa is the subspace of internal motion of the particles within the clusters of the decomposition a, and Xa is the subspace of the centreofmass motion of the clusters of a. As before, (8.18) leads to the decomposition L2(X) = L2(xa) ® L2(Xa) of L2(X) and the related decomposition of the Laplacian on X: ..dx = ..dxa ® lXa
+ lxa
® ..dxa
where ..dxa and ..dxa are the Laplacians on the spaces xa and Xa (or L2(xa) and L2(Xa)) in the metric (8.5). Again, if 7rXa is the orthogonal projection from X to X a , then
Va(X)
=
Va(7rxa X)
and consequently we have the decomposition
Ha = H a ® lXa where
fj,2
+ lxa
H a = 2..dxa
® T a,
+ Va(x)
is the Hamiltonian of the internal motion of the particles in the clusters of a, and Ta =  ~2 ..dxa is the Hamiltonian of the centreofmass motion of the clusters in a. Applying the HVZ theorem inductively, we arrive at the following representation of the continuous spectrum of H:
(Jc(H) =
U [A, (0),
)..Er(H)
where the discrete set T(H), called the threshold set of H, is defined as
T(H):=
U
(Jp(Ha) U{O},
#(a»l
the union of the point spectra of the breakup Hamiltonians and zero. The points of T(H) are called the thresholds. Thus one can think of the continuous spectrum of a manybody Hamiltonian H (in the centreofmass frame) as a union of branches starting at its thresholds and extending to infinity.
90
8 Manyparticle Systems
8.6 Existence of Bound States for Atoms and Molecules Bound states of an atom or molecule described by a manybody Hamiltonian, H, correspond to the point spectrum, rYp(H). Here we turn our attention to the study of rYp(H), addressing the question of how many bound states H produces. For N = 2, the results in Section 5.6, show that the point spectrum of H is finite if the potential V(x) is "shortrange", whereas a "longrange" attractive potential produces an infinite number of bound states. The borderline between short and longrange potentials is marked by the asymptotic behaviour V(x) rv Ixl 2 as Ixl ~ 00 (which is different than the borderline asymptotic behaviour of Ixl 1 which we encountered in scattering theory in Chapter 6). For N > 2, however, the question of whether rYp(H) is finite or infinite cannot be answered solely in terms of the asymptotic falloff of the intercluster potentials, fa (x); the nature of the threshold E at the bottom of the continuous spectrum plays a decisive role. Here we restrict our attention to the case where E is a twocluster threshold. This means that for energy E and slightly above, the system can only disintegrate into two bound clusters, C 1 and C 2 . (In the case of an atom, one of the clusters contains a single electron, and the other contains the remaining electrons and the nucleus.) This situation can be represented by a product wave function (8.19) where ¢ is the eigenfunction of Ha with eigenvalue E, Ha¢ = E¢, and Xa and x a denote the components of x along the subspaces Xa and xa respectively. Here u is chosen so that (u, TaU) is arbitrarily small. The condition that E is a twocluster threshold means that E is a discrete eigenvalue of Ha, and as a consequence it can be shown that ¢(xa) decays exponentially as Ixal ~ 00: (8.20) Ct > O. Using states of the form (8.19) as trial states to make ('IjJ, H'IjJ) < E, we can show that rYp(H) is infinite if the intercluster potential fa(x) has a longrange attractive part. For simplicity, we write this out in the case of Coulomb potentials:
for some
with 2: iE C 1 ,kEC2 eiek < 0, assuming that the clusters have opposite total charges. Using the exponential bound (8.20), it follows that
Observe that fa(xa) = qllxal with q < O. Since H and Ha¢ = E¢, the last inequality implies
= Ha 01a + 1 a 0Ta + fa
8.7 Scattering States
(7f;, (H  E)7f;! :::: / U, (Ta  qII \ Xa As in Section 5.6, we let
U
+ (COnst)IXal2) U)
E CO'(lR 3 ) satisfy
SUpp(U) C {xa
I 1
00. The wave operator Qd,m is an isometry from 1ia ,m to 1i. Moreover, the ranges 1id,m := Ran(Qd,m) satisfy Hd,m .l 1ib,n
if
a
=1=
b
or
m
=1=
n
8.8 Mathematical Supplement: Tensor Products
93
which follows from the fact that lim (eiHa,,,,t/Ii('l/Ja,m ® j), eiHb,nt/Ii('l/Jb,n ® g))
=
0
t>oo
if a =f. b or m subspace
=f. n. Therefore, the outgoing scattering states form a closed a,m
Under the condition (8.21) with f..t > 1, it has been proven that the property of asymptotic completeness holds  i.e., that H+ = L2(X) (see Chapter 17 for references). In other words, as t ; 00, every state approaches a superposition of channel evolutions and bound states (the bound states are the channel with #(a) = 1). Longrange systems: f..t ::::: 1. As in the onebody case, it is necessary to modify the form of the channel evolutions in the longrange case. The evolution eiHa,,,,t/1i with Ha,m = >..a,m + Ta is replaced by e iHa,,,,t/liiCl'.a,t (Pa) where Pa = iff'V X a. Here aa,t(Pa) is an adiabatic phase, arising from the fact that classically, the clusters are located at Xa = Pat(1 + O(rf1)) as t ; 00. The modification aa,t(Pa), whose precise form we will not give here, is similar to that for the onebody case (see Chapter 6). We refer the interested reader to the references listed in Chapter 17 for further details. We remark only that with this modification in place, the existence of the (modified) wave operators, and asymptotic c9mpleteness, have been proved for f..t not too small (see Chapter 17 for references).
8.8 Mathematical Supplement: Tensor Products We collect here a few facts about tensor products of Hilbert spaces, and tensor products of operators and their spectra (see [RSI] for details and proofs). Let HI and H2 be two separable Hilbert spaces. The tensor product of HI and H2 is a Hilbert space HI ® H2 constructed as follows. To 'l/Jl E HI and 'l/J2 E H2, we associate a map
'l/Jl ® 'l/J2 : HI x H2 ; C (ft, h)
f)
(ft, 'l/Jl)'Hl (12, 'l/J2)'H2
which is conjugate linear in each component ('l/Jl ® 'l/J2 (aft, h) = ti'I/Jl ® 'l/J2(ft, h), 'l/Jl ® 'l/J2(ft + gl, h) = 'l/Jl ® 'l/J2(ft, h) + 'l/Jl ® 'l/J2(gl, h), and the same for the second component). On the vector space, V, of all finite linear combinations of such conjugate bilinear maps, we define an innerproduct by setting
94
8 Manyparticle Systems (8.22)
and extending by linearity (it is straightforward to check that this is welldefined). Then HI Q9 H2 is defined to be the completion of V in the innerproduct determined by (8.22). A simple example, which appears in Section 8.1, is
for positive integers m, n. This Hilbert space isomorphism is determined by the map f Q9 g f+ f(x)g(y) (see, eg., [RSI] for details). Given bounded operators A and B acting on HI and H 2 , the operator A Q9 B, which acts on HI Q9 H2, is defined by setting
extending by linearity to all finite linear combinations of elements of this form, and then by density of these finite linear combinations, to HI Q9 H2. This produces a welldefined operator. This construction can be extended to unbounded selfadjoint operators A and B, yielding a selfadjoint operator A Q9 B (see, eg., [RSI]). Of particular interest for us are operators of the form A Q9 1 + 1 Q9 B, acting on H = HI Q9 H2, where A and B are operators acting on HI and H2 respectively. This is an abstract version of the "separation of variables" situation of differential equations. It is intuitively clear that we should be able to reconstruct characteristics of such operators from the corresponding characteristics of the operators A and B. As an example, we have the following important (and simple) description of the spectrum of A Q9 1 + 1 Q9 B under certain conditions on A and B, and, in particular, for A and B selfadjoint:
O"(A Q9 1 + 1 Q9 B) = O"(A) + O"(B) O"p(A Q9 1 + 1 Q9 B) C O"p(A) + O"p(B) C { ev's of A Q9 1 + 1 Q9 B} O"c(A Q91 + 1 Q9 B) = O"c(A) + O"c(B) U [O"c(A) + O"p(B)] U [O"p(A) + O"c(B)]. Rather than prove any such statements (an involved task, requiring further assumptions), let us just do a simple, suggestive computation. Suppose A7J!1 = AI7J!1 and B7J!2 = A27J!2. Then note that
(A Q9 1 + 1 Q9 B)7J!1 Q9 7J!2 = A7J!1 Q9 7J!2 + 7J!1 Q9 B7J!2 = AI7J!1 Q97J!2 +7J!1 Q9 (A27J!2) = (AI + A2)7J!1 Q9 7J!2, which shows, in particular, that O"p(A)
=
AI(7J!1 Q97J!2) +A2(7J!1 Q97J!2)
+ O"p(B) c { ev's of A Q9 1 + 1 Q9 B}.
9 Density Matrices
9.1 Introduction In this chapter we extend the basic notions of quantum mechanics to the situation of open systems and positive temperatures. This topic is called quantum statistical mechanics. Since the notion of temperature pertains to systems with an infinite number of degrees of freedom  infinite systems which are in states of thermal equilibrium, we should explain what the first sentence really means. By a quantum system at a positive temperature, we mean a quantum system coupled to an infinite system which is initially in a state of thermal equilibrium at a given temperature. In what follows, we develop a mathematical framework with which to handle such situations. We will see that we will have to replace the notion of wave function (i.e. a square integrable function of the particle coordinates  an element of L2 (JR3)) with the notion of density matrix, a positive, trace class operator on the state space L 2(JR3). The notions of ground state and ground state energy go over to the notions of Gibbs state and (Helmholtz) free energy. We consider some elementary examples and formulate some key problems. The notions of trace and trace class operators will be defined in the "mathematical supplement" at the end of this chapter.
9.2 States and Dynamics Consider a physical system described by a quantum Hamiltonian H acting on a Hilbert space 1{ (say, H = L1 + V(x) on L2(A), where A c JR3). Let {7/Jj} be an orthonormal basis in H. Then any state 7/J E 1{ can be expanded as 7/J = 'Laj7/Jj. Given an arbitrary observable A (say, position or a characteristic function of position), its average in the state 7/J is given by
(9.1) m,n
Now suppose that we know only that the system is in the state 7/Jn with a probability Pn for each n. We thus have much less information than before. Now the average, (A), of an observable A, is given by the expression
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
96
9 Density Matrices (9.2) n
This corresponds to the situation when the parameters an in (9.1) are independent random variables with zero mean, and variance E(lan 2 ) = Pn. Observe that (9.2) can be written as I
(A)
=
tr(Ap) ,
(9.3)
where p = 2:;n PnP,pn' Here P,p stands for the rankone orthogonal projection onto the vector 'l/J (P,pf = ('l/J, j)'l/J, or P,p = I'l/J)('l/JI in Dirac's notation). p is a trace class, positive (since Pn 2: 0 and P,pn 2: 0) operator, with trace 1: tr p = 2:;n Pn = 1. We extrapolate from this the assumption that generalized states are given by positive, trace class operators p on 1i, normalized such that trp = 1. Such operators are called density matrices. If the vectors 'l/Jn in the expression p = 2:;PnP,pn evolve according to the Schrodinger equation, ifiWf: = H'l/J, then the equation governing the state p is: (9.4) One takes this equation to be the basic dynamical equation of statistical mechanics. It is called the Landauvon Neumann equation, or the quantum Liouville equation. Problem 9.1 Derive equation (9.4) for p
= 2:;PnP,pn
with 'l/Jn as above.
If the spectrum of an operator H consists of isolated eigenvalues (of finite multiplicity) converging to 00, then an operator f(H) is trace class for any function f vanishing at (X) sufficiently fast (see Section 9.14). Thus, for any such positive function, f(H) is a density matrix (up to normalization). The operator f(H) commutes with H ([f(H), H] = 0), and therefore p := f(H) is a timeindependent solution of equation (9.4). It is a stationary solution. We summarize our conclusions. In the situation when only partial information about a quantum system is available  namely, we know only that the system occupies certain states with certain probabilities  it is natural to describe states of such a system by positive traceclass operators p 2: 0 (normalized by trp = 1), called density matrices, with the equation of motion given by (9.4), and averages of observables computed according to the prescription (9.3). If the operator H has purely point spectrum, then Equation (9.4) has an infinitedimensional family of timeindependent solutions stationary states  of the form f (H), where f is a positive function, decaying sufficiently fast at infinity.
9.3 Open Systems There are two situations which necessitate passing from wave functions to density matrices. In one case, one deals with a system of infinitely many de
9.3 Open Systems
97
grees of freedom. We consider this situation in the next section. In the other case, the system of interest is coupled to (or is a subsystem of) another, usually much larger, system, which we call the environment. Such a system is called an open system. Even if the total system is described by a wave function, say 'ljJ(x, y), where x and yare the coordinates of the system of interest and of the environment, respectively, the system itself cannot generally be described by a wave function. More specifically, given 'ljJ(x, y), there is, in general, no wave function cp(x) for our system such that, for any observable A = Ax associated with the system,
('ljJ, A'ljJ) = (cp, Acp) . However, there is a density matrix, p = P1j;, for our system, so that
('ljJ, A'ljJ)
=
tr(Ap)
(9.5)
for any system observable A. Indeed, the operator p is defined by its integral kernel (9.6) p(x, x') = 'ljJ(x,y)'ljJ(x',y)dy .
J
Problem 9.2 Check that (9.5) holds for any operator A acting on the variable x, provided p is given by (9.6).
One way to show (9.5) is as follows. Define the rankone projection operator Then (show this!) (9.7)
for any operator (observable) A. Now, if A is a system observable, i.e., an operator which acts only on the variables x, then taking the trace first with respect to the environment variables, and then over the system variables (see Section 9.14) one obtains
tr(AP1j;)
=
trsysttrenvir(AP1j;)
= tr syst [Atr en vir (P1j;)]. Denote p := tr env ir(P1j;). This is a positive, trace class operator, acting on the system's wave functions, normalized by
The integral kernel of p is precisely (9.6). Thus p is a density matrix, and we have shown
tr(AP1j;) = trsyst(Ap)
and, due to (9.7), also (9.5).
98
9 Density Matrices
The states (density matrices) which are rankone operators  i.e. which are of the form p = P'lj; for some normalized wave function 'Ij;  are called pure states. They are equivalent to wave functions in the sense that they are in onetoone correspondence with wave functions, up to a phase, and produce exactly the same expectations for arbitrary observables as the corresponding wave functions.
9.4 The Thermodynamic Limit The aim of this section is to provide an informal discussion of the notion of thermodynamic limit. Consider the concrete physical situation of a system of N particles (say, a gas) in a volume A c IR3 described by the Schrodinger operator
acting on L2(AN) with Dirichlet or periodic boundary conditions, where Vi and Wij are bounded (see Chapter 8). Our goal is to understand the main features of the corresponding dynamics (9.4), which are independent of N and A for Nand IAllarge and N IIAI fixed. First note that if A is bounded, then !(HN,A) is a trace class operator for any! which decays at 00 sufficiently fast (show this!  see Section 9.14). Hence if! is also positive, then !(HN,A) is a stationary state (up to normalization). A basic question is whether or not these stationary states are stable (in an appropriate sense) as solutions of the Landauvon Neumann equation (9.4). In general, such states are stable in the Lyapunov sense. Roughly speaking, this means that solutions of (9.4) which are close to the given stationary state at time t = 0, remain close to it for later times. A stationary state is said to be asymptotically stable if solutions of (9.4) which start close to the given stationary state, converge to it as t ; 00. The stationary states described above do not enjoy this stronger property of asymptotic stability. However, as a few examples accessible to analysis show, for N very large there is a finitedimensional subfamily of states which appear to be asymptotically stable on extremely large time intervals (of length eN where e is a large number), while convergence to them is very fast. Clearly these exceptional states playa key role in the description of large particle systems. To isolate such states, and to prove their longtime asymptotic stability, one passes to the idealized situation of an infinite number of particles in infinite volume, but with lim(NIIAI) finite and positive. This is the thermodynamic limit.
9.6 The T
>
0 Limit
99
9.5 Equilibrium States We are interested in stationary states which are asymptotically stable on long time intervals when the number of particles is very large. It is conjectured that (in the absence of transport) such states coincide with the states of thermal equilibrium ~ the (thermal) equilibrium states. The latter can be isolated as follows. Assume we have only one conserved quantity ~ the energy. Then, following the second law of thermodynamics, we can characterize the equilibrium states in a finite volume as follows: p is equilibrium iff p maximizes S(p), provided E(p) is fixed (E(p)
= E,
say) .
(9.8)
Here S(p) := tr(plnp) is the Gibbs entropy, and E(p) := tr(Hp) is the internal energy. The criterion above is called the principle of maximum entropy. This principle can be extended in an appropriate form to infinite systems. Variational problem (9.8) can be easily solved (see Chapter 12) to give the following oneparameter family of positive operators PT = e~H/T /Z(T) , where Z(T) := tre~H/T ,
as equilibrium states (for a definition of the operator e~H/T, see Section 2.2). These states are called the Gibbs states and T, which is the inverse of the Lagrange multiplier, is called the temperature. The quantity Z(T) = tre~H/T (or Z(f3) = tre~{3H for f3 = l/T) is called the partition function (of the system described by the Hamiltonian H, at temperature T). It is conjectured that in the absence of conserved quantities other than the energy, all equilibrium states of infinite systems can be obtained as (weak) limits of Gibbs states: that is, the expected value of any observable A (Le. a bounded, selfadjoint operator on our Hilbert space) in an equilibrium state WT (at temperature T) is given by wT(A) = lim tr(ApT A) Aiffi,3
'
where PT,A is the Gibbs state at temperature T for the Hamiltonian H restricted to a bounded region, A, of physical space (see [BR]).
9.6 The T
~
0 Limit
The Lagrange multiplier theorem of variational calculus implies that an equilibrium state minimizes the Helmholtz free energy FT(p) := E(p)  TS(p)
where T, or f3 = T~\ is the Lagrange multiplier to be found from the relation tr(HpT) = E.
100
9 Density Matrices
Problem 9.3 Show this.
By a straightforward computation, the equilibrium free energy, F(T) .FT(PT), is given by F(T) = TlnZ(T) . The next result connects Gibbs states and the free energy, to ground states and the ground state energy. Let 7/Jo be the (unique) ground state of the Hamiltonian H, and Eo the corresponding ground state energy. Let P'IjJ denote the rankone projection onto the vector 7/J. We have Theorem 9.4 (FeynmanKac Theorem) As T
PT
+
F(T)
and
P'ljJo
+
+
0,
Eo .
Proof Let Eo < El ~ E2 ~ ... be the eigenvalues of H (our standing assumption is that H has purely discrete spectrum, running off to 00), and let 7/Jo, 7/Jl,'" be corresponding orthonormal eigenstates. Then by completeness of the eigenstates, and the spectral mapping theorem (see Section 9.14),
where Pn = e En / T /Z(T). We can rewrite Pn as Pn
=
/2:= 00
e(EnEo)/T
e(En,Eo)/T .
n'=O
We see that Pn
~
1 and as T
0
+
n=O n?l
It follows easily that
F(T)
= TIn
we see that F(T)
IlpT 
P'ljJo II
(~eEn/T)
+
Eo as T
+
+
0 as T
+
= Eo  TIn
O. Furthermore, since
(1 + ~
e(EnEo)/T ) ,
O. 0
9.7 Example: a System of Harmonic Oscillators As an example, we compute the free energy and its thermodynamic limit for a system of N noninteracting onedimensional harmonic oscillators. The Hamiltonian of such a system is
9.7 Example: a System of Harmonic Oscillators
acting on
N
® L2(JR 1 )
~
101
L2(JR N ) (elements of manybody theory are described
1
in Chapter 8, and the symbol Q9 is explained in Section 8.8). Such oscillators describe quantized normal modes of lattice vibrations in solids (quantized elasticity waves, or phonons), or a quantized electromagnetic field in a cavity (photons). The operator HN has purely point spectrum. By separation of variables (see Section 7.5 and Section 8.8), the eigenvalues of HN are given by for
nj
=
0,1,2, ....
Consequently the partition function, ZN((3) = tre f3HN , is 00
j=l
=
n=O
II (1  eN
f3Wj
)l
j=l
and the free energy, F N ((3)
=
FN ((3)
~ In Z N ((3), is
1 N
=
!3 I)n(1  e f3Wj ). j=l
In order to pass to the thermodynamic limit N 7 00, we have to know how the frequencies Wj = wj(N) behave as N 7 00. In the case of phonons in a onedimensional crystal with N atoms spaced at a distance a from each other, one has 2nc j Wj = :;: N (9.9) where c is the speed of sound. Then, using
N
I:
j=l
f(wj) ~ 2~C J~7rc/a f(w)m.u for
V = N a large, we obtain an expression for the free energy per unit volume in the thermodynamic limit:
o as N
7
00.
102
9 Density Matrices
9.8 A Particle Coupled to a Reservoir Typically, one is interested in describing a small system interacting weakly with a large system (a small part of a gas interacting with the rest of the gas, for example, or an atom interacting with radiation). Often the large system is (or, for the given purpose, can be) described by a rather simple Hamiltonian. We call the large system an environment or (thermal) reservoir. The environment (the electromagnetic field, say) can be assumed to have an infinite number of degrees of freedom, and to be in a state of thermal equilibrium. We describe here such a model for the case of a single particle (in 3 dimensions, and in a potential V(x)) coupled to an environment. As a reservoir, we take a photon or phonon gas or, in general, a free, massless quasiparticle gas. In an approximation by a finite number of degrees of freedom, such a gas can be described by a collection of independent, onedimensional harmonic oscillators:
acting on
N
® L2(JR)
~
L 2(JRN). The simplest coupling of a particle to the
I
reservoir is given by the multiplication operator
IN
=
g(x)· x,
where g is a map from JR3 to JRN, and x = (Xl' ... ' XN) E JRN. Denoting the particle Hamiltonian by
("8" standing for "system") acting on JR3, we arrive at the following expression for the Hamiltonian of the total system  the particle and reservoir coupled together:
HN := H S
(9
1R
+ 1S
(9
HfJ
+ )"IN
acting on L2(JR3) (9 L 2(JRN). Eventually, when we study the states and dynamics ofthe Hamiltonian HN, we will want to send N to 00, while assuming a particular dependence of Wj on j (eg. (9.9)).
9.9 Quantum Systems We will distinguish three types of systems described by 8chrodinger operators H
=
li,2
,1
2m
+ V(x)
:
9.11 Hilbert Space Approach
confined systems
V(x)
+ 00
asymptotically free systems unstable systems
f7
V (x)
as
Ixl
V(x)
+  00
as x
103
+ 00
+
+ 00
canst as
Ixl
+ 00
in some direction
9.10 Problems One of the central questions of Statistical Mechanics is how the behaviour of a given system is affected by its interaction with an infinite environment which is in a state of equilibrium. There are three basic problems in this context: I. (Return to equilibrium). Show that a confined system is driven to a state of total equilibrium with the environment. II. (Thermal evaporation). Show that an asymptotically free system ends up eventually in a state of unbounded motion (diffusion);
III. (Metastable states). Find the probability of decay of unstable systems. These problems are extensively discussed in the physics literature. As far as rigorous results are concerned, at present there are partial answers to questions I, II, and III (I: [JP1, BFS, DJ, Mer]; II: [FMS, FM]; III: [SV]). This is a quickly developing area of (nonequilibrium) statistical mechanics, and we refer the interested reader to the original papers for more details. Topic III is discussed in Chapter 13.
9.11 Hilbert Space Approach Statistical dynamics can be put into a Hilbert space framework as follows. Consider the space 'HHS of HilbertSchmidt operators acting on the Hilbert space 'H. These are the bounded operators, K, such that K* K is traceclass (see Section 9.14). There is an innerproduct on 'HHS, defined by
(F, K) := tr(F* K). Problem 9.5 Show that (9.10) defines an innerproduct.
(9.10)
104
9 Density Matrices
This innerproduct makes HHS into a Hilbert space (see [BR, RSI]). On the space HHS, we define an operator L via 1
LK= n[H,Kl, where H is the Schrodinger operator of interest. The operator L is symmetric. Indeed,
n(F, LK) = tr(F* [H, K]).
Using the cyclicity of the trace, the right hand side can be written as
tr(F* HK  F* KH) = tr(F* HK  HF* K) = tr([F*, HlK) = tr([H, Fl* K) = n(LF, K) and so (F, LK) = (LF, K) as claimed. In fact, for selfadjoint Schrodinger operators, H, of interest, L is also selfadjoint. Now consider the Heisenberg equation i ak
at
= Lk
(9.11)
where k = k(t) E HHS. Since k(t) is a family of HilbertSchmidt operators, the operators p(t) = k*(t)k(t) are traceclass, positive operators. Because k(t) satisfies (9.11), the operators p(t) obey the equation
.ap
~ at
1
= Lp= n[H,pl·
(9.12)
If p is normalized  i.e., trp = 1  then p is a density matrix satisfying the Landauvon Neumann equation (9.12). The stationary solutions to (9.11) are just eigenvectors of the operator L with eigenvalue zero. To conclude, we have shown that instead of density matrices, we can consider HilbertSchmidt operators, which belong to a Hilbert space, and dynamical equations which are of the same form as for density matrices. Moreover, these equations can be written in the Schrodingertype form (9.11), with selfadjoint operator L, sometimes called the Liouville operator. Some key problems about the dynamics (see Section 9.10) can be reduced to spectral problems for the operator L. For more details, see [BR, Ha, HKT, JP1, BFSl.
9.12 Appendix: the Ideal Bose Gas Consider a system of identical, noninteracting particles subject to Bose statistics, in a box A C lRd . We assume periodic boundary conditions. If the number of particles in the box A is n, then their Hamiltonian is
9.12 Appendix: the Ideal Bose Gas
HAn ,
1
n
=
105
~   L1x ~ 2m ' i=l
acting on the space ®~ L2(A) := L;YID (An) with periodic boundary conditions. Here ® is the symmetric tensor product, and L§YID (An) is the L2 space of functions symmetric with respect to permutations of variables belonging to different factors in the product An. Computing the partition function
(9.13) is not a simple matter (try it!). Instead, we compute the generating function 00
00
n=O
n=O
(9.14) where z = e{3/L. In principle, the original partition function, ZA,n(J3), can be recovered by computing the nth derivative, ar; ZA({3, /1)lz=o, of ZA({3, /1) with respect to z at z = 0, if this derivative exists. There is a more natural way (from the point of view of statistical mechanics) to recover Z A,n ({3) from ZA({3, /1) which we discuss below in this section. Meanwhile, we compute ZA({3, /1), which is not difficult. Using separation of variables, and the result of Section 7.2 for a single particle in a box with periodic boundary conditions, we conclude that the spectrum of the operator HA,n is (9.15) where Ck = 2~ Ik1 2 . The eigenvalues have multiplicity 1 since the eigenfunctions lie in the subspace L;ym of L2(JRnd) (they are symmetric with respect to permutations of the variables Xi E JRd). Problem 9.6 Show (9.15).
The expression on the right hand side can be rewritten as
CJ(HA,n)
= {
L kE
2;; Zd
Cknk
1
nk
= 0,1, ... , , L
nk
= n}.
Using equation (9.28), which expresses the trace of an operator in terms of its eigenvalues, we obtain the following expression for Z A ({3, /1): 00
ZA({3,/1) = L
L
e{3~f:knkzn
The latter expression can be transformed as
=
Lz~nke{3~f:knk.
106
9 Density Matrices (9.16)
Note in passing that if for a selfadjoint operator A with a purely point spectrum P'j} accumulating at 1, we define detA:= I1~1 >'j whenever this is finite, then we can rewrite (9.16) as
where HI,A =  2~ L1x is the one particle Hamiltonian acting on L2(A) with periodic boundary conditions. Next, we consider the quantity P((3, Jl) := J(3 In ZA((3, Jl), called the pressure. We have
(9.17)
Using that as V
> 00,
f(k)
>
J
f(k)ddk,
we find the following approximation for the sum in (9.17):
P((3'Jl)=(27r~d(3 =
J
1 (27r)d (3dt2
In(lze(3ck)ddk
J
In (1  z e ck ) dd k
(9.18)
modulo terms vanishing as V > 00. To get the last integral, we changed variables k > k/V73. Now we show how to derive the partition function ZA,n((3) from the generating function ZA((3,Jl). The considerations below are heuristic, but can be made rigorous. Using expression (9.16) one can show that as a function of z = etJP" ZA((3, Jl) is analytic in the disk {Izl :::; c} for some c > O. Hence, due to (9.14), ZA,n can be computed by the Cauchy formula (9.19) By the definition of PA((3, Jl), we can write ZA((3, Jl) = eVtJPA «(3,p,). Writing also zn = e n1nz = eVvlnz, where v = n/V, (9.19) becomes
9.12 Appendix: the Ideal Bose Gas
107
Now we take V = IAI and n large, while v = njV remains fixed. Taking into account the fact that PA ((3, JL) has a limit as V + 00, and applying (formally) the method of steepest descent to the integral above, we find (9.20) where lIn cl is uniformly bounded in V, and JL solves the stationary phase equation JL: (3z8 z PA((3,JL) = njV or equivalently (passing from z to JL)
Define PA,n((3) := /3~ lnZA,n((3). Then relation (9.20) can be rewritten as
That is, PA,n((3), as a function of n, is (in the leading order as V + 00) the Legendre transform of PA((3, JL), considered as a function of JL. Similarly, we can pass from ZA((3, JL) to ZA,n((3) by taking the Legendre transform in the variable v = njV. Returning to the generating function Z A ((3, JL), we view it as a partition function with a variable number of particles, but with a fixed average number of particles. The latter is given by the formula fi
=
L 00
ntre/3HA,ne/3/"n jZA((3, JL),
(9.21 )
n=O
and depends on the value of the parameter JL (this viewpoint will be substantiated at the end of this section). The formula (9.21) implies the relation
from which we obtain the expression 1
njV=V
(9.22)
To have n 2': 0, we should take 0 :::; z :::; 1. The terms in the sum on the right hand side are, as can be easily checked, the average numbers of particles having momenta k:
To show this, one uses
108
9 Density Matrices
where ZA({3, JL) is considered as a functional of C = Ck. Equations (9.18) and (9.22) constitute the equation of state of the ideal Bose gas (parameterized by z). More precisely, solving Equation (9.22) for z as a function of the density p = n/V, and temperature T = 1/{3, and substituting the result into Equation (9.18), we find the pressure P as a function of p and T. However, if we seek the equation of state as a relation between P, E and V (which is, of course, equivalent to the expression involving P, p and T), the answer is much simpler. Indeed, using the definition of E
E
=
E({3, JL, V)
:=
tr (HA P{3,I',A)
and considering the generating function as a function, ZA({3, z), of {3 and z (and V) rather than of {3 and JL (and V), and similarly for the pressure, we find a a (9.23) E=a{3 InZA({3,z)=a{3 (V{3P({3,z)). Taking into account (9.18) and setting d E =
~ 2
=
3 we find
PV.
(9.24)
This is the equation of state of the ideal Bose gas. Problem 9.7 It is an instructive exercise to rederive the results of this section for the ideal Fermi gas.
Finally, we justify considering the generating function ZA({3,JL) as a partition function for a system with a variable number of particles. We pass to a space describing a variable number of particles  the (bosonic) Fock space:
F = EB~=o ®~ L2(A), where the n = 0 term is defined to be C. On this space we define the Hamiltonian H A := EB~=o HA,n, where, by definition, the n ZA({3, JL) can be written as
= 0 term is HA,O =
ZA({3,JL)
1. The generating function
= tre{3(HAI'NA) ,
where N A is the number of particles operator, defined as
9.13 Appendix: BoseEinstein Condensation
109
(n is the operator of multiplication by n). The partition function ZA((3,p) is associated with the Gibbs state
for the new system. This state is obtained by maximizing the entropy, while leaving the average energy and average number of particles fixed (hence two Lagrange multipliers, (3 and p, appear). The state Pfl,i' is called the grand canonical ensemble state, ZA((3, p) the grand canonical partition function, p the chemical potential, and z the fugacity.
9.13 Appendix: BoseEinstein Condensation We consider the ideal Bose gas, and analyze formula (9.22) for the average number of particles. We would like to pass to the thermodynamic limit, V t 00. The point is that Equation (9.22) is the relation between the average number of particles n (or the average density P = n/V), the temperature T = 1/(3, and the chemical potential p (or fugacity z = e fl i'). Recall also that 0 :::; z :::; 1. As long as P and (3 are such that z < 1, the right hand side of (9.22) converges to the integral
as V t 00. However, if the solution of Equation (9.22) for z yields, e.g., z = 1  O(V 1 ), then we have to consider the k = 0 term in the sum on the right hand side of (9.22) separately. In this case we rewrite (9.22) approximately as (from now on we set d = 3) (9.25) where we put
integration as k'
where ,\
=
a
no = =
l':z. Now using
Ck
= I;~, changing the variable of
k, and passing to spherical coordinates, we obtain
J27r (3/m (the thermal wave length), and
Thus Equation (9.25) can be rewritten as
llO
9 Density Matrices
no
P = V +,\
3
(9.26)
g3/2(Z).
Recall that no = l':'z' and that this equation connects the density p, the thermal wave length ,\ (or temperature T = 27r / m ,\2), and the fugacity z (or chemical potential J.L = ~ In z), and is supposed to be valid in the entire range of values, 0 < z < 1, of z. Can z really become very close to 1 (within O(l/V)), or is the precaution we took in the derivation of this equation by isolating the term no/V spurious? To answer this question we have to know the behaviour of the function g3/2(Z) for z E (0,1). One can see immediately that
g3/2(0)
=0,
g~/2(Z)
> 0 and g~/2(1) =
00.
The function g3/2(Z) is sketched below.
g 1 (z)
g 3 (1)
"2
~
_______2_____ _
~=~
____________
 L_ _~~
Z
1 Fig. 9.1. Sketch of g3/2(Z). We see now that if p,\3 < g3/2(1) (with g3/2(1)p,\3 ;::: a positive number, independent of V), then the equation (9.27)
'iJ,
has a which is obtained from (9.26) by omitting the Vdependent term unique solution for z which is less than 1, and is independent of V. Consequently, taking into account the term no/V = v(Lz) would lead to an adjustment of this solution by a term of order O(l/V), which disappears in the thermodynamic limit V . . . . , 00. However, for p,\3 = g3/2(1), the solution of this equation is, obviously, z = 1, and for p,\3 > g3/2(1), the above equation has no solutions at all. Thus for p,\3 ;::: g3/2(1) we do have to keep the term no/V. Moreover, we have an estimate
(here z* is the solution to Equation (9.26)) which shows that in the case p,\3 > g3/2(1), a macroscopically significant (i.e. proportional to the volume
9.13 Appendix: BoseEinstein Condensation
111
or the total number of particles) fraction of the particles is in the single, zero momentum  or condensed  state. This phenomenon is called BoseEinstein condensation. The critical temperature, T e , at which this phenomenon takes place can be found by solving the equation
describing the borderline case for A, and remembering that T result we have 2/3 2 T~ (   p  ) c m 93/2(1)
=
';~2' As a
From Equation (9.26) we can also find the fraction of particles, no/V, in the zero momentum (condensed) state as a function of temperature. This dependence is shown in the diagram below.
lr
T Fig. 9.2. In this elementary situation, we have stumbled upon one of the central phenomena in macrosystems  the phenomenon of phase transition. Indeed, the states for which all the particles are in the single quantum state corresponding to zero momentum, and those for which the macroscopic fraction of the particles in the quantum state of zero momentum (and consequently in every single quantum state) is zero, can be considered two distinct pure phases of ideal Bose matter (gas). The first pure phase  called the condensate  occurs at T = 0, while the second pure phase takes place for T ~ T e . In the interval 0 < T < Tc of temperatures, the Bose matter is in a mixed state in which both phases coexist. BoseEinstein condensation exhibits a typical property of phase transitions of the second kind: though all the thermodynamic functions and their first derivatives are continuous at the phase transition, some of the second derivatives are not. Typically one looks at the specific heat
c ._ v·
aE(T,fJ, V) aT '
112
9 Density Matrices
the change of heat or energy per unit of temperature. Using Equations (9.18) and (9.23), one can show that while C v is continuous at T = Te, its derivative with respect to T is not. C v as a function of T is plotted below (see [Hu], Sect. 12.3]): Cv/n
~
1
.J. _ _ _ _ _ _
2
I
IT C
.I
~==~~~T
::::. T 3/2
Fig. 9.3. Specific heat of the ideal Bose gas. One can show that the thermodynamic properties (eg., the equation of state  a relation between pressure, temperature and volume) of BoseEinstein condensation are the same as those of an ordinary gas  liquid condensation. The modern theory relates the phase transitions to superfiuid states in liquid helium (He 2 ) and to superconducting states in metals and alloys, to the phenomenon of BoseEinstein condensation. In the mean field description of the phenomena of superfiuidity and superconductivity, the wave function of the condensate  the fraction of particles (or pairs of particles, in superconductivity) in the quantum zero momentum state  called the order parameter, is the main object of investigation. Of course, in both cases one deals with interacting particles, and one has to argue that BoseEinstein condensation persists, at least for weakly interacting Bose matter. Problem 9.8 Extend the above analysis to an arbitrary dimension d.
9.14 Mathematical Supplement: the Trace, and Trace Class Operators This section gives a quick introduction to the notion of the trace of an operator, a generalization of the familiar trace of a matrix. More details and proofs can be found in [RSI], for example. Let p be a bounded operator on a (separable) Hilbert space, 1{. Since p* p 2:: 0, we can define the positive operator Ipi := vp*p (this operator can be defined by a power series  see [RSI]; see also Section 5.2). The operator p is said to be of trace class if
L(7/'j, Ipl7/'j! < 00 j
9.14 Mathematical Supplement: the Trace, and Trace Class Operators
113
for some orthonormal basis {'l/Jj} of H.. If p is a trace class operator, we define its trace to be trp = 2)'l/Jj, p'l/Jj) j
for some orthonormal basis {'l/Jj} of H.. This definition is independent of the choice of basis. Properties of trace class operators include . the trace class operators form a Banach space under the norm
IlpIII := tripi, and the trace is a linear functional on this space; that is, tr maps trace class operators to complex numbers, and it satisfies tr(aA+f3B) = atr A+ f3trB for any a, f3 E oo
where
JJ ( ...
eiSn / fi
2
·nt)
~ mn
nd/2
dXI ... dXn1
nl
Sn
:=
I)mnlxk+l  xkl 2/2t  V(Xk+l)t/n) k=O
= x, Xn = y. Define the piecewise linear function ¢n such that ¢n(O) = x, ¢n(t/n) = Xl, ... , ¢n(t) = Y (see Fig. 10.1).
with Xo
J
y
x
tin
2t1n· ..
s
Fig. 10.1. Piecewise linear function. Then
Note that Sn is a Riemann sum for the classical action
S(¢, t)
=
lot {; 1¢(sW  V(¢(s))} ds
10.1 The Feynman Path Integral
117
of the path ¢n. So we have shown
(10.2) where P:'y,t is the (n  I)dimensional space of paths ¢n with ¢n(O) = x, ¢n(t) = y, and which are linear on the intervals (kt/n, (k + I)t/n) for k = 0,1, ... , n  1, and D¢n = e~::t)nd/2d¢n(t/n) ... d¢n((n I)t/n). Heuristically, as n + 00 ¢n approaches a general path, ¢, from x to y (in time t), and Sn + S(¢). Thus we write
L
Ut(y,x) =
eiS(,t)/nD¢.
(10.3)
x,y,t
Here Px,y,t is a space of paths from x to y, defined as
Px,y,t:= {¢: [O,tj+
]R d1
l
t
11>12
A in the strong operator topology (written slimn>ooAn = A) iff IIAn¢  A¢II '> 0 for all ¢ E D(A). For bounded operators, we can take norm convergence. In the formula above, we used a uniform decomposition of the interval [0,1]. The formula still holds for a nonuniform decomposition. Proof for A,B bounded: We can assume Tn = eA/ne B/ n. Now by "telescoping",
snn _ Tn = snn _ T nn snl n
.x =
1. Let Sn
= e(A+B)/n
and
+ Tnsnl + ... _ Tnn n
nl
= L,T~(Sn  Tn)S;;;kl k=O
so nl
liS;;;  T:II ::::; L, IITnllkllSn  TnIIIISnllnkl k=O nl
::::; L,(max(IITnll, IISnll))n11ISn  Tnll k=O
::::; neiIAII+IIBIIIISn Tnll· Using a power series expansion, we see IISn  Tnll = O(1/n 2 ) and so IIS~ '> 0 as n '> 00. D A proof for unbounded operators can be found in [RSI].
T: II
11 Quasiclassical Analysis
In this chapter we investigate the connection between quantum and classical mechanics. More precisely, taking advantage of the fact that the Planck constant provides us with a small parameter, we compute some key quantum quantities  such as quantum energy levels  in terms of relevant classical quantities. This is called quasiclassical (or semiclassical) analysis. To do this, we use the Feynman path integral representation of the evolution operator (propagator) e iHt / n. This representation provides a nonrigorous but highly effective tool, as the path integral is expressed directly in terms of the key classical quantity  the classical action. The heuristic power of path integrals is that when treated as usual convergent integrals, they lead to meaningful and, as it turns out, correct answers. Thus to obtain a "quasiclassical approximation", we apply the method of stationary phase. Recall that the (ordinary) method of stationary phase expands the integral in question in terms of the values of the integrand at the critical points of the phase, divided by the square root of the determinant of the Hessian of the phase at those critical points. The difference here is that the phase  the classical action  is not a function of several variables, but rather a "functional", which (roughly speaking) is a function of an infinite number of variables, or a function on paths. Critical points of the classical action are the classical paths (solutions of Newton's equation) and the Hessians are differential operators. Thus we need some new pieces of mathematics: determinants of operators and elements of the calculus of variations. These are presented in supplementary Section 11.5 and Chapter 12 respectively. Below we consider a particle in ]Rd described by a Schrodinger operator fi2 H =  2m .1x
+ V(x).
(11.1)
Let L be a length scale for the potential, and 9 its size. So roughly, 9 sUPx lV(x) I and L = g(supx IV'V(x)l)l. Rescaling the variable as x + x' x/L, we find H = gH' where
H'
=
= =
fi'2   . 1 ,+ V'(x') 2m' x
where V'(x') = glV(Lx') and fi'/v'riI! = fi/(Lvmg). Now the potential V'(x') is essentially of unit size and varies on a unit length scale. The pa
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
122
11 Quasiclassical Analysis
rameter Ii/ is dimensionless. If h' / .;rn; < < 1, we can consider it a small parameter. As an example suppose V(x) is of the order lOOme 4 //1,2, where me4//1,2 is twice the ionization energy of the ground state of the hydrogen atom, and varies on the scale of the Bohr radius (of the hydrogen atom) L = /1,2/(me 2). Then /1,'/.;rn; = l/lO. In the expansions we carry out below, we always have in mind the operator H' and the dimensionless parameter n' with the primes omitted; that is, we think of (11.1) in dimensionless variables.
11.1 Quasiclassical Asymptotics of the Propagator The path integral (10.3) has the form of oscillatory integrals extensively studied in physics and mathematics. One uses the method of stationary phase in order to derive asymptotic expressions for such integrals. It is natural, then, to apply (formally) this method  with small parameter /1,  to the path integral, in order to derive a quasiclassical expression for the Schrodinger propagator eitH/n(y, x). We do this below. But first, we quickly review the basics of the method of stationary phase (in the finitedimensional setting, of course). The stationary phase method. We would like to determine the asymptotics of oscillatory integrals of the form
as /1, + 0 (here ¢ is a finite dimensional variable). The basic idea is that as /1, + 0, the integrand is highly oscillating and yields a small contribution except where 'VS(¢) = 0 (i.e., critical points). We study
1(/1,):= where!
E
r
JJRd
!(¢)eiS(e/»/nd¢
C[)(]Rd) and consider two cases:
1. supp(J) contains no critical points of S. Let 0 in ]Rd. Define
=
0e/> denote the gradient
where the innerproduct here is just the dot product in ]Rd. Note that LeiS(e/»/n = eiS(e/»/n, so integrating by parts gives
for any m.
11.1 Quasiclassical Asymptotics of the Propagator
123
2. supp(J) contains only one critical point, ¢ of S, which is nondegenerate (det S"(¢) i: 0). Expand S(¢) around ¢. Writing ¢  ¢ = VOO we obtain S(¢)//i
= S(¢)//i + ~(a, S"(¢)a) + O(v1ia3 )
where S"(¢) is the matrix of second derivatives of S: [S"(¢)]jk 8 2S(¢)/8¢j8¢k. So J(/i) = /id/2 e iS(f,)/li
r
JJRd
f(¢
+ Voo)ei (a,S"(f,)a)/2 eiO(Vfia
3
)da.
N ow we use the formula
(11.2) (We can derive this expression by analytically continuing J ea(a,S"a) from Re( a) > 0, though the integral is not absolutely convergent. Some care is needed in choosing the right branch of the square root function. An unambiguous expression for the right hand side is
where for a symmetric matrix A, sgn(A) denotes the number of positive eigenvalues minus the number of negative eigenvalues.) Noting that f( ¢+ voo) = f(¢) + O(VIi), we have the stationary phase expansion (for one critical point):
(11.3) We formally apply the method outlined above to the infinitedimensional integral (10.3). To this end, we simply plug the path integral expression (10.3) into the stationary phase expansion formula (11.3). The result is eiHt/li(y, x)
=
lx,y"
eiS(¢)/li D¢
L f, cp of
Mf,(det S,,(¢))1/2 e iS(f,)/li(1
+ O(v1i))
(11.4)
S
where Mf, is a normalization constant, and the sum is taken over all critical points, ¢, of the action S(¢), going from x to y in time t. Critical points and Hessians of functionals are discussed in the mathematical supplement, Chapter 12. Note that the Hessian S"(¢) is a differential operator. The problems of how to define and compute determinants of Hessians are discussed in the mathematical supplement Section 11.5. We will determine M := Mf" assuming it is independent of ¢ and V. For V = 0, we know the kernel of the propagator explicitly (see (2.15)):
124
11 Quasiclassical Analysis
eiHot/li(y, x) = (27rint/m)d/2eimlxyI2/2Iit. So in particular, eiHot/li(x, x) = (27rint/m)d/2. Now the righthand side of the expression (11.4) for eitHo/li(x, y) is (to leading order in n)
M( det S~ (¢o) )1/2 eiSo(¢o)/1i where the unique critical point is ¢o(s) = x + (y  x)s/t. Thus So(¢o) mly _x1 2 /2t, and S~(¢o) = ma;, an operator acting on functions satisfying Dirichlet boundary conditions. Comparison thus gives us
and therefore
eiHt/ li ( x) = '" (27ritn) d/2 (det( ma;)) 1/2 eiS(¢)/Ii(l y, _L m det S"(¢) ¢
cp
S
+
O(VYi))
(11.5)
as
n + O. This is precisely the quasiclassical expression we were looking for.
We now give a "semirigorous" derivation of this expression. We assume for simplicity that S has only one critical point, ¢, going from x to y in time t. Let {~j}~l be an orthonormal basis of eigenfunctions of S"(¢) acting on £2([0, t]) with zero boundary conditions (the eigenfunctions of such an operator are complete  see the remark in Section 9.14). So S"(¢)~j = /lj~j for eigenvalues /lj. For the nth order finite dimensional approximation to the space of paths in the path integral, we take the ndimensional space of functions of the form n
¢(n) =
¢ + Laj~j j=l
with aj E R Expanding s(¢(n)) around
¢ gives
where ~:= ¢(n) 
¢=
n
Laj~j.
j=l We also have
n
D¢(n)
= On
II daj j=l
(On some constant). Now using the fact that n
n
(~,S"(¢)~) = L aiaj(~i,S"(¢)~j) = L/lja;, i,j=l j=l
11.2 Quasiclassical Asymptotics of Green's Function
we have
(
i ¢(n):x+y
eiS(¢(n))/n Dq)n) = eiS(¢)/n
J
eiI: I" j a;/2n(1
125
+ O(a 3 /1i))Cn dn a
(as in Section 11.1, the integrals here are not absolutely convergent). Setting bj := aj / VIi this becomes
lin/2CneiS(¢)/n
J
ei I: I" j b;/2 (1
+ O(b3 V1i) )dnb
which is (see (11.2))
C n (27rili)n/2( det(S" (¢) IFn ))1/2e iS (¢)/n(1
+ O( VIi))
where Fn := {L:~ ajej} so that det(S"(¢)IFn ) = fG fJ,j. To avoid determining the constants Cn arising in the "measure" Dcp, we compare again with the free (V = 0) propagator. Taking a ratio gives us
which reproduces (11.5).
11.2 Quasiclassical Asymptotics of Green's Function Definition 11.1 Green's function GA(x, y, z) of an operator A is (A z) 1 (y, x), the integral kernel of the resolvent (A  z) 1 .
For A selfadjoint,
(A  z)l =
i
Ii
roo eiAt/n+izt/ndt
io
converges if Im(z) > O. Taking z = E f * 0, we define
+ if
(E real,
f
> 0 small), and letting
Note that the iO prescription is essential only for E E O'c(A), while for E E lR\O'c(A), it gives the same result as the +iO prescription. Here we are interested in the second case, and so we drop the iO from the notation. The above formula, together with our quasiclassical expression (11.5) for the propagator e iHt / \ yields in the leading order as Ii * 0 (dropping the iO from the notation)
126
11 Quasiclassical Analysis
(H  E)l(y,X)
=
~
1
00
o
L ?>
cp
K?>ei(S(?»+Et)/lidt S
where the sum is taken over the critical paths, ¢, from x to y in time t, and
K?>:=
(
m ) d/2 (det( mil;) ) detS"(¢)
1/2
27ritn
We would like to use the stationary phase approximation again, but this time in the variable t. Denote by [ = [(x, y, E) the critical points of the phase S(¢) + Et. They satisfy the equation
8S(¢)/8t = E. The path WE := ¢It=t is a classical path at energy E (see Lemma 12.36 of Section 12.6). Introduce the notation So(x, y, t) := S(¢) for a classical path going from x to y in time t. Then the stationary phase formula gives (in the leading order as II, * 0)
(H  E)l(y,x) =
~ LDYE2eiWWE/Ii,
(11.6)
WE
where the sum is taken over classical paths going from x to y at energy E. Here we have used the notation DYE2 := KWE(27rin)1/2(82S0/8t2)1/2It=t and we have defined WWE (x, y, E) := (So(x, y, t) + Et)lt=t (so WWE is the Legendre transform of So in the variable t). Lemma 11.2
D
WE
= (_1_8 2 S 0 )d1 det [8 2W WE 8 2 WWE _ 8 2WWE 8 2 WWE ] 27rin 8t2
8x8y
8E2
8x8E 8y8E
.
(
11.7
)
Proof We just sketch the proof. The first step is to establish
m8';) (m)
det( det(S"(w)) =
T
d
det
(8 S
2 0 )
8x8y
(11.8)
(we drop the subscript E for ease of notation). To see this, we use the fact that if for an operator A we denote by J A the d x d matrix solving AJ = 0 (the Jacobi equation) with J(O) = 0 and j(O) = 1, then (see (11.12)) det(m8';)
det(S"(w))
det Lma; (t) det JSIf(W)(t)·
Next, we use 1 JSIf(W)(t) = m
2S (88x8y 0)1
11.2 Quasiclassical Asymptotics of Green's Function (Equation (12.6) of Chapter 12), and for the free classical path x)(s/t),
~Lma2(t) = m
({;2S(1>O))l 8x8y
s
=_
127
1>0 = x + (y
(m)11 t
to arrive at (11.8). We can then show that
equals the determinant on the right hand side in equation (11. 7) (see the appendix to this section for details). D We will show in Section 12.6 (see Lemma 12.37) that the function WWE (the action at energy E) satisfies the HamiltonJ aco bi equation
h(x, 
8~;E ) = E.
Differentiating this equation with respect to y gives
8h 8 2 WWE _ 0 8k 8x8y  , and we see that the matrix (8 2 W WE /8x8y) has a zeroeigenvalue. Thus its determinant is zero. So if d = 1, (11.7) yields D
__ 8 2 W WE 8 2 W WE 8x8E 8y8E WE 
(d = 1).
(11.9)
Formula (11.6), together with (11.7) or (11.9), is our desired quasiclassical expression for Green's function (H  E)l(y, x). 11.2.1 Appendix
Proposition 11.3 At t
= t,
Proof We drop the subscripts from So and WWE to simplify the notation. Differentiating W = S + Etlt=f with respect to x, we obtain 8W 8x
= 8S + 8S 8t + E 8t 8x
which due to the relation 8S/8t
8t 8x
=  E gives
8x '
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11 Quasiclassical Analysis
aw ax Similarly,
as ax'
aw aE =t. This last equation, together with as/at =  E yields aw ay
as ay
and
aaE2w = ~ = _ (a s)1 aE at 2 2
2
Furthermore,
a2s axay
a2w a2w aE at axay axaE at ay a2s [ a2wa2w a2w a2w] = at 2  axay aE2 + axaE aEay
=+
and the result follows. 0
11.3 BohrSommerfeld Semiclassical Quantization In this section we derive a semiclassical expression for the eigenvalues (energy + V. We use the Green's levels) of the Schrodinger operator H = function expansion (11.6) from the last chapter. For simplicity, we will assume d = 1. Application of the expression (11.6) requires a study of the classical paths at fixed energy. Consider the trajectories from x to y at energy E. We can write them (using informal notation) as
;:Ll
cPn = cPxy ± no: where 0: is a periodic trajectory (from y to y) of minimal period, at the energy E, while cPxy is one of the four "primitive" paths from x to y at energy E, sketched in Fig. 11.1. V( )
\
V(;) /
x/~~~
E~ E~_~ x
y
x
Fig. 11.1. Primitive paths at energy E.
y
11.3 BohrSommerfeld Semiclassical Quantization
129
All these paths are treated in the same way, so we consider only one, say the shortest one. The space time picture of 00 as Ixl > 00 (as fast as some power of lxi, say); i.e., V(x) is a confining potential. We first define a couple of quantities which are familiar from statistical mechanics (see Chapter 9; we repeat some of the definitions here).
Definition 11.5 The partition function, Z(f3), at inverse temperature (3 is Z((3):= tr e{3H (the trace is welldefined since O"(H) fast).
= {En}o with En
> 00
>0
sufficiently
11.4 Quasiclassical Asymptotics for the Ground State Energy
131
Definition 11.6 The free energy, F, is
F(j3) :=
131 InZ(j3).
The free energy is a useful quantity for us here because of the following connection with the ground state energy of the Schrodinger operator H: lim F(j3) = Eo.
(3+=
This is the FeynmanKac theorem of Section 9.6. Our goal is to find the semiclassical asymptotics for Eo by deriving an asymptotic expression for Z(j3) from a path integral. As we have seen (Problem 10.1), the path integral expression for Z(j3) is
Z(T/n) = { } 0 =} det A = etr(inA) 5. det A = fL ev of A A We would like to define the determinant of a Schrodinger operator. Example 11.8 Let H = .::1 + V on [0, L] with zero boundary conditions (assume V is bounded and continuous). Then using the fact that for nEZ, sin(7rnx/L) is an eigenfunction of .::1 with eigenvalue (7rn/L)2, we obtain
II (H 
(7rn/ L )2) sin(7rnx/ L) II
=
IIV(x) sin(7rnx/ L) II
::;
(sup IVI) I sin(7rnx/ L) II,
and spectral theory tells us that H has an eigenvalue in the interval [(7rn/ L )2_ sup lVI, (7rn/ L)2 + sup IVI]. Since { J2/ L sin(7rnx/ L) I n = 1,2, ... } is an orthonormal basis in L2[0, L], we have
a(H) = {(7rn/L)2
+ 0(1) In E Z}.
fL ev H A = 00. For a positive matrix, A, we can define (A (s) := tr A = LA ev of A A
So trying to compute the determinant directly, we get
8
Problem 11.9 Show in this case that det(A) = e(~(O).
8.
11.5 Mathematical Supplement: Operator Determinants
Now for H = L1
133
+ Von [0, L]d with zero boundary conditions, A
ev of
H
exists for Re(s) > 1/2 (see Example 11.8 for d = 1). If (H has an analytic continuation into a neighbourhood of s = 0, then we define
So defined, det H enjoys Properties 14 above, but not Property 5. It turns out that for H = L1 + V, (H does have an analytic continuation to a neighbourhood of 0, and this definition applies. It is difficult, however, to compute a determinant from this definition. In what follows, we describe some useful techniques for computation of determinants. U sing the formula
('ex)
1
A s = r(s) Jo
tS1etAdt
for each An E u(H) leads to
Now A is an eigenvalue of H iff e tA is an eigenvalue of e tH (this is an example of the spectral mapping theorem  see Section 5.2), and so e tAn is the nth eigenvalue of e tH . Thus
This formula can be useful, as it may be easier to deal with tr( e tH ) c tH (x, x)dx than tr(H S).
J
Example 11.10 We consider H = L1 in a box B periodic boundary conditions. In this case
[L/2, L/2]d with
in B x B, if B is very large, and so tr e tH
=
r
JBXB
e tH (x, x)dx
~
r (21ft)d/2 = (21ft)d/2 vo l(B).
JB
But calculation of det H by this method is still a problem.
134
11 Quasiclassical Analysis
Remark 11.11 Often (and in all cases we consider here), we have to compute a ratio, ~:~ ~ , of determinants of two operators A and B, such that det A and det B must be defined through a regularization procedure such as the one described above, but the determinant of the ratio AB 1 can be defined directly. Since ~:~~ = det(AB 1 ), we can make sense of the ratio on the left hand side without going to regularization.
The most useful calculational technique for us is as follows. Let A and B be Schrodinger operators on £2([0, T]; JRd) with Dirichlet boundary conditions. Denote by JA the solution to AJA = 0 with JA(O) = 0, jA(O) = 1 (J a d x d matrix valued function on [0, TD. Then one can show (see, eg., [LS, KID detA det B
det JA(T) det JB(T)·
Remark 11.12 If A = SI/(¢;) for a critical path matrix along ¢; (see Section 12.5).
(11.12)
¢;,
then JA is the Jacobi
Problem 11.13 Let A(T) be the operator a;+q(t) defined on £2([0, T]; JRd) with Dirichlet boundary conditions, and let JA be the corresponding Jacobi matrix. Show that the functions det A(T) and det J(T) have the same zeros of the same multiplicities (to is a zero of f (t) of multiplicity n if a k / at k f (to) = 0 for k = 0, ... , n  1 and an /atnf(to) i= 0).
12 Mathematical Supplement: the Calculus of Variations
The calculus of variations, an extensive mathematical theory in its own right, plays a fundamental role throughout physics. This chapter contains an overview of some of the basic aspects of the variational calculus. This material in used in Chapters 11 and 13, in conjunction with the path integral introduced in the previous chapter, to obtain useful quantitative results about quantum systems.
12.1 Functionals The basic objects of study in the calculus of variations are functionals, which are just functions defined on Banach spaces (usually spaces of functions). If we specify a space, X, then functionals on X are just maps S : X + R In the calculus of variations, one often uses spaces other than L2(]Rd), and which are not necessarily Hilbert spaces. Among the most frequently encountered spaces are the LP spaces (p ?: 1)
and the Sobolev spaces HS(]Rd), s = 1,2, ... , introduced in Section 1.5. Recall that the Sobolev spaces are Hilbert spaces. The space LP(]Rd), endowed with the norm
II¢IILP
:=
(ld 1¢(X)lPdX)
liP,
is a Banach space, but not (for p 1= 2) a Hilbert space. We will also encounter Sobolev spaces of vectorvalued functions,
(s = 1,2, ... ) where f2 is an open domain in ]Rd, and aa was defined in Section 1.5. Spaces of continuously differentiable functions also arise frequently. For k E {O, 1,2, ... }, and an open set f2 C ]Rd, we define C k (f2; ]Rm) to be the set of all functions ¢ : f2 + ]Rm such that aa¢ is continuous in f2 for all letl :::; k, and for which the norm
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
136
12 Mathematical Supplement: the Calculus of Variations
11¢IICk(l?;IR=)
L
sup la"'¢(x)1 lal::;k xED
:=
is finite. Equipped with this norm, C k (Q; JRm) is a Banach space. Example 12.1 Here are some common examples of functionals, S, and spaces, X, on which they are defined.
= L2([a, bj; JR), f E X* = X is fixed (the meaning of the notation X* will be explained in the next section), and
1. X
S: ¢
lb
f+
f¢.
Note that S(¢) is welldefined, by the CauchySchwarz inequality. 2. Evaluation functional: X = C([a, b]), Xo E (a, b) fixed, and S: ¢
f+
¢(xo)
Compare this with the first example by taking f(x) = 8(x  xo) (E X*; in this case X* i X). 3. Let V : JRm + JR be continuous. Set X = {¢ : JRd + JRm I V(¢) E LI(JR d)}, and take S:¢f+
4. Dirichlet functional: X
HI(JRd), and
=
S:
5. Classical action: fix a,b a, ¢(T) = b}, and define S:
r V(¢).
JIRd
E
¢
f+
¢
f+
~
2
r
JIRd
IV'¢12.
JR m , set X = {¢
E
C I ([o,T];JRm) I ¢(O)
loT {~ml¢12  V(¢)}.
6. Classical action: Let L : JRm x JRm (a Lagrangian function), and
S: ¢
f+
+
JR be a twice differentiable function
loT L(¢, ¢)dt.
Here, X is as in Example 5. 7. Action of a classical field theory: fix X
f, g, E HI (JRd; JR m), set = {¢ E HI(JR d X [0, T]; JRm) I ¢(x,O) = f(x), ¢(x, T) = g(x)},
and define S:
where V : JRm
¢ +
f+
loT ld {~lat¢12 + ~IV'x¢12 + V(¢)}.
JR is a continuous function.
12.2 The First Variation and Critical Points
137
8. Quadratic form: let B be a a selfadjoint operator on a Hilbert space, X = D(B), and set
s: 9. Action of an EM field: X
S:A
f4
cP
1
f4
= {A
E
~ loT
l3
Hl(1R3 x [0, TJ; 1R3 )
{IAI2 + 1\7
10. Lagrangian functional: Suppose £ : IR x IRd is a twice differentiable function, and set
S : cP
f4
(12.1)
2(cp,Bcp).
+
X
\7. A
= O}
and
AI2}.
IR (a Lagrangian density)
J
12(cp(x, t), \7cp(x, t))dxdt
(here X is a space of functions on spacetime IR~ x IRt, the form of which depends on the form of 12). We will encounter many of these functionals in applications to quantum mechanics and quantum field theory.
12.2 The First Variation and Critical Points The notion of a critical point of a functional is a central one. It is a direct extension of the usual notion of a critical point of a function of finitely many variables (Le., a place where the derivative vanishes). The solutions of many physical equations are critical points of certain functionals (action or energy functionals) . In what follows, the spaces X on which our functionals are defined will generally be linear (i.e. vector) spaces or affine spaces. By an affine space, we mean a space of the form X = {CPo+CP cp E X o}, where CPo is a fixed element of X, and Xo is a vector space. We will encounter examples of functionals defined on nonlinear spaces when we study constrained variational problems in Section 12.3. Let X be a Banach space, or an affine space based on a Banach space. 1
Definition 12.2 A path, CP>., in X is a differentiable function I ::1 A from an interval I C IR containing 0, into X.
f4
CP>.
Definition 12.3 The tangent space, Tq,X, to X at cp E X is the space of all "velocity vectors" at cp:
Tq,X:=
{%>.I>.=o
1
CPA is a path in X,CPo = cp}.
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12 Mathematical Supplement: the Calculus of Variations
Problem 12.4 Check that Tq,X is a vector subspace of X (or Xo in the affine case).
In fact, if X is a Banach space, then Tq,X = X (and if X is an affine space based on the Banach space X o , Tq,X = Xo). To see this, just note that for any ~ E X, ¢).. := ¢ +,\~ is a path in X, satisfying ¢o = ¢ and o¢)../o'\ = ~ (the reader is invited to check the corresponding statement for affine spaces). So the tangent space to a linear space is not very interesting. The notion of a tangent space is useful when working in nonlinear spaces (we will see an example of this shortly).
Definition 12.5 A variation of ¢ E X along such that ¢o = ¢ and 0¢)../0'\1)..=0 =~.
~ E
Tq,X is a path, ¢).. in X,
Example 12.6 Define X xy = {¢ E C([a,b],X) I ¢(a) = x,¢(b) = y} (an affine space). Then Tq,Xxy = Xoo (see Fig. 12.1), and an example of a variation of ¢ E X xy in the direction ~ E Xoo is ¢).. = ¢ +,\~ E X xy . y
x
Fig. 12.1. Variations of a path. We wish to define a notion of differentiation of functionals which is a direct extension of usual differentiation of functions of finitely many variables. To do so, it is useful to introduce a new concept ~ that of the dual space.
Definition 12.7 1. A bounded linear functional on a Banach space, B, is a map l : B  ',I" 1>'=1"=0 = ~, and
tl" ¢>',I" 1>'=1"=0 = 'TJ.
t>.
Definition 12.18 Let 8 : X  JR be a functional. We say 8 is twice differentiable is there is a bounded linear map 8 2 S(¢) : Tq,X  (Tq,X)* (called the Hessian or second variation of 8 at ¢) such that
(12.3) for all
~,'TJ E
Tq,X and all variations ¢>',I" of ¢ along
~
and 'TJ.
Remark 12.19 1. The Hessian 8 2 S(¢) can also be defined as the second derivative of 8(¢), i.e., 8 2 8(¢) = 8· 88(¢). That is, we consider the map ¢ 17 88(¢) and define, for'TJ E Tq,X, 8 2 8(¢)'TJ := 88(¢>.), where ¢>. is a variation of ¢ along 'TJ. 2. We will often use the notation 8"(¢) to denote 8 2 8(¢).
t>.
Computations of the second derivatives of the functionals in our list of examples are left as an exercise (again, we suppose for simplicity that the action of the dual space is just given by integration).
Problem 12.20 Continuing with our list of examples of functionals, show that
= D2V(¢) (a matrix multiplication operator). = ,,1 (the Laplacian). 5. S"(¢) = m81 D2V(¢) (a Schrodinger operator) acting on functions satisfying Dirichlet boundary conditions: ~(O) = ~(T) = O.
3. 8"(¢) 4. 8"(¢)
6.
S"(¢) = d/dt(8JL)d/dt  (d/dta~¢L)
+ 8~L,
(12.4)
with Dirichlet boundary conditions. 7. 8" (¢ ) = 0 + V" (¢), with Dirichlet boundary conditions: ~ (x, 0) ~(x, T) = O.
=
12.5 Conjugate Points and Jacobi Fields
143
8. S"(¢) = B.
9. S"(¢)
= D (the D'Alembertian).
10.
The following criterion for a critical point to be a minimizer is similar to the finitedimensional version, and the proof is left as an exercise. Theorem 12.21 Let ¢ be a critical point of a twice continuously differentiable functional S : X + lR. 1. If ¢ locally minimizes S, then S" (¢) 2 0 (meaning (S" ( ¢) ~,~) 2 0 for all ~ E
Tq,X).
2. If S"(¢) > c, for some constant c > 0 (i.e. (S"(¢)~,~) 2 cll~ll~ for all ~ E Tq,X), then ¢ is a local minimizer of S. Problem 12.22 Prove this.
Let us now pursue the question of whether or not a critical point of the classical action functional S(¢) =
lT
L(¢(s), ¢(s))ds
(which is a solution of the EulerLagrange equation  i.e., a classical path) minimizes the action. As we have seen, the Hessian S"(¢) is given by (12.4). We call a~L the generalized mass.
a;L
Theorem 12.23 Suppose a~2L > o. Suppose further that is a bounded function. Then there is a To > 0, such that S" (¢) > 0 for T ::; To.
Proof for L = !fj¢2  V(¢). In this case S"(¢) = md 2/ds 2  V"(¢), acting on L2([0, T]) with Dirichlet boundary conditions. Since inf a( _d 2/ ds 2) = (7f/T)2, we have, by Theorem 5.25, d2/ds 2 2 (7f/T)2. So S"(¢) 2 m(7f/T)2sup IV"I, which is positive for T sufficiently small. D Corollary 12.24 For T sufficiently small, a critical point of S (i.e., a classical path) locally minimizes the action, S.
12.5 Conjugate Points and Jacobi Fields In the remainder of this chapter, we study the classical action functional and its critical points (classical paths) in some detail. While such a study is of obvious importance in classical mechanics, it will also prove useful in the quasiclassical analysis of quantum systems that we undertake in Chapters 11 and 13.
144
12 Mathematical Supplement: the Calculus of Variations Thus we consider the action functional S(¢)
=
lt
L(¢(s), ¢(s))ds.
We have shown above that ift is sufficiently small, then S"(¢) > 0, provided (fP L/8¢2) > 0. So in this case, if ¢ is a critical path, it minimizes S(¢). On the other hand, Theorem 12.21 implies that if ¢ is a critical path such that S" (¢) has negative spectrum, then ¢ is not a minimizer. We will show later that eigenvalues of S"(¢) decrease monotonically as t increases. So the point to when the smallest eigenvalue of S" (¢) becomes zero, separates the t's for which ¢ is a minimizer, from those for which ¢ has lost this property. The points at which one of the eigenvalues of S"(¢) becomes zero playa special role in the analysis of classical paths. They are considered in this section. In this discussion we have used implicitly the fact that because S"(¢) is a Schrodinger operator defined on L2 ([0, t]) with Dirichlet (zero) boundary conditions, it has a purely point spectrum running off to +00. We denote this spectrum by {An(t)}]'" with .An(t) + +00 as n + 00. Note that if ¢ is a critical point of S on [0, tJ, then for T < t, ¢T := ¢I[O,T] is a ~ritical po~nt of Son [O,T]. Thus for T :::; t, {.An(T)} is the spectrum of S"(¢T) = S"(¢) on [0, T] with zero boundary conditions. We specialize now to the classical action functional S(¢)
=
lt{; 1¢1
2 
V(¢)}
on the space X = {¢ E C 1 ([0,t];lRd) I ¢(O) = x, ¢(t) = y}, and continue to denote by ¢, a critical point ofthis functional (classical path). Theorem 12.25 The eigenvalues .An(T) are monotonically decreasing in T.
Sketch of proof Consider .AI(T), and let its normalized eigenfunction be 'l/J1. Define ¢1 to be 'l/J1 extended to [0, T+ E] by 0. So by the variational principle,
Further, equality here is impossible by uniqueness for the Cauchy problem for ordinary differential equations, which states the following: if a solution of a linear, homogeneous, secondorder equation is zero at some point, and its derivative is also zero at that point, then the solution is everywhere zero. To extend the proof to higher eigenvalues, one can use the minmax principle. 0
°for some n is called a So c = ¢( To) is a conjugate point to x, then S" (¢) on [0, TO] has a ° eigenvalue. That is, there is some nonzero L2([0, TO]) with = = °such that Definition 12.26 A point ¢( TO) such that .An (TO) = conjugate point to ¢(O) = x along ¢. if
~ E
~(O)
~(TO)
12.5 Conjugate Points and Jacobi Fields
s" (¢)~ = O.
145
(12.5)
This is the Jacobi equation. A solution of this equation with called a Jacobi vector field.
~(O) =
0 will be
Definition 12.27 The index of S"(¢) is the number of negative eigenvalues it has (counting multiplicity) on L2([0, t]) with zero boundary conditions. We recall that for T small, S"(¢) has no zero eigenvalues on [0, T] (Theorem 12.23). Combining this fact with Theorem 12.25 gives the following result. Theorem 12.28 (Morse) The index of S"(¢) is equal to the number of points conjugate to ¢(O) along ¢, counting multiplicity (see Fig. 12.2).
Fig. 12.2. Index =
#
of conjugate points.
The picture that has emerged is as follows. For sufficiently small times, a classical path ¢(O) locally minimizes the action. As time increases, the path might lose this property. This happens if there is a point in the path conjugate to ¢(O). Example 12.29 An example of a conjugate point is a turning point in a onedimensional potential (see Fig. 12.3, and remember that we are working with the functional of Example 12.1, no. 5). V(x) ',j
x
a Fig. 12.3. A turning point. The classical path ¢ starts at a, and turns back after hitting b at time
T.
Now
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12 Mathematical Supplement: the Calculus of Variations
SI/(¢) = ma;  vl/(¢) and it is easy to check that SI/(¢)¢ = 0 (just differentiate Newton's equation). Since ¢(O) = ¢(T) = 0 (the velocity at a turning point is zero), b is conjugate to a. We return to the Jacobi equation (12.5), and consider its fundamental solution, J(s). J(s) is the d x d matrix satisfying
SI/(¢)J
=0
with the initial conditions
J(O) = 0
and
j(O) = 1.
J is called the Jacobi matrix. Proposition 12.30 The Jacobi matrix has the following properties 1. For any h E ]Rd, Jh is a Jacobi field. Conversely, any Jacobi field is of the form J h for some h E ]Rd.
2. ¢(TO) is a conjugate point to ¢(O) iff J(TO) has a zeroeigenvalue, i.e. det J(TO) = O. Proof. 1. The first part is obvious. To prove the second part let ~ be a Jacobi field, and let h = ~(O). Then ~ := Jh satisfies the same differential equation as ~ with the same initial conditions. Hence ~ = ~, and therefore ~ = Jh. 2. We have shown above that ¢( TO) is a conjugate point iff there is a Jacobi field ~ such that ~(TO) = O. By the previous statement, there is h 1= 0 such that ~ = Jh, which implies J(To)h = O. So J(TO) has a zero eigenvalue (with eigenvector h), and det J (TO) = O.
o
Now we give the defining geometric/dynamic interpretation of J. Consider a family of critical paths ¢v (s) starting at ¢( 0) with various initial velocities v E ]Rd. Denote v = ¢(O). Then
J(s) =
a¢;~s) Iv=v
is the Jacobi matrix (along ¢). Indeed, ¢v satisfies the equation as(¢v) = O. Differentiating this equation with respect to v, and using that SI/(¢) a¢aS(¢), we find
o=
a u¢S '" (¢v) = S1/ ( ¢v ) a¢v av av·
Thus, a¢v/avlv=v satisfies the Jacobi equation. Next,
12.6 The Action of the Critical Path
and
8 . 8v
8 8v
= v =
¢v(O)
147
1
which completes the proof.
12.6 The Action of the Critical Path In this section we consider the situation of Example 12.1 no. 6, and its special case, Example 12.1 no. 5. Thus we set
x
= {¢ E C 1 ([0, t];ll~m) I ¢(O) = x, ¢(O) = y},
and S(¢) =
lot L(¢(s), ¢(s))ds.
Suppose ¢ is a critical path for S with ¢(O) = x and ¢(t) = y. We will denote the action of ¢ by So(x, y, t) := S(¢) (the action from x to y in time t). The momentum at time s, is k(s) := (8L/8¢)(¢(s), ¢(s)). Lemma 12.31 We have
8So 8x
= k(O)
Proof. Again, we specialize to L = integration by parts, we find 8S(¢)/8x
~~o
and
ml¢1 2/2 
=
k(t).
V(¢). Using the chain rule and
lot {m¢. 8¢/8x  V'V(¢)· 8¢/8x}ds = lot {( m¢  V'V(¢)) . 8¢/8x} + m¢. 8¢/8xl~ =
which, si~ce ¢ is a critical point, (8¢/8x)(t) = 0, and (8¢/8x)(0) = 1, is just m¢(O) = k(O), as claimed. The corresponding statement for 8So/8y is proved in the same way, using (8¢!8y)(0) = 0, and (8¢/8y)(t) = 1. 0 This lemma implies 8k(0)/8y = 8 2 S0 (x, y, t)/8x8y. On the other hand, for L = ~1¢12_ V(¢), 8k(0)/8y = (8y/8k(0))1 = mJ 1(t) (as m(8y/8k(0)) is the deriv?tive of the classical path ¢ at t with respect to the initial velocity k(O)/m = ¢(O)). This gives
8 2 S 0 (x, y, t) 8x8y which establishes the following result:
= _
m
J1() t
(12.6)
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12 Mathematical Supplement: the Calculus of Variations
Proposition 12.32 If y is a conjugate point to x then det( 8 2 s;;~::,t)) =
00.
The following exercise illustrates this result for the example of the classical harmonic oscillator. Problem 12.33 Consider the onedimensional harmonic oscillator, whose • 2 Lagrangian is L = !!}¢2  m~ ¢2. Compute
So(x, y, t)
=
2 . ~ ) [(x 2 sm wt
+ y2) cos(wt) 
2xy]
and so compute
fPSo(x,y,t)/8x8y
=
~.
smwt Note that this is infinite for t = mf / w for all integers n. Thus the points ¢(mf/w) are conjugate to ¢(O). Lemma 12.34 (HamiltonJacobi equation) The action So(x, y, t) satisfies the HamiltonJacobi equation
8So/8t = h(y, 8So/8y)
(12.7)
where h is the classical Hamiltonian function associated with L. Proof. The integrands below depend on s (as well as the parameters x,y, and t), and ¢ denotes 8¢/8s. Since So = S(¢) = J~ L(¢, ¢)ds, we have
8S(¢)/8t = L(¢, ¢)Is=t
+ lot (8L/8¢. 8¢/8t + 8L/8¢. 8¢/8t)ds
= L(¢, ¢)Is=t + 8L/8¢. 8¢/8tl:~&
+ lot (8L/8¢ Since ¢(s) = y + ¢(t)(s  t)
+ O((s 8¢
at Is=t
Using this, the fact that
d/ds(8L/8¢» . 8¢/at.
t)2) (here we used ¢(t) = y), we have =
"
¢(t).
¢ is a critical point of S,
and 8¢/8tl s=0 = 0, we find
8S(¢)/8t = ((8L/8¢) . ¢  L(¢, ¢»Is=t
= h(¢,8L/8¢)ls=t. Since ¢Is=t follows. D
=
y and, by Lemma 12.31, (8L/8¢)ls=t
=
8So/8y, the result
12.6 The Action of the Critical Path
149
Lemma 12.35 (Conservation of energy) (8L. energy (¢):= .' ¢  L)
8¢
I
_ = const.
=
Proof We compute
Since for ¢ = ¢, the expression on the right hand side vanishes, ~~ . J;  LI is a constant (which we will denote by E). Now
hence E is the energy of ¢. 0 We want to pass from a timedependent to a timeindependent picture of classical motion. We perform a Legendre transform on the function So(x, y, t) to obtain the function W(x, y, E) via W(x, y, E)
=
(So(x, y, t)
+ Et)lt:8So /8t=E'
(12.8)
We denote by [ = [(x, y, E), solutions of
8So/8tl t=t = E.
(12.9)
There may be many such solutions, so in the notation W, we record the classical path ¢ we are concerned with (for which ¢(O) = x, ¢(f) = y). Note that from the energy conservation law
we have W(x, y, E)
=
i
t
8L
~
o 8¢
h
'. ¢ds = _k . dx,
where f k· dx := f~ k(s) . ¢(s)ds and, recall, k(s) = 8L/8J;1=· Lemma 12.36 ¢It=t is a classical path at energy E. Proof By the HamiltonJacobi equation (12.7) and the conservation of energy (Lemma 12.35), ¢It=t is a classical path with energy 8So(x, y, t)/8tlt=t, which, by (12.9), is just E. 0
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12 Mathematical Supplement: the Calculus of Variations
Lemma 12.37 W¢ satisfies the HamiltonJacobi equation
(12.10)
h(x, oW¢/ox) = E. Proof. Using (12.9) and Lemma 12.31, we compute oW¢ = oSo It=t + (oSo ax ax at
+ E)lt=t of = ax
So by conservation of energy, E
oSo It=t = k(O). ax
(12.11)
= h(x,k(O» = h(x, oW¢/ox). D
12.7 Appendix: Connection to Geodesics The next theorem gives a geometric reinterpretation of classical motion. We consider a classical particle in ]Rd with a potential V(x). Recall the notation f(x)+ := max(f(x) , 0). Theorem 12.38 (Jacobi theorem) The classical trajectory of a particle at an energy E is a geodesic in the Riemannian metric
(u, v)x = 2(E  V(x»+u· v (where u· v is the inner product in ]Rn) on the set {x classically allowed region).
E
]RnlV(x) ::; E} (the
Proof. By the conserv3;tion of energy (Lemma 12.35), a classical path ¢ has a fixed energy E = ml¢1 2/2 + V(¢). Hence ¢ is a critical point of the action S(¢) = J(ml¢1 2/2 V(¢»ds among paths in M := {¢ I ml¢1 2/2+ V(¢) == E}. Using the relation ml¢1 2/2 + V(¢) = E, we can write ml¢1 2/2  V(¢) as
ml¢1 2/2  V(¢) Hence
¢ is
=
ml¢1 2 = myh(E  V(¢»/ml¢l.
a critical point of the functional L(¢) :=
Jml¢IV
2(E  V(¢»/mds
on M. This functional gives the length of the path in the metric above. On the other hand, we can reparameterize any path with V(¢) < E so that it satisfies ml¢1 2/2 + V(¢) = E. Indeed, replacing ¢(s) with ¢(A(S», we note 1¢(A)1 2 = ('\(s»21¢(A(S»12. We must solve ('\(S»2
which we can rewrite as
= ~ E .V(¢(A(S») m 1¢(A(S»12
12.7 Appendix: Connection to Geodesics
r
io
1¢('\(s'))I~(s')
JE  V(¢('\(s')))
ds'
=
151
(2s
V:;;:
which can be solved. Since the functional L( ¢) is invariant under reparameterizations (if'\ = a(s), a' > 0, then 1¢(s)lds = I¢('\)I~; gld,\ = 1¢('\)ld'\), if ¢ is a critical point of L(¢), then so are different reparameterizations of ¢, and in particular the one, ¢, with the energy E. This ¢ is also a critical point of L(¢) on M, and, by the above, a critical point of S(¢) at energy E. Thus classical paths are geodesics up to reparameterization (so they coincide as curves). D Problem 12.39 Check that the EulerLagrange equation for critical points of L(¢) on M yields Newton's equation.
13 Resonances
The notion of a resonance is a key notion in quantum physics. It refers to metastable states  i.e., to states which behave like stationary states for long time intervals, but which eventually break up. In other words, these are states of the continuous spectrum (i.e. scattering states), which for a long time behave as if they were bound states. In fact, the notion of a bound state is an idealization: most of the states which are (taken to be) bound states in certain models, turn out to be resonance states in a more realistic description of the system. We sketch briefly the mathematical theory of resonance states. To characterize them in terms of spectral properties, we have to use more detailed information than just the spectrum, such as spectral densities, or else generalize the notion of the spectrum altogether. The resulting theory, which is elegant and powerful, is not considered here. Rather, we study the particular but central case of resonances due to tunneling, on which we illustrate some of the mathematics and physics involved.
13.1 Tunneling and Resonances Consider a particle in a potential V(x), of the form shown in Fig. 13.1; i.e., V(x) has a local minimum at some point xo, and V(xo) > limsupx+(X) V(x), for x in some cone, say. If V(x) + 00 as x + 00 (in some cone of directions), then the corresponding Schrodinger operator, H, is not bounded from below. "would be
boun~d state" Vex)
tunnels under barrier ro
x
Fig. 13.1. Unstable potential.
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
154
13 Resonances
If the barrier is very thick, then a particle initially located in the well spends lots of time there, and behaves as if it were a bound state. However, it eventually tunnels through the barrier (quantum tunneling) and escapes to infinity. Thus the state of the particle is a scattering state. It is intuitively reasonable that
1. the energy of the resonance ::::; the energy of a bound state in the well 2. the resonance lifetime is determined by the barrier thickness and height, and n. Since the resonances are very close to bound states if the barrier is large or n is small (there is no tunneling in classical mechanics), we try to mimic our treatment of the ground state (Section 11.4). But right away we run into a problem: if V(x) 1+ 00 as x + 00 in some directions, then
Z(j3) = tr e/3H =
00.
The paradigm for this problem is the divergence of the integral
for ,\ :::; O. However, we can define this integral by an analytic continuation. Z('\) is welldefined for Re(,\) > 0, and it can be continued analytically into ,\ E IR as follows. Move ,\ from Re('\) > 0 into Re('\) :::; 0, at the same time deforming the contour of integration in such a way that Re('\a 2 ) > 0 (see Fig. 13.2).
Fig. 13.2. Contour deformation. Of course, in this particular case we know the result:
Z('\) =
( 2,\)1/2 = z. (21'\1)1/2 7r
for ,\ < 0 (which is purely imaginary!).
7r
13.2 The Free Resonance Energy
155
There is a powerful method of rotating the contour which is applicable much beyond the simple integral we consider. It goes as follows. For 8 E lR, we change variables via a = eob. This gives Z(A) = eO
1
00
e)..e 28 b2 /2db.
(13.1)
The integral here is convergent and analytic in 8 as long as (13.2) We continue it analytically in 8 and A, preserving this condition. In particular, for A E lR, we should have 7r/4 < Im(8) < 37r/4. Now observe that the right hand side of (13.1) is independent of 8. Indeed, it is analytic in 8 as long as (13.2) holds, and is independent of Re(8) since the latter can be changed without changing the integral, by changing the variable of integration (b ft Co' b, 8' E lR). Thus we have constructed an analytic continuation of Z(A) with Re(A) > 0 into a region with Re(A) < O. In fact, we have continued this function onto the second Riemann sheet! Finally, we define Z(A), for A < 0, by (13.1) with 8 obeying (13.2).
13.2 The Free Resonance Energy With a bit of wisdom gained, we return to the problem of defining the partition function Z({3) and free energy F({3) in the case when V(x) 00, as x + 00, in some directions (or more precisely, sUPxEr V(x) < 00 for some cone r). We begin with a definition.
+
Definition 13.1 A family of operators, H(8), for 8 in a complex disc {181 ::::: E}, will be called a complex deformation of H if H (0) = H, H (8) is analytic in {I 8 I : : : E}, and H (8) is an analytic continuation of the family
H(8)
for 8
E
lR, where U(8), 8
E
= U(8)1 HU(8)
(13.3)
lR, is a oneparameter unitary group.
If V (x) has certain analytic properties, then H (8) can be constructed by changing the variables in H as x ft xeo for 8 E lR, and then continuing the result analytically in 8. Let, for example, V(x) = Cx3 as x + +00. Then V(eOx) = Ce3 0x 3 . Take 8 = i7r/3. Then V(eOx) = Cx 3 is positive in the direction x + +00. The group U(8) here is the group of dilations:
for 7/J E L2(lRn). Assume that we can construct a complex deformation, H(8), of H, such that
156
13 Resonances
Z((3) = tre!3 H (O)
0, or more generally for I()I :S f, Im(()) > O. (In fact, it is not a simple matter to define the exponential e!3 H (O) rigorously  see [SV]. Below we will deal formally with e!3 H (O), assuming it has all the properties which can be derived from the power series expression for the exponential.) Proposition 13.2 If tre!3 H (O) is independent of ().
T e , so the transition is discontinuous: decay jumps from W3 to a trajectory at the bottom of the barrier.
no trajectories for 't>'tc
't
= co
Fig. 13.6. Discontinuous transition. Thus for intermediate temperatures, the nature of decay depends radically on the geometry of the barrier.
13.5 Preexponential Factor for the Bounce
161
The results above support the following physical picture of the tunneling process. With the Boltzmann probability (canst)e E/ T , the particle is at an energy level E. The probability of tunneling from an energy level E is (canst)e SE / Ii where SE is the action of the minimal path at energy E. The probability of this process is (canst)eE/TSE/Ii. Thus the total probability of tunneling is (canst)
J
eE/TSE/Ii
~
(canst)eEo/TSEo/1i
where Eo solves the stationary point equation fJ E fJE (T
SE
1
1 fJSE fJE
+ h) = T + Ii
= O.
Here fJSE I fJE is the period of the trajectory under the barrier at the energy level E, and SEo + nEolT is the action of a particle (in imaginary time) at energy Eo corresponding to the period niT = T.
13.5 Preexponential Factor for the Bounce The bounce solution, Wb, presents some subtleties. Since Wb breaks the translational symmetry of Se(¢), uh is a zeromode of S~(Wb): S~/(Wb)uh
= o.
To establish this fact, simply differentiate the equation fJSe(Wb) respect to s and use the fact that S~' = fJ2Se (see Section 12.4). As a result, we have two problems:
o with
1. S~/(Wb) has a zero eigenvalue, so formally
[detS~/(wb)rl/2 =
(canst)
Je(f"S~'(Wb)f,)j2IiD~
=
00
(13.7)
2. Wb has one zero (see Fig. 13.7), and so the SturmLiouville theory (from the study of ordinary differential equations) tells us that, in fact, S~(Wb) has exactly one negative eigenvalue.
cOb s
Fig. 13.7. This gives a second reason for the integral (13.7) to diverge.
162
13 Resonances
To illustrate these divergences, we change variables. Let {~d be an orthonormal basis of eigenfunctions of S~(Wb) with eigenvalues )..k, in increasing order. For ¢ near Wb, write ¢ = Wb + ~ with
Then
+L
00
Se(¢) ~ Se(Wb)
But
)..0
'oa~/2Iidao
=
00
00
(2)..0)1/2 trn
=
i 12 )..0 7rn
1 1/ 2
The second integral, correctly treated, is shown to contribute (see the following section) S;I/2TV27rn (13.8) where Sb is the action of the "bounce", Se(Wb). Hence
r
Jnear
eSe(
1fe/> = we/> + "2 Consequently,
fI
:=
UHU 1 =
'P'
J:~(1ff)2 +
V(¢o
+~) :
acting on L2 (Q, d/Lc(~)). Hence
fI = ~
J:
{(1ff)2 + ~A2~ + O(>,e)} : +V(¢o)
where
A2
:=
HessV(¢o)
=
L1x
+ F"(>"¢o).
N ow we change the covariance
To this end we write
d/LC = iPd/LC A, where
iP =
ZCAZcle(~,(C1CA1)~)/2.
We define another unitary operator
U1 : L 2 (Q,d/Lc)
+
L 2 (Q,d/LCA)
by (U1F)(~) := iPl/2 F(~). Then
Define on L 2 (Q,d/LCA)' Then
H new = Hquad where
+
J:
O(>..e) : +V(¢o)
(14.32)
14.8 Quadratic Approximation
Hquad =
~
J:('if~ + ~A2~)
: .
193 (14.33)
Passing now to the corresponding new creation and annihilation operators (see Section 14.6), we have Hquad
=
J
a*(x)Aa(x)dx.
(14.34)
Thus we have a new free theory with covariance ~Al, and with oneparticle operator A. A determines entirely the spectral properties of Hquad. Let us summarize the above analysis. We began with a Hamiltonian of the form (14.28), where the functional V(¢) has a minimizer ¢o, acting on the space L2( Q, d/Lc). We constructed a unitarily equivalent Hamiltonian (14.32) acting on the space L 2 (Q,d/LCA) where the covariance is CA = (2A)1 with A = (HessV(¢0))1/2. Up to the additive constant V(¢o) (the classical energy of ¢o), the new Hamiltonian is a perturbation of the quadratic Hamiltonian (14.33) or (14.34) by a term which is of higher order  O(e)  in the new quantum field ~. Consequently, one expects that the lowenergy spectrum of the original (and new) Hamiltonian is determined by the spectrum of this quadratic Hamiltonian (14.33) or (14.34), which, in turn, is determined by the spectrum of A, the square root of the Hessian of V (¢) at ¢o. Now we briefly discuss the translationinvariant case, continuing to work with real fields. First, we take another look at the minimizer ¢o. As a critical point of the potential functional V(¢), it is a stationary solution of the classical equations of motion for the classical Hamiltonian functional H(¢, 'if)
=
J~'if(X)2 +
V(¢)
(see Section 14.2.4). The corresponding Hamilton's equations are equivalent to the "Newton equation" (14.35) ¢ = aV(¢), which is nothing but the familiar nonlinear wave equation D¢ + '\F'('\¢)
=0
with 0 = al  ,1x the D'Alembert operator (this is the classical dynamics corresponding to the quantum dynamics given by the Schrodinger equation i8'IjJ / at = H'IjJ). This equation is invariant with respect to the relativistic group of motions  the Poincare group. In particular, because of translational invariance, for any y E ]Rd, ¢o(x  y) is also a stationary solution of the equation of motion (14.35). Thus we have a ddimensional manifold
where Th is the "shift operator" mapping ¢(x) to ¢(x  h), each point of which is a stationary solution of (14.35).
194
14 Introduction to Quantum Field Theory
Now we expand the field ¢(x) "around" the manifold Me. In other words, we decompose any field ¢(x) as
¢(x) = ¢o(x  y)
+ ~(x 
y)
where ¢o(x  y) is the projection of ¢(x) onto the manifold Me, and ~(x  y) is the projection of ¢(x) in the orthogonal direction; i.e. ~y .l T¢yMe where we have used the notation ~y(x) = ~(x  y), ¢y(x) = ¢(x  y), etc. Note that the tangent space, T¢yMc is spanned by the functions a¢o(x  y)/axj, j = 1, ... , d. Consequently, the above orthogonality condition becomes
J~(x)"\l¢o(x)dx
=
o.
(14.36)
Observe that the functions a¢o / ax j are zeromodes (eigenfunctions with eigenvalue zero) of the Hessian operator Hess V (¢o). Indeed, the functions ¢h(X) := ¢o(x  h) are all critical points of the functional V:
Differentiating this equation with respect to h j at h = 0, we find
Hess(V(¢o))a¢o/axj =
°
which establishes our assertion. The condition (14.36) states that the fluctuations are orthogonal to the zeromodes of HessV(¢o). Thus we pass from the field ¢(x) to the pair (y,~(x)) where y E]Rd and ~(x) satisfies (14.36). That is, y = y(¢) satisfies the equation
J
{¢(x
+ y)  ¢o(x)}"\l¢o(x)dx =
and ~(x)
°
= ¢(x + y(¢))  ¢o(x).
Conversely, the field ¢(x) is reconstructed from y and equation
¢(x) = ¢o(x  y)
+ ~(x 
~(x)
according to the
y).
The transformation of the momentum field, 7r( x), can be found in the Lagrangian formalism. Then the transformations of both ¢(x) and 7r(x) yield a canonical transformation. The analysis presented above for the non translationinvariant case can be extended to the translationinvariant one. We do not pursue it here.
14.8 Quadratic Approximation
195
14.8.1 Further Discussion 1. Quantization: Physically, we can reinterpret what we have done above as quantization of the classical solution ¢o(x) in the first case, or the manifold of solutions ¢o(x  y) in the second. Indeed, we can arrive at the results above by first making the canonical transformation
¢(x)
f+
~(x)
= ¢(x)  ¢o(x)
and
7r(x)
f+
7r(x)
(14.37)
and then performing canonical quantization. In the translation invariant case, we have to quantize in the presence of a symmetry group. Have we encountered such a problem before? In fact we have, in quantizing the Maxwell equations. There we had to deal with gauge symmetries. As a result, we quantize the electromagnetic field in the direction transverse to the orbits of the gauge group. Passing to the field ~ in (14.37) can be considered as quantization in the direction transverse to the group of translations. 2. Relativistic invariance. The classical equation (14.35) is invariant with respect to the relativistic group of motions  the Poincare group. Therefore any stationary solution, ¢o, of this equation can be "boosted" by a Lorentz transformation into a traveling wave. For d = 1, the boost is given by
¢v(x,t)
= ¢o (x ~o).
(14.38)
Problem 14.34 Check by direct calculation that if d = 1, the function ¢v satisfies (14.35) provided ¢o is a critical point of V(¢). Thus in the translationally invariant case, a single stationary solution leads to an entire 2ddimensional manifold of timedependent solutions. These solutions are traveling waves, or solitons (in physics terminology). They are parameterized by their initial positions, Xo and their velocities, v. 3. Momentum conservation. The physical quantity which is conserved due to translation invariance is the field momentum. In the classical case, the latter is given by
P(¢,7r) =
J
7r(x)\l¢(x)dx.
The quantum field momentum, P, is obtained by quantizing P(¢,7r) in the standard way. Problem 14.35 Show that in the translationinvariant case, P(¢,7r), and its quantum counterpart, P, are conserved  i.e.
{P(¢, 7r), H(¢, 7r)}
=0
and
1
n[P,H] = O.
196
14 Introduction to Quantum Field Theory
Suppose d = 1. For soliton (14.38), the energy Esol = H(pr;art = h ort ogona '1'0 0 f H part,
(16.16) art = l1jJgart)(1jJgart I in the Dirac notation (we have assumed tacitly i.e. that the operator Hpart has a unique ground state 1jJgart). We want to apply the map Dpp to the operators H(e)  z ·1 for z in a neighbourhood of the ground state energy Eo. Recall that the decimation map Dp is defined on operators H, s.t. D(H)n RanP is dense in RanP, provided 0 E p(Hp) and
pg
IIPHRpl1
1. In physics terminology, Hoo is
relevant marginal irrelevant
if
a 1.
if
We record also the action of the group Sp on the Banach space BJ (remember that J = ~wV). The following equation follows from Equations (16.49)(16.50): (16.51 ) Applying these equalities to operators of the form H =
L
L
H rs and Hl =
r+s~O
H rs , we find
r+s~l
(16.52) for v :::;
1
2.
(16.53)
16.8 The Renormalization Map
225
16.8 The Renorrnalization Map We define the renormalization map as a composition of a decimation map and a rescaling map. Let Dp := D Ep ' where Ep = Ep(Hf), and Dp is the decimation map which is defined in (16.3). We define the renormalization map by (16.54) Without specifying a domain of definition of the map R p , expression (16.54) is purely formal. To describe such a domain we note first that the map Rp is defined on operators of the form w· Hf, w E C, acting on RanEI(Hf). Indeed, one can explicitly compute (16.55) In other words, the operators
W·
Hf, w E C are fixed points of Rp.
Problem 16.15 Show (16.55).
L'?:.l ::; 1, and XA9 
if A ::; ~ 0 if A> 1 (and the corresponding
relation for x>.>d. We do not explain here how this works, but refer the interested reader to [BeFS, BFS4] for details.
234
16 Supplement: Renormalization Group
16.11 Appendix In this appendix we show Theorem 16.23 The map Rp is analytic on Uc .
Proof. Let P := Ep and P = 1  P. By the definition of R p, and equation (16.3), it suffices to show that R(H) := H f P(PHP)1PHf is analytic as a map from Uc to the Banach space B4p l/2J. The argument we use below is the same as the one used in showing that the operator PH P is invertible on RanP. Let H E Uc and assume 118HII ::; c, where 8H = H  H f , i.e. we take r = 1 here. Define the operator K = Hi1/2(8H)pHi1/2. Take c > 0 sufficiently small so that )..( J, 1) :=
(J ,,:)
1/2 ::;
i. Then equation (16.42) implies that
w::;1
IIKII ::; 4118HIIJ .
i,
Thus for 118HIIJ < the operator 1 PH P on RanP and we have R
+K
is invertible, and therefore so is
= H}/2 P[l + Kj1 P H}/2
.
The operator [1 + Kj1 can be expanded in a Neumann series. Now Proposition 15.1 of [BFS4j implies that
:E (
K)n Hy2 converges in the Banach space B4p l/2J. n=1 1/2 1/2 Since K = Hi P8H P Hi the claim follows. 0
Thus the series H}i2
1 7 Comments on Missing Topics, Literature,
and Further Reading
General references There is an extensive literature on quantum mechanics. Standard books include [Bay, LL, Me, Schi]. More advanced treatments can be found in [Sa, BJ, DR]. For rigorous treatments of quantum mechanics see [BS, T3]. Mathematical background is developed in the following texts: [Ar, BiS, CFKS, Fo, HS, LiL, RSI, RSII]. Further mathematical developments and open problems are reviewed in [Fe, HuS, Lil, Li5, Sim4]. Chapter 3 For a discussion of the relation between quantization and pseudo differential operators, see [Fe, BS, DS, Ho, Shu]. Chapter 4 For more discussion about the relation between the uncertainty principle and the stability of atoms, see [Lil]. Chapter 6 Chapter 6 can serve as an introduction to (onebody) scattering theory. References to books on scattering theory can be found at the end of this chapter. The wave operators for shortrange interactions were introduced in [Mol, and for longrange interactions, in [Do, BuM]. Chapter 8 Chapter 8 is somewhat more advanced than the chapters preceeding it. Recommended reading for this chapter includes [CFKS, HuS, Ag]. Recently, mathematical manybody theory, especially scattering theory for manyparticle systems, has undergone rapid and radical development. The latter is covered in [H uS, D J, G L] .
S. J. Gustafson et al., Mathematical Concepts of Quantum Mechanics © SpringerVerlag Berlin Heidelberg 2003
236
17 Comments on Missing Topics, Literature, and Further Reading
Certain aspects of the theory of Coulomb systems (atoms, molecules, and aggregates thereof) are reviewed in [Fe, Lil, Li3, Li4, Sigl, T3]. Chapter 9
In the chapter on density matrices, we touch upon the rich and fastdeveloping subject of nonequilibrium quantum statistics, or quantum mechanics at positive temperatures. For a review of this subject see [Sig3]. The original papers mentioned in the text are [BFS, DJ, JPl, JP2, JP3, Mer] More material on density matrices can be found in [T4]. Standard texts covering thermodynamics and statistical mechanics are [Fey, Hu, LL2, Ma2]. Rigorous treatments of equilibrium statistical mechanics can be found in [T4, Is, Ru, Sim6, BR, Ha]. Chapter 10
Standard references on path integrals are [FH, Kl, Schu, Ra]. Rigorous results can be found in [Sim3]. Important original papers are [La, Co, LS]. Chapter 11
An extensive rigorous treatment of quasiclassical asymptotics can be found in [Iv, Mas, MF, Ro]. Section 11.5
For a discussion of determinants, see [BuN, BR, Kl, LS, Ra, Schu, Schw] and references therein. Chapter 12
For elementary facts on the calculus of variations, one can consult [GF, BB]. Chapter 13
The mathematical theory of resonances started with the papers [AC, BC, SimI]. A classic, influential review of the theory of resonances is [Sim2]. Recent developments are reviewed in [HS]. Since the latter book appeared, there has been important progress on the timedependent theory of resonances: see [BZ, CS, MS, SW]. Our approach to the theory of resonances is close to [SV], and its physical predecessors are [La, Co, Af, LOl, L02]. For complex classical trajectories, see, eg., [Bu, BuF, BuG, Vol, Vo2]. Our
17 Comments on Missing Topics, Literature, and Further Reading
237
treatment of positive temperatures is close to [L01, L02]. A more careful treatment involves the coupling of a quantum system to a thermal reservoir (see Sections 9.99.10). This, as well as many other relevant ideas can be found in [CL, L01, L02]. Metastable states at zero and positive temperatures are a subject of intensive study in condensed matter physics, quantum chemistry, nuclear physics, and cosmology. See [BFGLV, HTB, VS, SV, DHIKSZ] for reviews and further references. Chapter 14
Appropriate texts for further reading in quantum field theory are [Be, CDG, FS, Frl, Ge, GJ, Ja, LeB, Ra, Ry, Schw, Sim5, Zi]. Some of the recent developments are reviewed in [Fr2]. Chapter 15
The results on the theory of radiation, as well as the renormalization group approach, are taken from [BFS2, BFS3, BFS4, BCFS]. The existence of the ground state for the physical range of the parameters was proved in [GLL]. See reviews in [BFS1, HSp, Sig2]. Chapter 16
Texts on the standard renormalization group method include [LeB, Zi, FFS, Mal, BDH, FKT, Sal]. Missing topics
As was mentioned in the introduction, this book is not a standard text on Quantum Mechanics for  say  physics undergraduates. As a result, many standard (and sometimes key) topics are not included. The most important missing topics are 
Perturbation theory Spin Scattering theory The theory of atoms and molecules Perturbation theory for resonances and the Fermi Golden Rule HartreeFock and ThomasFermi approximations
Fortunately, these topics are well covered in the literature: see [Bay, LL, Me, Schi, T3], and, for perturbation theory, [Ba, Ka, Re, RSIV]. Moreover, [RSIV] gives an excellent exposition of the perturbation theory of resonances and mathematical questions of manybody theory. For the latter two topics, see also the review [HuS].
238
17 Comments on Missing Topics, Literature, and Further Reading
For a modern exposition of the HartreeFock and ThomasFermi theories see the seminal original papers [LiS1, LiS2], with further mathematical developments in [Lio], the book [T4], and the reviews [Fe, Lil, Li2]. The review [Lil] also describes the connection between ThomasFermi theory and the problem of stability of matter. The articles [LiSl, LiS2, Lil, Li2] mentioned above, can also be found in [Li4]. Scattering theory is presented in the textbooks mentioned above, as well as in [GW, Mel, Ne, Ta]. Rigorous results on scattering are reviewed in [HuS] (the manybody case), and in textbook form appear in [DG, RSIII, Ya].
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S.K. Ma, Modern Theory of Critical Phenomena. Benjamin: New York, 1976. [Ma2] S.K. Ma, Statistical Mechanics. World Scientific, 1985. [Mas] V.P. Maslov, Theorie des Perturbations et Methods Asymtotiques. Dunod, 1972. V.P. Maslov, M.V. Fedoryuk Semiclassical approximation in quantum me[MF] chanics. Contemporary mathematics, 5. Reidel: Boston, 1981. R. Melrose, Geometric Scattering Theory. Cambridge, 1995. [Mel] [Mer] M. Merkli, Positive commutators in nonequilibrium quantum statistical mechanics: return to equilibrium. Comm. Math. Phys. 223 no. 2 (2001) 327362. M. Merkli, I.M. Sigal, A timedependent theory of quantum resonances. [MS] Comm. Math. Phys. 201 (1999) 549576. A. Messiah, Quantum Mechanics. Interscience, 1961. [Me] C. M!2iller, General properties of the characteristic matrix in the theory [M!2i] of elementary particles I. Danske. Vid. Selske. Mat.Fys. Medd. 23 (1945) 148. R.G. Newton, Scattering Theory of Waves and Particles. McGrawHill: [Ne] New York, 1966. R. Rajaraman, Solitons and Instantons. North Holland, 1982. [Ra] [RSI] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol I. Functional Analysis. Academic Press, 1972. [RSII] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. II. Fourier Analysis and SelfAdjointness. Academic Press, 1972. [RSIII] M. Reed and B. Simon, Methods of Modern Mathematical Physics III. Scattering Theory. Academic Press, 1979. [RSIV] M. Reed, B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators. Academic Press, 1978. F. Rellich, Perturbation Theory of Eigenvalue Problems. Gordon and [Re] Breach: New York, 1969. D. Robert, Autour de l'Approximation Semiclassique. Birkhauser, Basel, [Ro] 1983. G. Rapstorff, Path integral approach to quantum physics: an introduction. [Ra] Springer, 1994. D. Ruelle, Statistical Mechanics. Benjamin: New York, 1969. [Ru] L.H. Ryder. Quantum Field Theory. Cambridge, 1996. [Ry] J.J. Sakurai, Advanced Quantum Mechanics. AddisonWesley: Reading, [Sa] 1987. M. Salmhofer, Renormalization Group. SpringerVerlag, 2000. [Sal] [Schul L.S. Schulman, Techniques and Applications of Path Integration. John Wiley & Sons., 1981. [Schw] A. Schwarz, Quantum Field Theory and Topology. SpringerVerlag, 1993. [Schi] L.I. Schiff, Quantum Mechanics. McGrawHill: New York, 1955. [Shu] M. Shubin, Pseudodifferential Operators and Spectral Theory. Springer, 1987. [Sig1] I.M. Sigal, Lectures on Large Coulomb Systems, CRM Proceedings and Lecture Notes Vol. 8, 1995 73107. [Sig2] I.M. Sigal, Renormalization group approach in spectral analysis and the problem of radiation, CMS Notes 33 no.3 (2001) 39. [Mal]
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LM. Sigal, Quantum mechanics at positive temperatures. Preprint, 2002. LM. Sigal, B. Vasilijevic, Mathematical theory of quantum tunneling at positive temperature. Ann. Inst. H. Poincare (2002) (to appear). B. Simon, The theory of resonances for dilation analytic potentials and the foundations of timedependent perturbation theory. Ann. Math. 97 (1973) 247274. B. Simon, Resonances and complex scaling: a rigorous overview. Int. J. Quant. Chern. 14 (1978) 529542. B. Simon, Functional Integration and Quantum Physics. Academic Press, 1979. B. Simon, Fifteen problems in mathematical physics. Perspectives in Mathematics, 423454. Birkhiiuser, 1984. B. Simon, The P(¢h Euclidean (Quantum) Field Theory. Princeton University Press: Princeton, 1984. B. Simon, Statistical Mechanics of Lattice Gases, Vol. 1 Princeton Series in Physics, Princeton University Press, 1993. Va. Sinai, Theory of Phase Transitions: Rigorous Results. Pergamon: Oxford, 1982. A. Soffer, M. Weinstein, Timedependent resonance theory, Geom. Fun. Anal. 8 (1998) 10861128. J. R. Taylor, Scattering Theory: The Quantum Theory on Nonrelativistic Collisions. John Wiley & Sons: New York, 1972. W. Thirring, A Course in Mathematical Physics, Vol. 3: Quantum Mechanics of Atoms and Molecules. Springer, 1980. W. Thirring, A Course in Mathematical Physics, Vol. 4. Quantum Mechanics of Large Systems. SpringerVerlag, 1983. A. Vilenkin, E.P.S. Shellard, Cosmic Strings and other Topological Defects. Cambridge, 1994. A. Voros, Spectre de l'equation de Schrodinger et method BKW. Publications Mathematiques d'Orsay 81 (1982). A. Voros, The return of the quartic oscillator: the complex WKB method. Ann. Inst. H. Poincare Sect. A 39 (1983) no. 3, 211338. F. Wegner, Ann. Physik (Leipzig) 3 (1994) 555559. S. Weinberg, The quantum theory of fields. Vols. 13. Cambridge University Press: Cambridge, 1996. U. Weiss, Quantum Dissipative Systems. World Scientific, 1993. A.S. Wightman, Introduction to R. Israel, Convexity in the Theory of Lattice Gases Princeton University Press: Princeton, 1979. D. Yafaev, Scattering Theory: Some Old and New Problems. Springer Lecture Notes in Mathematics No. 1937, SpringerVerlag, 2000. J. ZinnJustin, Quantum Field Theory and Critical Phenomena. Oxford, 1996.
[SimI]
[Sim2] [Sim3] [Sim4] [Sim5] [Sim6] [Sin] [SW] [Ta] [T3] [T4] [VS]
[Vol] [Vo2] [Weg] [We] [Wei] [Wi]
[Val [Zi]
Index
c k (n),138
o
C (D),8 C (lR d ),8 HS(R d), 137 H S (D),137
o
£l(R d), 23, 67 £2(R d), 4, 7
U(Rd), 137 nparticle sector, 184 action, 5, 118, 138, 146, 149, 168 adiabatic partition function, 156 adjoint, 19 affine space, 139 analytic continuation, 154, 162 angular momentum, 28, 76 annihilation operator, 74, 179, 183, 186, 198 asymptotic completeness, 64, 65, 95 asymptotically free systems, 105 asymptotically stable, 100 average value, 97 Balmer series, 78 Banach space, 9, 137 basis, 8 BirmanSchwinger  operator, 60  principle, 60 BohrSommerfeld quantization, 132, 163, 166 BornOppenheimer approximation, 82 BoseEinstein condensation, 113 boson, 85, 106, 185 bound state, 48, 54, 63, 92 bounded operator, 9 canonical
 commutation relations, 30  coordinates, 172  operators, 30  variables, 30, 175 Cauchy problem, 13 CauchySchwarz inequality, 8 causality, 5 central subspace, 227 centreofmass motion, 8385 channel evolution, 94 chemical potential, 111 classical  action, 145  field theory, 138  path, 130, 145 cluster decomposition, 87, 93 commutator, 11, 27, 30, 175 complex deformation, 155, 156 condensate, 113 configuration space, 169 confined systems, 105 confining potential, 43 conjugate point, 146, 147 conservation of energy, 29, 150 conservation of probability, 13 constraint, 143 continuous spectrum, 38 convolution, 23, 187 correspondence principle, 5 Coulomb gauge, 80, 173 Coulomb potential, 55, 77, 92 covariance operator, 175, 182, 186, 192 covariant derivative, 198 creation operator, 74, 179, 183, 186, 198 critical point, 124, 125, 139, 142 critical temperature, 113
246
Index
D'Alembertian, 145 decimation map, 204, 210 delta function, 24, 38 dense set, 8 density matrix, 9799 determinant  of an operator, 133, 135 dilation, 54, 155 Dirac delta function, 24 direct sum, 85 Dirichlet boundary conditions, 69 Dirichlet functional, 138 dispersion , 33  law, 180 distribution, 24 domain of an operator, 8 dual space, 140 eigenvalue, 37 electric field, 79, 173 electromagnetic field, 139 entropy, 101 environment, 99, 104 equation of state, 110 equilibrium  state, 101  thermal, 97 ergodic mean convergence, 48 EulerLagrange equation, 142, 168 evaluation functional, 171 evolution operator, 17, 117 excited state, 201 existence of dynamics, 14 expected value, 176 exponential decay, 92  of an operator, 14 Fermi golden rule, 197 fermion, 85 Feshbach map, 204, 210 Feynman path integral, 119 FeynmanKac theorem, 102, 133 field, 168 fine structure constant, 197 first resolvent identity, 42 fixed point, 205, 225 Fock space, 110, 184, 197 Fourier inversion formula, 23
Fourier transform, 22 Fnlchet derivative, 141 free  evolution, 63, 65  Hamiltonian, 18  propagator, 18, 125 free energy, 97, 101, 103, 133, 155  Helmholtz, 101 free propagator, 18 fugacity, 111 functional , 137  linear, 140 Gateaux derivative, 141 gauge  invariance, 79  transformation, 79, 173 Gaussian, 18 Gaussian measure, 175, 191 generalized eigenfunction, 66 generalized normal form, 216 generator of dilations, 54 Gibbs entropy, 101 Gibbs state, 97, 101 grand canonical ensemble, 111 Green's function, 127, 129, 163 ground state, 97, 201 ground state energy, 132, 134 group property, 17 Hamilton's equations, 29, 30, 171 HamiltonJacobi equation, 5, 129, 150, 151 Hamiltonian  function, 5, 30, 150  functional, 170  operator, 28, 30, 185  system, 171 harmonic function, 142 harmonic oscillator, 73, 102, 104, 134, 150 Heisenberg  equation, 29, 30  representation, 29  uncertainty principle, 33 Hessian, 120  of a functional, 144 Hilbert space, 7, 137 HilbertSchmidt operator, 116
Index HVZ theorem, 88, 91 hydrogen atom, 35, 55, 76 ideal Bose gas, 106, 110 identical particle symmetry, 84 index, 147 infinite systems, 97 initial condition, 13, 37 initial value problem, 13 inner product, 6, 7 instanton, 158, 159 integrable function, 22 integral kernel, 9, 58, 60, 66, 99, 117, 176 integral operator, 9, 60, 116 intercluster distance, 94 intercluster potential, 87, 94 internal energy, 101 internal motion, 86 intertwining relations, 66 intracluster potential, 87 invariant subspace, 47 inverse  of an operator, 10 irrelevant term, 224 isometry, 15, 16, 64 isospectral map, 208, 226 Jacobi equation, 128, 147 identity, 170 matrix, 136, 148 vector field, 147 Josephson junction, 159 Kato's inequality, 79 kernel  of an integral operator, 9 KleinGordon equation, 142, 168, 171 Lagrange multiplier, 143 Lagrangian, 121, 139  density, 168  functional, 168 Lamb shift, 197 Landau levels, 80 Landauvon Neumann equation, 98, 100 Laplace method, 133
247
LaplaceBeltrami operator, 76, 77 Laplacian, 6, 9, 24, 144 lattice approximation, 188 Lebesgue measure, 175 Legendre function, 76 Legendre transform, 128, 151, 169 Lie algebra, 170 linear functional, 140 linear operator, 8 linearization, 226 LippmannSchwinger equation, 57 longrange interaction, 66  potential, 92, 95 magnetic field, 79, 173 marginal term, 224 Maxwell's equations, 173, 182 mean value, 27 metastable state, 105, 153, 157 minmax principle, 52, 56, 93 minimal coupling, 198 minimizer of a functional, 142, 145 momentum operator, 8, 24, 28 monopole, 191 Morse theorem, 147 multiindex, 7, 63 multiplication operator, 8 multiplicity of an eigenvalue, 38 Neumann series, 42, 218 Newton's equation, 29, 142 nonequilibrium statistical mechanics, 97 norm, 7 normal form, 74 observable, 27, 28, 97, 175, 181 oneparameter unitary group, 18, 155 open system, 97, 99 operator ,8 angular momentum, 28 bounded, 9 evolution, 17, 117 Hamiltonian, 28, 30, 185 HilbertSchmidt, 116 integral, 9, 60, 116 Laplacian, 9, 24 linear, 8
248
Index
momentum, 8, 24, 28 multiplication, 8 particle number, 185 position, 27 positive, 21, 98 Schrodinger, 6, 9, 13, 54, 117, 144, 175 selfadjoint, 20 symmetric, 20 trace class, 98, 114 unitary, 16 operatorvalued distributions, 179 order parameter, 114 orthogonal projection, 86, 98, 208 orthonormal set, 8 Parseval relation, 8 particle number operator, 74, 110, 185 partition function, 101, 103, 107, 132, 155 partition of unity, 89 path integral, 119, 125, 133, 157, 163 periodic boundary conditions, 70 permutation, 184 phase space, 30 phase transition, 113 phonon, 103, 104 photon, 103, 104, 202 Plancherel theorem, 23 Planck's constant, 6, 179 plane wave, 24, 40 Poincare group, 193, 195 point spectrum, 38, 92 Poisson bracket, 30,170,171,174,175 polarization identity, 14 position operator, 27 positive operator, 21, 98 positive temperatures, 97 potential, 6 pressure, 108 principle  of maximum entropy, 101  of minimal action, 143, 168 probability distribution, 27 projection, 206 orthogonal, 86, 98  rankone, 98, 99, 215  spectral, 204, 205 propagator, 17, 117
 free, 18 pseudodifferential operators, 31, 175 pure state, 100 quadratic form, 139 quantization, 29, 175, 195 quantum statistical mechanics, 97 quasiclassical analysis, 129, 134, 157 range  of an operator, 21 rankone projection, 98, 99 reduced mass, 83 reflection coefficient, 71 relative motion, 84 relevant term, 224 renormalization group, 207 renormalization map, 204, 225 repulsive potential, 54 reservoir, 104 resolvent  of an operator, 37, 42  set, 37, 57 resonance, 73, 153, 166, 197, 203 decay probability, 156 eigenvalue, 156 energy, 73, 154, 156 free energy, 156 lifetime, 154, 156 width, 156 return to equilibrium, 105 Riesz integral, 43 Ruelle theorem, 48 Rydberg states, 93 scalar potential, 79 scaling dimension, 224 scattering channel,93 eigenfunction, 66 operator, 65 state, 48, 63, 64, 153, 201 Schrodinger eigenvalue equation, 69 equation, 5, 6, 13, 37, 63, 117 operator, 6, 9, 13, 54, 117, 144, 175 representation, 29 second law of thermodynamics, 101 second variation of a functional, 144
Index selfadjoint operator, 20 semiclassical analysis, 129, 134, 157 semigroup, 205, 210, 226 separation of variables, 96, 200 shortrange  interaction, 64  potential, 92, 94 Sobolev space, 7, 137 soliton, 195 specific heat, 113 spectral projection, 204, 205 spectrum  continuous, 38  of an operator, 37, 185  point, 38, 92 spherical coordinates, 77 spherical harmonics, 76 spin, 84 spreading sequence, 41 stable  asymptotically, 100  in Lyapunov sense, 100  manifold, 230  subspace, 227 state space, 4, 30, 170, 175 state vector, 4 stationary phase method, 125, 128, 133, 163 stationary state, 98, 100 statistical mechanics, 97 strong operator topology, 122 superposition principle, 5 symmetric operator, 20 symmetrization, 184 symplectic  form, 171  matrix, 175 operator, 170 tangent space, 139 temperature, 101 tensor product  of Hilbert spaces, 95  of operators, 96 thermal equilibrium, 97, 101 thermal evaporation, 105 thermodynamic limit, 100, 111
threshold, 91, 92  set, 91  twocluster, 92 topological (anti) solitons, 191  charge, 191  conservation law, 190  sector, 190 trace  of an operator, 59, 115 trace class operator, 98, 114 transmission coefficient, 73 transverse vectorfield, 173 triangle inequality, 7, 44 Trotter product formula, 117, 121 tunneling, 154, 159, 163 twocluster threshold, 92 ultraviolet cutoff, 187, 198 uncertainty principle, 33, 56 unitary  operator, 16 unstable manifold, 230 subspace, 227 systems, 105 vacuum, 183, 201 variational derivative, 141, 226 vector potential, 79, 121, 197 virial  relation, 54, 196  theorem, 162 vortex, 191 wave equation, 142, 173  nonlinear, 142 wave function, 4, 27, 97 wave operator, 64, 65, 67, 94 weak convergence, 37 Weyl  criterion, 38  sequence, 37 Wick  ordering, 178, 181  quantization, 181
249