PROCEEDINGS OF THE
NATIONAL ACADEMY OF SCIENCES Number 12
December 15, 1926
Volume 12
MAXWELL'S EQUATIONS AND ATOMIC DYNAMICS By ARTHUR BRAMLZY BARToL Rgs5ARCH FOUNDATION Communicated November 10, 1926
The electromagnetic equations have generally been considered as determining the nature of the field when the motion of the charge is known. If, however, we were in a position to represent the electromagnetic forces on the surface and within the charge as given functions of the position and velocities of the charge then Maxwell's equations could be regarded as the equation of motion of the electric particle since these electromagnetic equations hold not only for free space but also within the boundary of the electric charge. In this paper, a certain form is taken as representing the electric and magnetic forces within the charge as given functions of the velocities of the moving body and of its position. This solution satisfies the first of Maxwell's equations identically and gives the second set as the equations of motion of the charge for which this solution is valid. Among the solutions of the. differential equations of this motion, there are a certain set for which the energy integral is constantthis set of paths, which we shall term nonradiating, being determined by the Sommerfeld quantum conditions. Using the results obtained by this analysis, we have been able, following DeBroglies' ideas, to obtain the Bohr frequency condition as a consequence of these ideas. The electromagnetic equations for any coordinate system in the abbreviated tensor form can be written: +
bFtj aF,*
bXk
byxi
+
aF,
bx
=
o
0(1)
and
(O) a
J (2) where the Greek indices repeated will be understood to be summed from 1 to 4, XI, x2 and x3 representing the spatial and X4 the time coordinate, where F, represents the components of the electric and magnetic intensities with their proper signs and J' the current density (Eddington=
PHYSICS: A. BRA MLE Y
654
PROC. N. A. S.
"Math. Theory of Relativity," Cambridge Univ. Press, pp. 171 to 175). The operation of covariant differentiation, indicated above, acting on a field force
Fa' is defined in general coordinates by the relation (F)a
+ raFa + reat
= a
where Fj
=/a
=
giagi0Fa
(agja+ t)gka
_
igjk
and ai j = 5$ giag. icgg~~ 1 =jj i
the g's being the coefficients in the energy tensor H
Hg~ dxa
dx"
gpis ds(3
(3)
where H is the energy of a charge of mass m and charge e which moves in the space according to the Coulomb Law neglecting the reaction of the field of the moving particle on itself. This energy function H corresponds to the energy of the electron in the method used by Bohr and others in the treatment of quantum dynamics. The current density function Ji has the form Ji = p dxt/ds, while the continuity equation gives the condition Ja (p dxa/ds)a. Since the only case we can solve completely is when the equations of motion can be expressed in such a form that the variables are separable, we shall limit the discussion to this case, then
gia dxt/ds = Qi(x) where Qi(x') is a function of the coordinate x' only. We shall take as the value of the electric and magnetic forces within and on the boundary of the moving charge Fij = QiQj,ij, where the quantities 1ij have the skewsymmetric property that Iij =Iji and Iii = 0 (not summed) the equations of motion being determined in such a manner that Maxwell's equations are satisfied for this form of the electromagnetic forces. The first set of Maxwell's equations (1) are satisfied for this form of the forces and the second set (2), namely
(QiQa)a Ii = pQi Qi = giaQ
(4)
PHYSICS: A. BRAMLEY
VOL. 12., 1926
655
become the equations of motion of the moving charge under the influence of its own electromagnetic field and the external field subject to the two conditions: (I) That the relation Ja = 0 is satisfied. (II) That the differential equations (4) are of such a form that the momenta are functions of the corresponding coordinates only. If we now limit ourselves thus further to the case where Q, = 0 (not summed) and make a contact transformation on the momenta only such that the energy and action integral are invariants for this transformation, then equation (4) takes the form Qa'Q. = GQ"Q,Qi, where G is a scalar, an equation which is identical with the equation of the geodesics in a Riemann space where the vector appearing there is now the momentum corresponding to the motion, the first integral of the equations of motion being of the form dxaYdx g dx ds e  constant. (5)
When, however, the paths are periodic, the above integral must satisfy the condition 
.rG g_l
ddxcdxa 2nni {ninteger =
(6)
the integral being taken over a complete cycle. In the case where the variables are separable. this condition reduces to the Sommerfeld quantum conditions with the addition, however, of the quantum integral with respect to the time coordinate. This extra time coordinate has been used by the writer to give an account of the multiplet structure of lines (Phil. Mag., June, 1925) and the relation of the multiplet and zeeman structures (J. Frank. Inst., Jan., 1926) giving formula for these phenomena which agrees well with experiment. The fourth quantum condition has also been used by P. A. M. Dirac (Proc. Royal Soc., June, 1926) to account for the Compton effect on the wave theory. From (5) we see that the energy is constant or periodic along a path for which the conditions (6) are true, so that the path of a nonradiating electron obeying Maxwell's equations must satisfy the equations of motion of a Riemannian geodesic (with arbitrary gauge factor) in a four dimensional space subject to the condition Jfpi dqt = nh h = constant; pi being the momentum corresponding to the coordinate q' where the equations of motion are in such a form that the variables are separable. In moving from one qu7antized path with energy H' and momenta
PROC. N. A. S.
PHYSICS: A. BRAMLEY
656
p' to another with energy H" and momenta pi, the electron will emit radiation of energy H'  H", the direction of the electromagnetic wave being determined by the electromagnetic momentum vector. The radiation wave, moreover, will also be determined by the electromagnetic potentials k,, found as solutions of Maxwell's equations for free space, which have the form at a great distance from the charge system of K, = Aielxl + lax2 + lax$  14x4 (7) where the directions cosines in the four dimensional space, 11, 12, hs and l4 will be proportional to the components of the electromagnetic momentum vector Mi of the wave. Since this momentum vector is equal to the energy flux divided by onehalf the square of the velocity of propagation, we have
, k=M = k
(8)
gM). gjj
gjj
From this conclusion we see that in order to have interference the direction of the two waves in the fourdimensional space must be identical, the nature of the interference total or partial depending on the relative place of the two waves. If we write (7) in the familiar form
ki
=
Ai e2niv(cosax + cos
y
+ cosys5

V)
v = frequency of the radiation where V = velocity of propagation and we are lead to the following theorem: THZORZM. The frequency of v of an electromagnetic wave is equal to the direction cosine between the time axis and the wave normal. We shall now apply this method for the determination of the frequency of an electromagnetic wave to the case of an electron revolving about a positive charge E. The energy H of a nonradiating electron moving in a plane orbit is
2e) dr 2 fdo\ dr\2 + r2 (d_) (H ~2eE\ \S \ds/ r
~~H~(l1

(s\ 2 r_ E\fdt
2e _. (1I' r
fdXa\2 =gaa ds which determines the g's. The equations of motion of the electron in the special case of a circular d'r dr are: orbit, i.e., dS2 dS
dsds
Vor.. 12, 1926
V1,12PHYSICS: A. BRAMLEY
r(1_2eE)
[email protected])
e E (I
d (dr\(dO\ d20 ds2 +r ds
G
d(1 d2t + 2 TS dr
2d)= O
dXa\2&d
ds
andand
2eE)
657
tt
2E 1
rdxa d 2d G g( aa , i\s/ ds
r//d/ds/ =
whose solutions are,
d=dO=
fG
gaa(ds )dxa
= Geaad(ds)dxa p= (  ) Lt p4s
and
g xaa( )= d)dsa H2 e fG gaa( 2
where P3, P4 and H2 are constants of integration.
H=
As

( dsa d
H2 e2fG gaa (Qdx)dxa
we find after simplification and evaluation of the constants of integration that
M4 = k
Fp12 P421
= k(H'H")
but since the frequency of the electromagnetic wave is proportional to the time component of the wave momentum M4, then the frequency v of the light emitted by the radiating electron in moving from one stationary state to another is given by the relation nvra h ~~constant) =h(H' v ( H"t)) (
the Bohr frequency condition. A detailed discussion of these results appears in the December issue of the Journal of the Franklin Institute.