SLAC-PUB-513
October 1968
VW
SPIN-ONE PARTICLE
IN AN EXTERNAL
ELECTROMAGNETIC
FIELD*
D. Shays’ and R. H. Good, Jr.tT
Institute
for Atomic Research and Department of Physics
Iowa State University, Ames, Iowa
ABSTRACT
In this paper the electrodynamics
antiparticle
is developed using a (1, 0) 0 (0, 1) six-component
wave function.
Anomalous magnetic dipole and electric
rupole effects are included.
covariant
integral
of a spin-one particle-
and has no auxiliary
quad-
The wave equation is manifestly
conditions.
The invariant
for the system is derived and the nonrelativistic
limit
is discussed.
(submitted to Phys. Rev.)
*
This research was done in the Ames Laboratory
Energy Commission.
Contribution No. 2237.
t Present address: Physics Department,
Norman, Oklahoma, 73069 o
of the U.S. Atomic
University
tt On leave 1968-9 at Stanford Linear Accelerator
University, Stanford, California, 94305.
of Oklahoma,
Center,
Stanford
I,
INTRODUCTION
Recently Joos ,
new descriptions
scriptions
Weinberg,
2
and Weaver,
of a free particle
Hammer and Good’ have developed
with spin s = 0, l/2,
1, . . . . These de-
are of interest because they are closely analogous to the Dirac theory
for a spin-l/2
the spin-l/2
spin.
1
particle
and they permit many of the well-known
theory to be extended to apply uniformly
Joos’ and Weinberg’
Covariantly
gave their description
defined matrices
appear as the generalization
to particles
of arbitrary
in a manifestly
as developed by Barut,
of the Dirac
discussions for
Muzinich,
yP matrices.
covariant form.
and Williams4
Weaver, Hammer,
and
n
Good’ gave their description
generalizing
in Hamiltonian
the Dirac Hamiltonian
2
l
form and found an algorithm
p + pm to any spin.
N
in these two approaches are identical for odd-half-integral
for
The wave functions
spin and are equivalent,
in the sense of being related by an operator that has an inverse,
for integral
spin.
In any case the wave function forms the basis for the (s, o) 9 (0, s) representation
of the Lorentz group.
Also the wave function corresponds
wave function used by Pursey’
in his treatment
In later works most of the properties
worked out.
Sankaranarayanan
to the momentum-space
of free particles
with spin.
of the free-particle
theory have been
and Good’ studied the spin-l
case in detail and
Shay, Song, and Good7 the spin-3/2
case.
eral discussions
of the polarization
operators6
density matrices
for describing
Sankaranarayanan
and Good gave gen-
and the position operators.
8
The
orientational
properties were set up by
7
Sankaranarayanang and by Shay, Song, and Good.
Mathews lo and Williams,
11
Draayer, and Weber
obtained definite formulas for the Hamiltonian for any
spin.
The descriptions
have been applied so far only to free particles
is how to include effects of an external electromagnetic
-2-
field.
and a question
In view of the
success in treating
all these properties
of the free particle
spins, one might hope that electromagnetic
for any spin.
increases,
multipole
interactions
uniformly
could also be introduced
The problem becomes more and more complicated
since a particle
of spin s can have anomalous electric
moments up to the 2
2s
by a (1, 0) 0 (0, 1) wave function,
and having arbitrary
This new formulation
treatment
as the spin
and magnetic
order.
The purpose of this paper is to give the theory of a spin-l
field,
for all
interacting
particle,
described
with an external electromagnetic
magnetic dipole and electric
turns out to be worthwhile
quadrupole moments.
because it permits
a complete
of the system (some aspects involving the anomalous quadrupole mo-
ment were not covered before).
The results
have not so far suggested a generalization
The spin-l
particle
apply exclusively
to higher spins.
in an external field was originally
and Kemmer l3 using a lo-component
to spin one and
wave function.
studied by Proca
Corben and Schwinger
12
14
showed how to include an anomalous magnetic dipole term in Proca’s theory and
Young and Bludman
cializing
15
took account of an anomalous electric
to time-independent
electric
fields and space-time
fields in the anomalous quadrupole terms,
Sakata-Taketani
This Hamiltonian
complicated
16
Spe-
independent magnetic
they obtained a Hamiltonian
of the
type which included the effects of the anomalous moments,,
formulation
involves a 6-component wave function which has
Lorentz transformation
properties.
The wave equation given here is manifestly
iary conditions on the wave function.
respect to space reflection,
the definition
quadrupole.
from the anomalous quadrupole term.
and requires
no auxil-
The equation has the usual symmetries
time reflection,
of a Lorentz-invariant
covariant
and charge conjugation.
It leads to
inner product that includes a contribution
It was found that there are two possible
-3-
with
choices for the anomalous quadrupole term in this wave equation,
correct
transformation
nonrelativistic
properties
each having the
and giving the same type of contribution
in the
limit to order m -2 .
For any spin of particle,
the values of the normal electric
and magnetic
moments depend on the wave equation used to describe the particle.
normal magnetic moment g-factor
is l/2 and the normal electric
Here the
quadrupole
moment is -li2/2m
22
c . The values of the moments were found by making a
Foldy-Wouthuysen
type of transformation,
leading to a nonrelativistic
Hamil-
tonian correct to order m -2 ,
II.
THE WAVE EQUATION
The equation is
yap + raTa! + 2m2+(eh/12)
+ tedm2)
where na! is -i( a/ax,)
y6
, ap , TV m&/axJ
- eAQ! and F
F
@P
F..
1J
= wpa)
=E
ijk
ys,ap
aP
Fc@
1
“,,I e = 0 ,
is the field tensor,
- wpp
s,
Fi4 = - Fqi = -iEi,
F44 = 0 o
The Latin indices run from 1 to 3, Greek from 1 to 4 with x4 = it.
c andh are omitted.
intrinsic
moments,
are six-by-six
0
ij
Factors of
The constants A and q are real and adjust the sizes of the
defined in terms of
Y.. =
1J
(1)
-s.s -s.s.
ij
~1
-4-
matrices
I
The other matrices
are defined in terms of the yap by
Ys,o!p = i [%* ) 7jiJ
Y~,(Y/?,~Y = paps
rpv],
(3)
’
+ 2 %P %v
(4)
The operators
wave function.
(aF,/&J,
satisfy
satisfy
no are understood to act on everything
The gradient operators
to their right,
inside brackets,
including the
such as in the factor
The y
d
Yap = Ypa, and yola! - 0 so there are 9 of them independent; the Y
5,oP
Y5,ap = - Y5,pa and there are 6 of them independent; the symmetry
properties
act on the fields FaP only and not on the wave function.
of Y6 crP ~1/ are
9 ,
y6,~/3,~~
= - y6,@! ,/JV
‘f-5, N~,Pv
= yQ-wd
Y~,oL&w
and there are therefore
+ y6,~~,v~
(5a)
’
(5b)
,
+ y6,~v,pv.
10 of them independent.
= ’
’
(SC)
One defines Y5 by
Y5 =
and then there are 36 independent Hermitian
‘6, G,PV
1, Y5, y
@’ iY5 L!p ’ Y5,@’
Some other properties are
which form a complete set of 6 x 6 matrices.
given in Ref. 6.
-5-
matrices
III.
LORENTZ
TRANSFORMATIONS
The Lorentz transformation
AND CHARGE CONJUGATION
properties
be the same as that of the free particle
of the wave function are assigned to
wave function.
However this assignment
does not settle the question because in the free-particle
wave function is used in the Hamiltonian
variant formulation.
formulation
discussion a different
than in the manifestly
co-
The relation between the two functions was given in Eq. (62)
of Ref. 6. For the transformations
continuous with the identity the two functions
behave the same and the notation of Refs. 3 and 6 is used.
For the reflections
and charge conjugation there is a difference.
As shown later,
Eq. (1) specializes
and so his assignments for the dis-
to Weinberg’s formulation
continuous transformation
properties
For Lorentz transformations
the wave function transformation
are the appropriate
ones to use.
continuous with the identity
rule is
+I’Go=W(x)
where x1 denotes z’, t’ and the matrix
A+ Y44 = 344 11-l
CA =A”C
= y5
(6)
A satisfies
A-1%pA= aacl“pvYpv
11-ly5"
for zero fields
)
o
-6-
,
,
(79
(w
UC)
(8)
Here C is the charge-conjugation
matrix
defined by
(9)
where C, is a unitary matrix
such that
csg=
The C matrix
-E*c
S
.
(10)
has the property
= y?: , ’ Y4i ’ -1 = -YXi,
1J
CYij c-l
‘Y44’
In consequence of Eqs. (6) and (7a) every Greek subscript
same sense as in Dirac’s
-1
=Y$4
(11)
l
is a vector index in the
theory and Eq. (1) is evidently covariant.
For the space reflection
Xf = -xi, t’ = t
Equations (6) and (7) apply again with A chosen to be r,40 Since ro and FaP are
regular under space reflection
formation,
the covariance
is again evident.
instead of Eq. (8)) the equation
blr,A
= -y5
applies D By including factors of y, the parity noninvariant
formed.
For this trans-
For example (eh*/12)
y5 y5 op FaP is an electric
,
(12)
interactions
can be
dipole interaction
term.
For the time reflection
xi = xi, t’ = -t
the wave function transformation
V(x’,
rule is
= A[C~@>] *
-7-
(13)
where again A is y44 and satisfies
andF
Eqs. (7).
By explicit
calculation,
using Aa
to be pseudo, one verifies that Eq. (1) is covariant.
QP
The charge conjugate wave function is defined by
fjF = (cl))*
It satisfies
(14)
an equation the same as Eq. (1) but with all terms proportional
changed in sign.
matrices
0
to e
It follows from the fact that Eq. (7~) applies to all transformation
A that $I’ has the same Lorentz transformation
properties
as tip. The
charge conjugation has period two,
as follows from the fact that CECs is unity.
IV. INVARIANT
18
INTEGRAL
Let the adjoint wave function be defined by
F = 4+ Y44
The equation it satisfies
l
(15)
is
(16)
where ??o! is -i (3/axa) I- eA,! D It follows from Eq. (7b) that the adjoint function
transforms
according to
p (x’) = 7 (x) A-l
for isochronous
Lorentz transformations
(17)
and according to
it’(x’)
=[F(x)
c-y*A-l
for time reflections.
-8-
(18)
If @)
is a solution of Eq. (16) and fi(@ of Eq. (1)) then the current
@(n)
_
#Q)
y
tn)
cd%
- (eq/6m2) gQ) (aF
PV
@
($4
/ax ) y
P
~,Pv,PQ
(19)
is conserved,
aJr’“‘/axcy
=0
.
-71-tQ)
@
indicate that the ? acts only on the
( P
>
P
J($ n, is a Lorentz four-vector so the integral of J4(Q,4 over
Here the brackets in a factor like
$‘I’ .
Evidently
space is a time-independent
Lorentz scalar.
The invariant
integral
is therefore
defined by
(Q$~), &“)
= i(4m)-lJ
d3x [(F@?“)
- (eq/6m2) qcQ) (aF
An alternative
PV
Y4P4”)
/ax ) y
P
6,~,/34
lp
1
(20)
1
(21)
.
form is
(@(Q),@(“) = i(4m)-‘/d3x
[-2 #Q)’
- (eq/6m2) q(‘) (aF
-9-
PV
Y44 yqini $(n)
/ax ) y
P 6,w,P4
$@)
.
The factor is chosen to give the right nonrelativistic
As in the Dirac theory,
the invariant
limit,
as discussed below.
the anomalous magnetic moment term does not influence
integral formula.
The integral
For the time-rate-of-change
of matrix
is not positive definite.
elements of any operator T, one finds
that
d(,(“,
T @(n))/dt = i (4m)-l/d3x
$(Q) [W, T]-$(m
(22)
where W is the operator inside the square brackets on the left in Eq. (1).
applies in general,
even when the fields and T are time dependent.
This
Equation (22)
can be easily derived by operating on the equation
W (T @@)) = [W, T] _ @)
from the left with $(‘),
on Eq. (16) from the right with (-Tfi)
on the left can be rearranged
q(Q) and T ,@).
and adding; the terms
into i times the divergence of a current
From Eq. (22) it is seen that matrix
operation of the system are time independent.
Jo built from
elements of a symmetry
The point is that if T is a symmetry
operation and $(n) satisfies the equations of motion then T @(n) also is a solution.
Then W fi@) and WT $(n) are both zero and the right-hand-side
V.
NONRELATIVISTIC
of Eq. (22) is zero.
LIMIT
As was first emphasized by Foldy and Wouthuysen in the spin one-half case,
the nonrelativistic
Dirac particle
Hamiltonian
approximation
corresponds
to an expansion on m-l,
For a
they developed the series by making unitary transformations
which removed odd parts of the Hamiltonian
of the
to higher and higher
.
orders of m -I.
Their process cannot be applied directly
isn’t a Hamiltonian
to begin with,
here because there
In the Dirac case it is appropriate
unitary transformations
because the invariant
unitary transformations
preserve this form into the nonrelativistic
- 10 -
integral
to consider
is J d3x @(‘)? @(@ and the
limit.
With
the different
invariant
integral that applies here, Eq. (21), it is not appropriate
to keep wave function transformations
the limit here is to take out the rest-energy
non-unitary
The basic idea used in finding
unitary.
part of the wave function,
transformation
that removes odd parts of the wave equation in a
-1
and then expand in powers of m e
certain order,
The first step is to convert the wave equation,
type of notation.
The appropriate
@= y44=
a.=iy
1
matrices
0
1
1
0
()
’
44 yi4 = - y5si
In terms of them the y5,0p matrices
‘5,ij
‘5, i4
y6, aP,w
= -6r
ijk ‘k
= -6~~
type of matrix
Eq. (1)) into a nonrelativistic
are
=
s.1
0
0
- s.1
)
(
The
make a
e
are
’
.
is easily converted into terms of 2 and 2 by using
the result
k
~P,Yv
= -P/12)
y5,/JvI +
+4(8
a/J %
-6 o!v $3/L) - 4%!ppv y5 ’
(23)
It is assumed that the external fields F
Q!p
satisfy the- homogeneous Maxwell
equations
c0rpp.JaFcYp’axp= O
so the third term in Eq. (23) doesn’t contribute
term in Eq. (23) leads to terms proportional
- 11 -
in the wave equation.
to (aFav /ax,)
The second
in the wave equation;
this type of term is retained so that all the results below apply even in the case
when the wave function overlaps the sources of the external fields. With this
-2
notation change and after multiplication by m the wave equation reads
- (e/m2) (P-I-h) 2. (EJ+iy,EJ + 2
(24)
The second step is to make the substitution
,
+ = exp[(2 0$n> - imtJ til
multiply
through by
exp [- (g 0z/m) + imt]
and expand for small m
-1
. The effect of the time-factors
im -k 7r4e The expansion needs to be carried
contribution.
The calculation
(1-p)+2i(7r4/m) (l-t@-2i(g*z/rn)
L
+(n2/m2)(W)
,
- (e/m2)(ptA)~-
is just that 7r4 becomes
out to order m -3 to get the quadrupole
leads to
(7r4/m) + 2i(7r4/m) (g*drn)
(E+iY5E,) + i(‘y-+)2(~4/m)(l-p)
(1-p) + (r4/m)2(
l+@
- 2i(a*~/m)(~4/m)(~.~m)
lz/m)3
P
- ((u-&n)(a2/m2)tl+P)
+(~2/m2)(g-~hn)(l-P)
+(e/m2)(g-~hn)(P+A)
E*(@iy,g)
+ i( r4/m) (g -2/mj2( l+p) - (g *z/m) (n4hQ2( 11-P)+ (r4/m)2(g -z/m) (1-p) + (4/3) (2
- (e/m2)(p+A) g*(EJ+’iy513)(g*~/m)+i(eq/m3)(s
The odd terms only begin in the m -2 order O
- 12 -
s +s s -4/3spk) y5 atBp+i y5E#a%]
pk
kp
(25)
b$ = o o
As a third step one makes a similarity
transformation
so that the same
equation holds but with /3, y5, g, I/I,- replaced by p’, v;, g’, @i where
p’ zz
( )
-1
0
0
1’
’=01
y5
(10’ )
The point of this is that, if $i is considered as two three-component
functions,
then the upper part of Eq. (25) is
2Gs -I- O(mm2) es + O(mm2) *L
=0 ,
where O(m-2) denotes terms of order m -2 . The small components are thus of
-2
order m
in the
compared to the large. There is considerable simplification
lower half of Eq, (25), leading to the result
1
4i(n4/m) + 2(n4/m)2+ 2(n2/m2) - (e/m’)(l+Qg*
- 2i(Ez/m)
( r4/m) (52/m)
+ i(e/m2)(&*7r/m)(
B,
+ 2i(r4/m) (ga/m?
1-h) k Et- i(e/m2J(l+h)
- (eq/m3) ( spsk+sksp- 4/36pk)taEp/aXk)
- 13 -
20 g(kE/m)
1
eL = ’
o
(26)
This can be rearranged
into the form
(-ir4 - (2m)-’ $
fi, = (H-e@ tiL
(27)
3
where H is given by
H = e$ + ( r2/2m) - (e/4m) ( l+A) 20 B,
+ (e/8m2)(l-k2q)(sisj+s
- (e/8m2)(1-A)s-*(x
j s i - 4/3 sij)(aEj/aXi)
x~-~x~)
+ (1/6)(e/m2)(1-A)(V*lZ)
9 (28)
and where $Jis -iA4 .
As a fourth and final step in finding the nonrelativistic
reorganized
into Hamiltonian
form.
$f=
then, to order m
-2
[
the equation is
If the function W is defined by
l+ (2m)-’ (H-e+)
1
eL
, the $ term cancels out, Eq. (27) becomes
i&&/at = H*
and &is identified
limit,
as the nonrelativistic
,
(30)
wave function.
It is clear that, from what has been said so far, this identification
unique.
For example,
fiL might be taken as the nonrelativistic
the ?T: term manipulated
However the identification
integral..
into a contribution
to the Hamiltonian
above is supported by the limiting
Starting from Eq. (21) and disregarding
($‘tQ)d’tn’) = i(4m)-l/h3x
[2i@(Q)t,.,$n)
is not
wave function and
in the m -2 order.
value of the invariant
terms of order m -3 , one finds
+ k4!$QJ)t (l+p)@(n) - @(“)? (1+-P,?r4@(ril
=
= d3xwfQ)t,tn)
/
o
(31)
- 14 -
Therefore,
except perhaps for further
nonrelativistic
VI
unitary transformations,
ti is the correct
wave function.
DISCUSSION
The magnetic dipole term in H can be written
g-factor
is l/2,
electric
is (l+E\)/2.
Thus for a particle
described by Eq. (1) the normal g-factor
the same as for a Dirac particle.
quadrupole interaction
H
eq
zz-
as -g(e/2m) & B, where the
The conventional form for a spin s
term is
Q
4s(2ZZ - 1) (sisj+sjsi
where Q is the quadrupole moment.
dE.
- 2/3Sij s2) &
j
By comparison
the quadrupole moment of this particle
,
with Eq. (28) one sees that
is
Q = (-1+ A+ 2q)/(2m2)
,
the normal moment being then - 1/(2m2).
An alternative
way to include the anomalous quadrupole contribution
the term (qe/m2)(aF,dax,)
variant integral
rp $1/
in place of the ys term in Eq. (1).
is to use
The in-
can still be defined all right and the same nonrelativistic
except for a different
factor in the VoE
NN term.
term used in Eq. (1) has a universal
limit applies
However the type of quadrupole
application because the y
6, aP,w
matrices,
Lorentz type (2,0) 8 (0,2), exist for all spins greater than one-half whereas
matrices
like
y,, , Lorentz type (1, l), exist only for spin one.
The connection between this formulation
formulations
is found by specializing
and other spin-one free-particle
Eq. (1) to the case e = 0 and rewriting
it as
(32)
- 15 -
where PQ! is -is/ax@.
Here one can operate with -P P y and use the matrix
a! P@P
property
This leads to
(P,P,j$b = (Papa+ 2mT2+
and so gives the Klein-Gordon
equation
PoPa+
Furthermore
= -m2@.
(34)
this combines with Eq. (32) to yield Weinberg’s
equation2
(35)
Thus Eqs. (34) and (35) together are equivalent to Eq. (32).
In Ref. 6, Section 6,
the relations
between Eqs. (34) and (35) and the other free-particle
formulations
were given.
Just as in the spin-l/2
principle
case, the polarization
be followed throughout the interaction.
polarization
= t-i/6@
This is a gauge-independent
can in
One defines the four-vector
operators.
6
E
pvpcr y5, vp TIT
y5 r5 , luc+nc
.
(36)
notion and it fits in with the scheme of free-particle
The fourth component is
T4=(i/m)g*a
for a particle
particle
operator by
TP = (i/12m)
polarization
of the spin-l
spin-one
in an electrostatic
;
field, this is (ip/m) times the helicity
(s-0 x/p) which is central to the discussion in the next section.
- 16-
operator
VII.
APPLICATION:
SEMICLASSICAL
SOLUTION OF THE GENERAL ELECTRO-
STATIC PROBLEM
As an example of the usefulness of this theory the semiclassical
tion for the solutions of Eq. (1) will be set up in the case of a spin-l
an electrostatic
field.
fields.
WKB method to the relativistic
those developed by Pauli
21
The approximation
spin-l
case.
in
in the relativistic
definite helicities
the expansion coefficients.
is the generalization
of the
Some of the ideas are the same as
spin-l/2
solutions are expressed as linear combinations
contribute
particle
This applies for example to a deuteron moving relativis-
tically through laboratory
approximation,
approxima-
problem.
The approximate
of functions that have, in first
and a set of differential
equations is given for
The anomalous quadrupole moment term does not
to the wave function in first approximation.
The equation to be solved is
+
+2m2c2+eAia!*E Nhr
C
eii3
+ m2c3
’ ‘5 ‘jke
(37)
2
axiaxj
-ix
q (BiSj + SjSi - $ Cij AL
m2c3
This is just Eq. (1) specialized to the case A, = 0, A4 = i$ time-independent.
factors of h and c have been reinserted
and it has been written
The
in terms of the
s- and /3- rather than the y-matrices.
The results are conveniently
particle
problem.
expressed in terms of the solutions of the free-
In this case one considers solutions of the form
$ = w exp 1%-l(p*
E-Et)]
- 17 -
e
(33)
The equation determining
w is then
p2tI+P) - 2 (g-g)2 /3 - 2 (E/c) go&p - (E/c)~ (l+p) + 2m2c2 3 w=o
C
By looking at the problem in the rest frame,
0
(39)
& = 0, one sees clearly that there
are six solutions for each fixed N0
p It is easy to find them by supposing they are
eigenstates of the six-by-six
helicity
operator,
(ZE/P) w=uw
say
,
(40)
Then Eq. (39) reduces to a two-by-two
where u=O,*l.
problem.
There are
solutions only if
E = eW
where E= f 1 and W is an abbreviation
particle/antiparticle
the solutions
w&j
2 2 112 ,
for c(p2 + m c )
the positive root.
,o
as E = f 1. The final formulas
solutions are identified
w C,(T(~) of the free-particle
= -$
(41)
uO
we, *&I
=
uO
Ll
22mc2
(E- E/P) u=uu
\
(42)
(W=P)u*l
c
where u,(g) are the solutions of the three-by-three
normalized
helicity
’
)
eigenvalue problem
,
so that
In the representation
for
problem are
(W=P)u*I
’
The
(s.). = i E.. explicit formulas
1 Jk
Ilk
- 18 -
for these functions are
The factors in Eqs. (42) are chosen so as to give a normalization
Eq. (21). If the q-term can be disregarded
-1
Et) then
exp (-ih
appropriate
to
and if v has time-dependence
where
(4%
The normalization
is such that I = E/mc2 for the free-particle
The semiclassical
approximation
solutions.
is obtained by substituting
(is/LPI)
1exp
$ = a0 -t- (B/i) a1 + . . .
[
into Eq. (37) and formally
cability
obtaining a solution to first order for small A. Appli-
of this approximation
is discussed below.
g= 7s ,
(this is a different
In terms of the abbreviations
E = -%/at
use of the symbol g than before) the terms in h give
C
P2( It-P) -, z(~.p)~p
- 2cW1(E-e$)g*p@
- c-2(E-e$)2(1+P)+2m2c2
Solutions of this equation are known by comparison
Eq. (39).
1
a9 = 0. (45)
with the free-particle
problem,
According to Eq. (41) it is necessary that
E - e@ = EC (p2 + m2c 3 l/2
In the following
only the particle
equation for the classical
solutions,
Hamilton-Jacobi
.
E = + 1, are considered;
(46)
then this is the
function S. Also only the solutions with
definite energy E are found so that
s=s-Et
where 3 is time-independent.
Suppose the classical
,
(47)
This means that p is Es and is also time-independent.
problem is solved so that s as a function of 5 and three constants,
- 19 -
values of integrals
of the motion,
is known.
The only problem then
is to deter-
Equation (45) implies that
mine a0’
c
ao=u=O,*l
(48)
*ow*lo- , (9
where A, are three functions of position still to be determined.
on A, is that the approximation
The limitation
should be solvable to next order to get a1. Thus
Eq. (37) for the terms in I? gives
- 2(zz)2p-2c
1
-’ (E-e+) g*EP - cm2 (E-e$)2 (11-p)+ 2m2c2 a1
(4%
(Even in this order,
the q term does not contribute.)
The matrix
of a1 is the sarne as in Eq. (45) and has zero determinant.
of coefficients
Equations (49) have
a solution for a1 only if the vector on the right is orthogonal to the solutions of
the homogeneous equations formed with the Hermitian
on the left.
-(E-e+)
Taking that Hermitian
conjugate is the same as replacing
so those solutions are just
Wtl,T
[(p.v+
Nm
conjugate of the matrix
w-1, $)
and the condition for solvability
v~~)(~+P) -2p(s+
sep+s*p
s-v) - 2c-‘W#WV
-N-m
NM--h.
- ec
-1 E*o!P-ec
NN
- 20 -
(E-e+) by
-‘Ag-g
1
~A,w+~
‘VW
,
u =O
P
.
is
Here 1 acts on everything to the right,
as differential
co-
including the 5 dependence in p. Expressed
equations for A, this reads
-1
WT
-1 T L2p(l+P) -2PP(7-+@)+2c
we2
,
1 ,
W+l
(T
l
5%
2 21/2
Here W still denotes c(p2+ m c )
. By using the explicit formulas
Eqs. (42)) one can verify that the left-hand side simplifies
Eq. (50) be written
for w6 u,
,
to 4p. V A . Let
5T
as
(51)
where the coefficients
principal
C 7u can be found given the potential
$(xJ and choice of
function S. Since at every point p is normal to the surface 3 = constant,
Eqs. (51) determine Ar everywhere
In this respect the semiclassical
if the A7 are given on one particular
approximation
is like the classical
which one can have various numbers of particles
~trajectories.
For any particular
is a set of total differential
streaming
surface.
problem in
on the various allowed
orbit s(t), since N
p is c -2 (E - e$) dxJdt, Eq. (51)
equations
(E -e+) dAr/dt
= c2 cC,,A,
U
(52)
which determine the amplitudes A ~ if they are known at the start.
In the one-dimensional
problem,
when $ and s depend on a single coordinate,
say z, one can solve for the A7. explicitly.
The principal
S =$pdz - Et
function is
(53)
where
-’ (E -e$J2
p=C?
C
- m2 c4
- 21-
l/2
1
(54)
for particles
moving in the positive z-direction.
spin matrices
occur in Eq. (50) so it is appropriate
which sz is diagonal.
to use the representation
to
P(d.Ar/dz) = - ;*Ttdp/W
dropping out.
The amplitudes
,
are then just p
-l/2
and the semi-
solutions are
Go = (z)l”
(11)
exp ix-’
(jpdz
- Et)
,
exp iliB1(jpdz
These are eigenstates of the helicity
static one-dimensional
(559
-Et)
for discussing the normalization,
order in Fi they amount to the same thing and I is E/cp.
The semiclassical
(55b)
sz, as are the exact solutions of the electro-
p denotes -i A 1 whereas in Eqs. (55) p is given by Eq. (54).
since it is inversely
.
problem.
Equation (44) is still appropriate
there,
in
are zero.
Equation (51) uncouple and simplify
classical
of the
In place of Eqs. (43) one has
and their derivatives
the A-term
Only the z-components
proportional
to the classical
approximation
although .
To first
This is a sensible result
velocity.
is expected to apply when terms marked by
higher powers of iEiare smaller than those marked by lower powers.
Typically
the AI&
Eqs. (45) and
terms are considered one higher order than p2 in deriving
- 22 -
(49).
Using Eq. (46) and considering
relative
tric field.
E - e$ of order mc2, one finds that the
size II~z/P~ is of order (II eEf /m2c3),
This parameter
also measures the size of the g
to P2, as long as h is of order unity.
terms are considered.
Another parameter
Their size relative
(RqYEf /mc Ef 1. The approximation
two parameters
where Ef is the size of the elec-
are small.
to the 2
l
l
g terms relative
enters in when the q
g terms is about
is expected to apply, then, when the above
For example,
a deuteron has A= .7, q = 25 and if
Ef 2 lo4 esu, EEf /Ef z 1 cm -1 , the first param-17
eter is about 10
and the .second about 10-13 . The approximation would surely
it moves in laboratory
fields,
apply in that case.
- 23 -
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