Semiclassical matrix-mechanics. II. Angular momentum operators

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Semiclassical matrix-mechanics. II. Angular momentum operators

Richard More

To cite this version:

Richard More. Semiclassical matrix-mechanics. II. Angular momentum operators. Journal de

Physique II, EDP Sciences, 1991, 1 (2), pp.97-121. �10.1051/jp2:1991150�. �jpa-00247513�

J

Phys.

II1 (1991) 97-121 FtVRIER 1991, PAGE 97

Classification

Physics

Abstracts 03 655

Semiclassical matrix-mechanics.

II. Angular momentum operators

RJchard M More

Lawrence Liverrnore National

Laboratory

L-321, Livermore, CA 94550, US A

(Received

27 June J990,

accepted

_m

final

form 29 October J990)

Abstract. Semiclassical

angular

momentum matrices are calculated using a new contour-

integral

formula for matrix-elements WKB

sphencal

harrnonic functions _are found to be

exactly

orthonormal with the

contour-integral inner-product

Matnx-elements obtained from these wave-

functions _{are accurate} to about 29b and the matrices

obey

the cornrnutation relations

expected

of quantum

angular

momentum operators The semiclassical wave-functions _are related to a

superposition of allowed classical orbits and this illustrates the connection to the

Feynman path-

summation formulation of quantum mechanics for electrons _m a

spherically

syrnrnetnc

potential.

1.

Innoducdon.

A

key step

in the

development

Of

quantum

^mechanics was the introduction Of matrices to

represent dynamical

Variables such _as position x Or momentum p

[I]

Matrices

appeared

_as

generalizations

Of Founer components

x(w), p(w)

used _m the classical calculation Of

radiation from _atoms From _a modern point of _mew the matrices

X,

P _are given

by quantum mechanics,

and the connection with Fourier

components

^is an

approximate

_or

limiting correspondence

with quantities defined _m the classical

theory [2]

The

correspondence

_is Valid

in the limit of

large quantum

^{numbers for} transitions with small

changes

_in

quantum number,

i e

,

for a small subset of matrix-elements _near the

diagonal

One _can ask how _to

interpret

the rest of the

matrix,

Or more

exactly,

how tO relate the

general

matnx-element tO a semiclassical

quantity

Recently,

More and Warren

proposed

_{an answer} tO th~s question, a semiclassical formula _in which the matnx-element _is determined

by

a contour

integral

Over WKB wave-functions

[3].

This formula has _a

Simple physical interpretation

for radiative transitions Of _atomic electrons the transition _is associated with _a second-Order Orbit intersection which _Occurs at a

uniquely

,

defined radius for transitions

having

substantial

matrix-elements,

Such _as

ls-2p, 2s-3p, 2p-3d.

The matrix-element calculated _in this way is accurate to _a few percent,

quite generally,

for

large

_Or Small

changes

_m the

quantum

numbers The method also works for radial

integrals

associated with

quadrupole

transitions and for the

photoelectric

effect

When

applied

tO the harmonic

oscillator,

the

contour-integral

formula g1Ves a surprising result for the first time a semiclassical

theory

_agrees in almost _every respect ^with the

98 JOURNAL DE

PHYSIQUE

II M 2

quantum ^treatment

[4].

The semiclassical matrices have the _correct commutator

[P, X]

=

ihl, diagonahze

the Hamlltoman and

exactly obey

the matrix

equations

^of motion

(e

_g,

[H,

P _oc

li~.

However the matrices are not Hermitian and the excellent results described hold

only

for matrix-elements

F~,~

^with mm n.

In this paper the contour

integral

method _is

applied

to the

angular

momentum of _a

nonrelativistic electron _{in a}

Spherically Symmetric potential.

The most

interesting

questions will be _: How accurate are the semiclassical results ? Do the semiclassical matnces

obey

the

algebraic

relations of the

Heisenberg

matrix mechanics ? What _is the

physical

interpretation of semiclassical

Spherical

harnlonic functions ?

Section 2 outlines the

contour-integral

method Semiclassical

sphencal harmonii

_functions

are discussed _in Section 3

Angular

momentum matnces

L~, _{L~, L~}

are calculated in Section 4

They

_are within 2-3 fb of the

quantum

matnces and

exactly satisfy

the

important algebraic

relations _:

~~X,

~y] ~~=

jj)

and CyChC permUtabons

The semiclassical matrices

L~

_and

_L~

are

exactly

correct however

L~

and

L~

^are

Slightly

_non-

Hermitian. Thus the semiclassical

theory

_agrees with quantum

theory

for all but _one

algebraic property

of the

angular

momentum matrices.

These

unexpected

results _are

probably

related to the fact that the WKB

angular

functions

are

exactly

orthonormal when their _inner

product

_is

flefined by

the

contour-integral

formula

(Sect. 4)

Because the semiclassical _matrix elements _are

analytic

functions of the quantum numbers

I,

m it is

possible

to evaluate them with

half-integer

Values such _as

I

=

1/2 (Sect. 5). Angular

momentum matnces obtained th~s way are

exactly

the Pauh matrices

Semiclassical

angular

functions

Yi~(@, 4

_can be combined with WKB radial functions

R~i(r)

to make _a three-dimensional wave-function

p~i~(r).

This function has _a direct geometnc

interpretation

related to

Feynman's

quantum ^mechanics : for _an electron _{moving in}

a

sphencally

_symmetnc

potential

the semiclassical wavefunction _is obtained

by

superposition of allowed

classjcal ^irbits

^of a

family

Such that _one orbit passes

throu)h

each

point

_{m Space}

(Sect 6)."

Orbits of th~s

family

_are

generated by

_{rotating a}

Single

orbit

through

_Various

ankles

the

phase assigned

to each orbit _is determined

by

the rotation

angle

Th~s constructio6 _{g1Ves a} detailed connection between classical

paths,

rotational symmetry ^{and the WKB} wave-

function.

lihe

_overall

_goal

_of

_this

_work

is to

develoj

_a

complete

semiclassical

theory

which will describe the

entire'range

of

phenomena

encountered in quantum

mechanics,,as

far _as

possible.

Of course

iriany

_aspects of the desired

theory

_are

already

known

[5, 6].

In

searching

for the rules of

thq,semiclassical theory

_{we are} assisted

by

comparisons with the

quantum theory

At the

saine

_{time, the} semiclassical

theory

also

gives

_a better intuitive

undersfianding

of

qujntum theory.

2.

Contour-integral

fornlulafion.

The usual semiclassical

theory

_is based upon thd Bohr-Sommerfeld

quantization

rule for _one- dimensional motion,

ip~ _ix)

dx

=

2 ark

(n

^{+ '}

⁽²⁾

c~ 2

M 2 SEMICLASSICAL MATRIX-M.ECHANICS II 99

where p~

ix)

= momentum _oc

[E~ U(x)

^"~ with

E~

= energy

eigenvalue

,

x is a

coordinate,

allowed to be _a

complex number,

and

U(x)

_is the

potential,

assumed -to be _an

analytic

function of _x The

positive

and

negative

signs of p~

ix) correspond

to the two directions of motion _in the _± x-direction

mathematically

_p~

ix)

has _a branch-cut

along

the real _x-ax~s in the

classically

allowed range where

E~

_m

U(x) Assuming

this allowed _{range is a}

single interval,

_{x~~~ w x wx~~~, p~}

ix)

will be

positive

^{and real}

just

above the branch-cut and the contour

C~

^{encircles the allowed} range once in the clockwise sense. With these

conventions,

the Bohr-Sommerfeld rule _{is a} contour

integral

The WKB

travelhng

_wave

>

~ ^± p~(x') dx'/h ± iw/4

~~

_~~~

fi

~ ^'~ ~~~

is

typically

_an

analytic

function of

x'ha~ing

a branch-cut

along

the allowed range

x~~~~ ^{w x w} x~~~~. Both p~

ix), pi ix)

_may be

sipgular

at

the.singularitips

of

U(x)

The

prefacior

I/ fi

_and

(he

_{action p}

ix')

dx' have additional

branch-cuts,

but these cancel _in the

wave-function because of

equktion (2) Specific analytic properties

, ^{of the WKB}

spherical

harmonic functions _are summanzed _m

appendix

B.

One _can construct a real wave-function

p~ ix)

^{defined for real} x

by

the equation

~niX)

=

lllll Re

~t IX

_{+ t}

El

+

~i IX

+ I

E)1 14)

This function oscillates _in the allowed _{range x~~~~ w x wx~~~} and has the correct bounded behavior _in the forbidden ranges It g1Ves a

good approximation

to the quantum wave-

function aside from

(weak) singularities

at the

turning

points _x~~~~, x~~~~

Equation (4)

_can be used _m

place

of the usual connection formulas.

The matrix-element of _a

dynamical

Variable

fix, p)

will be calculated

by

the contour-

integral ^for~nula [3]

Fnm

_"

j

~i IX) / IX, > ~i IX)

dX

15)

where C encloses the allowed _range for both states n, ^{m in} the clockwise _sense Here

fix, fl)

is the

quantum

operator expression, but equation

(5)

_is semiclassical because the WKB wave-functions _are

explicit

combinations of classical momentum and action functions.

Although equation (5)

_is s1mllar to the

quantum

^formula for a matr~x element there _are also

specific

differences The quantum ^formula is not a contour

integral

and contains additional

tennis associated with the

integrals

of

p( fp(, _pi fpj.

The additional tennis _cause

oscillatory (interference)

modulation of the

probability density

and their _{omission can} be called _a restricted

interference

approximation ^In

Appendix

A it _is shown that

equation (5)

converges with the quantum expression in the semiclassical limit where the

oscillatory integrals

become small The best

agreement

with the

quantum theory

is obtained when the

p~

wave is taken for the state

having

the smaller allowed range of classical motion.

Equation (5)

_g1Ves

satisfactory

results for matrix-elements of several systems

[3, 4]

and _in section 4 _we will _see that it succeeds again for the

spherical

harnlomc functions. An extension to three-dimensional systems is indicated _m section 6

(see especially Eq (78) below).

Temporarily adopting

equation

(5),

One can ask whether it fOrn1s the basis for _a consistent

100 JOURNAL DE

PHYSIQUE

II M 2

semiclassical

theory

For

example,

for the

special

_case

f

=

I,

_equation

(5) implies

a contour-

integral inner-product

of semiclassical

wave-functions,

Un,

m ^"

j ~i IX) ~i IX)

dX

16)

In part ^{I of th~s} paper, equation

(6)

was evaluated with WKB wave-functions for the harnlonic oscillator with the

unexpected

result _U~~>

= 3~~> for all states n'm _n For radial motion _in the Coulomb

potential

and also _{in a} screened Coulomb

potential,

it was found that

integrals U()

were 3

~~. + O

(10~

^~

[3].

With the usual _inner

product,

WKB wave-functions _are not _even

approximately

orthonor- mal and therefore these results _are evidence that equations

(5), (6)

can

give

a more

satisfactory

semiclassical

theory

For the WKB

spherical

harnlonic

functions,

the

orthonornlahty

relation _is exact,

U(j~

=

Sit.

for all

i'm I (see Eq (30) below)

U~~ =

corresponds

to

nornlalizing

the wave-function _in tennis of the classical

round-tnp travel-time,

this _{is a} well-known normalization of WKB functions

[6]

^{However this}

nornlahzation _agrees with the quantum

normalization, p~

^{~ dx} ₌

I, only

in the semiclassi- cal limit

Another

special

_case of equation

(5)

_{is n}

= m, the semiclassical expectation ^Value

In

"

j ~i /~t

dX

17)

When the

quantity f

_{is a} function

fix),

this _is

simply

~ _j2 j ^/lx)

~

j8)

n ^" ~~

w

^~'

The

right-hand

side _is the classical time-average

off,

which _is

usually

a

good

approximation to the quantum

expectation-Value

_even for low

quantum-numbers.

For

example,

results from equation

(8)

_are

exactly

correct for the expectation Values

(I/r), (1/r~)

for electrons _{in a}

Coulomb

potential [7]

We _now compare equation

(5)

to the

Heisenberg correspondence principle

In the limit of

large

and

nearly-equ41

_quantum

numbers,

_equation

(5)

_gives for the _case

f

=

fix)

d~ ^'(En E~)

$

_dX'IA

Fnm

" ~ ~

j ^{/ IX)}

^e

P X

=

la1~ j _{$ fix)}

e- ~"~~f~ cc

flxlt))

e~ ^~"~ dt

19)

Here

quantities

without _a

subscnpt

_are evaluated _{on an} average orbit

(e.g., p(x)m

p~

ix)

m p~

ix)

for

large

and

nearly-equal

_n,

m).

This _{is a} well-known calculation

[6]

In order to denve equation

(9)

it is necessary ^{to omit} the

oscillatory

terms

p( fp(, pi fpj

and these

are

already

omitted from equation

(5)

_so

F~~ automatically

reduces to the Founer component

off

for the _case

in

m « n. In

general

when

in

_{m is} not

small,

_one cannot

speak

of Founer

components

^{because the} two orbits _are too different _to be

approximated by

_a

single

space-time trajectory.

M 2 SEMICLASSICAL MATRIX-MECHANICS II 101

Th~s _is

especially

evident _m calculations of the

photoelectric

matrix-element where _one orbit _is bound and the other extends out to infinite distance _so there _{is no}

meaningful

_average

trajectory Equation (5)

still gives

good

results for these matrix-elements

[3]

Thus equation

(5)

does not

disagree

with the

Heisenberg correspondence relation,

but rather

generalizes

it

by

_{giving a} semiclassical

approximation

to the entire matnx of

f.

Equation (5)

for the matrix-elements _is a step ^toward a

complete

semiclassical

theory,

but many

questions'remain

_concerning the interpretation and the

generalization

to three-

dimensional systems

and/or time-dependent

atomic

dynamics

With _some

effort,

_{one can}

employ

_equations

(5)-(8)

to denve solutions for certain

specific

three-dimensional and time-

dependent systems.

^{A fresh}

approach

to the _more

general

_{case is given}

by

the orbit

superposition

method discussed in section 6.

3.

Spherical

harmonic functions.

Semiclassical

sphencal

harmonic functions _are

easily

constructed

by

the WKB method

[6, 8, 9]

^{We sketch th~s} calculation _in order to draw attention to a few

important

^technical points.

The associated

Legendre

_{equation may} be written

~

ill -x~)~~j

+

~A- ^'~~

^{&= 0.}

⁽¹⁰⁾

dx dx

1-x~

In

quantum mechanics,

this equation arises

by

separation of variables and the

eigenvalue

A _is shared with the radial equation. ^The quantum

eigenvalue

is A

=

iii +1),

where

I

is the

(integer) angular

momentum However _in the semiclassical

theory

it is normal to assume

A

=

(I

^{+ ~}

^ill)

2

where

I

is an integer. ^{With this}

choice, tie

_WKB wave-functions

&i~

turn out to be

single-

valued

analytic functions,

_a stnct

requirement

for the

method

of this paper The _same separation constant, equation

(11),

is

normally

used _in the WKB solution of the radial

equation [3, 6-9]. Surprinsingly,

the semiclassical matrix

L~

_{has the}

_quantum eigenvalues iii

₊

1) despite

the _use of

equation ill)

_m the _wave functions

(Sect 4)

For _{now we} consider

positive integer

m(0

_< m w

I).

For _a solution & of the form &

= exp

Ii

_a

ix) ],

_equation

(10)

becomes

~~

~~~~[i _{j~~ ~~~~ j~~1~~2}

~~

~ ~~ ~~~i _~'

_~~~~

For

large

_quantum numbers the

right-hand

side _is

relatively

small when it _is omitted _one has the zero-order equation for _«:

~~ ~~~

~~ ~"

(~+~ )~_ ^m2

~ l

x2

The

right-hand

side of equation

(12)

_is

equated

to the first-order term _on the left to give

«~i~ =

~ log

₂

ill ^{x~) ~~°}

_dX

⁽¹⁴⁾

Thls is the usual WKB

technique.

Planck's constant does not appear in equations

(10)-(14).

In

physical

terms the expansion

parameter

^{is I}

lit

₊

1/2)

and therefore it is necessary ^to

carefully inspect

the results for small

102 JOURNAL DE

PHYSIQUE

II N 2

I Mathematically

the expansion is not

uniformly,valid

because of

singular

behavior at

x = ± I ai~d at the _zeroes of

d«o/dx.

The _zeroes of

d«o/dx

_are

turning

_points, located at x ₌ ± J~,

,

~2

~ i ^2' ~~~~

~

~

The classical

motiin

is

confinid

_{to the}

region xi

=

(cos ii

w xo, that _is, excluded

from~a

cone around the

iz-axis.

_The

physical significance

of th~s restriction becomes deaf _in section 6.

It is useful to wnte

q(x)

=

/~ ₍₁₆₎

With this notation equation

(13)

becomes

~°

=

(i

+

~~~~~. (17)

X I _x

Equation (17)

_is

explicitly

solved _m tennis of functions

fix),

_~

ix)

defined

by

sin

f

=

~

cos

f

=

~~~~

(18)

J~ J~

Sin ~ =

~

@

CDS ~ =

jh II 9)

xo i-x J~

i-x~

The definitions _are appropriate because the trigonometric ^relations

sin~ @'+ cos~

=1 _are

obdyed

for

a(I

_x. The

boundary

values of

f,

_{~ are}

ii± xo)

= ~

i± xo)

= ±

]. ₁₂₀₎

The derivatives _are

~~ q/~)

~

~) _[

ji

~) ~X)

~~~~

For either sign of _m it follows that

«oix)

=

(f

+

fix) im1

~

ix) 122)

At this point, equation

(22)

_appears to be _a mathematical relation

having

little intuitive content. In section 6 _we will _see that it has _a

simple

geometnc

interpretation

in effect it is the

transformation of _a

simple

_wave

exp(i (I

₊

1/2)

_y defined _{in a} rotated coordinate system.

We combine equations

(14)-(22)

to define the semiclassical

travelling-wave

for all

m(- ^I

_{w mm}

^I by

~+ ~~~ ^aim

qix)

+ ix

±

l~

+2

q

ix)

_ix

/fi

±'~'

~~

fi

~~~~

,' xo

~

fi2

M 2 SEMICLASSICAL MATRIX-MECHANICS II 103

The

nght-hand

side

depends

_on the sign of _m

only through

the coefficient _ci~, which _is taken to be

~fm "

) e~~~~+m)

^w/2

~ "

(24) Equations (23), (24) imply

At

m

lx)

=

)~ et

m

lx) 125)

The

magnitude

of ci~ is chosen ^to normalize the functions with the inner

product

of

equation (6).

The

phase

_agrees with _a well-known definition of the

spherical

harmonic functions

[10, 11].

The formulas

simplify

for _m

=

0 The appropriate

special

_case of

equation (23)

_may be written

~Xp ^±1 + ~ ^~

~~

_~~~~ ~

~ _{~~~~~~ ~~}

j2

i'+~

sin

~~~~

2

These _are

travelhng

waves to compare ^to the

Legendre polynomial

_we fornl.

Pf

=

Re

[Pi

₊

Pi ]. (27)

Comparisons

_{are given in}

figure

,

the

agreement

is

good

_away from the

tu£rung

points ^which

occur at x ₌ ± I

Equation (27)

_{is a} known

asymptotic

^{fornlula for the}

Legendre

function

[12].

..

The WKB approximation to the

spherical

harnlonic function _is obtained from

l'lLl@, 4

₌ ^Re

18im

+

8iml ~Pml4

~im4

=

Re

18im

+

8iml /~

₂

¹²⁸⁾

gr

Numerical _compansons show that

equation (28)

_{is a}

good approximation

_apart from the

singularity

at the

turning points

± J~ A

sample

_{comparison is given in} table1

Table I

Comparison of

WKB and exact

Spherical

Harmonics

for

cos@

=005,

4

=0

f

_m WKB Exact Difference

0 0 0.3184 0.2821 12 8 fb

0 0 0239 0.0244 2 5 fb

0.3685 0.3451 6 8 fb

2 0 0.3160 0.3130 0 fb

2 0 0381 0.0386 7 fb

2 2 0 4105 0.3853 6 5 fb

4. Matrix-elements and

matrix-algebra.

In this section the

contour-integral

formula _is used to calculate the _inner

product

Of WKB

.wave-functions and matrix-elements of x=cos@

Angular

momentum matrices

L~,

L~, L~

are constructed and their

algebraic properties

examined

104 JOURNAL DE PHYSIQUE II, M 2

s

=

i o

Legendre function

Exaci Legendre function Po ^{o exact}

WKB

>,

os os

a H

W W

8 ₀ 8 ₀

d _<

.0 s -o s

.i .i o

'~

.08 -06 -04 .02_ 0 02 04 06 0810

'~~10

_{.08 -06 .04--02 0 02 04 06 0810}

cos8 cos0

a)

b)

s S

=2

y

= 3

Legendre function

Legendre function

i o ^exact

exact

WKB WKB

os

~

m #

8 ₀ 8 ₀

$ $

~ ~

.os .os

-i o -1

.15 .1

-10 -08 -06 -04 .02 0 02 04 06 08 10 -10 .08 -06 -04,02 0 02 04 06 08 10

cos0 cos0

c) d)

Fig I Companson of quantum and semiclassical Legendre functions for several angular momenta

The semiclassical functions

~'J$

are

obviously

orthonormal with

respect

to the index _m because the

#-dependence

_{is a}

simple

factor

exp(im 4 )

_as m quantum mechanics Thus it is

only

necessary to calculate the inner

product

of functions

having

the _{same m}

Ui[~

=

Aim lx) elm lx)

dx

129)

From

equation (23)

_{we see} that the factor

(I -x~)

_appears with _powers

(m(,

+

[m

in

8+,

fJ~

respectively

and therefore cancels _in

equation (29)

This _means that the

mtegrand

_is

analytic _except

for branch-cuts

along

the allowed _ranges

(-xo, xJ)

and

N 2 SEMICLASSICAL MATRIX-MECHANICS II 105

( go, go),

where xJ,

go

are turning

points

^for quantum numbers

if, _m)

_and

if',

_m

).

In

particular

the

integrand

of

equation (29)

has _no

branch-points

at x ₌ ±1 Therefore the contour of

integration

_may be

displaced

to

large ix

and the

integral

^is

easily

evaluated

by residues,

_giving :

U)j~

=

an,

for all

f'

m

I (30)

The

diagonal

value

iU)Q)

is

unity

^{because of the} normalization chosen in

equation (24)

The surpnse is the

orthogonality

for all pairs

f, I'

with

f

<

I'

The calculation _can be

repeated

for the

special

_{case m}

=

0 where the

simpler

functional fornl of

equation (26)

_{gives an}

elementary

tngonometnc

integral

lpi _{lx) PI lx)}

_dx

= ~

~ 3fi 131)

This is

exactly

the

orthonornlallty

relation for

Legendre polynomials

It would be difficult to obtain

anything

like this result for WKB functions without _use of the contour

integral

_inner

product.

Next _{we examine} the matrix-element of _x

= cos between states

ii, _m)

_and

ii', _m).

Contour

integration

shows this matnx-element _{is zero} for

I=I'

and for all

i'm

I+1

Thls _means the selection rules agree

exactly

with quantum ^{mechanics. For}

I'= I

₊ I _we find

~ ~

i

go

^{~ ~'} +

/fi

~'

~~~~

~~~~

~ ~~

'~ l ₊

+/I _Xl

^~~~~

Table II shows that this agrees

closely

with the

quantum qatrix-element

_even for samll

quantum

^numbers

Next _we fornlmatnx-elements of the

angular-momentum operators L~ L~, L~, using

the usual

representations

i~

= -1

~

d4

I+

=

i~

_±

i~

=

e~~~ (± ^I

^{+ i cot}

^{I (33)}

°@

°4

Table II Test

of

_equation

(32) for

the matrix-elements

of

_x

= cos

QM

and SC denote

quantum and semiclassical tiatrix-elements

I,

m

I

+ 1, m

QM

SC difference

1,0 2,0

0.5164 0.5000 3 2 fb

1,1 2,1

0.4472 0.4269 4.6 fb

2,0 3,0

0 5071 0.5000 4 fb

2,1 3,1

0 4781 0 4705 6 fb

2,2 3,2

0.3780 0 3657 3 2 fb

3,0 4,0

0.5040 0.5000 0 8 fb

3,1 4,1

0 4880 0 4839 0 8 fb

3,2 4,2

0 4364 0.4316 fb

3,3 4,3

0 3333 0 3240 2 8 fb

106 JOURNAL DE PHYSIQUE ^{II N 2}

Substitutihg

_{x,= cos} and

performing

the integration over the

angle 4,

we -have

(I"~~

⁺

l~(iS i''~~)

~

j

8im+1~~ ~)~ fi~ Aim

^d~

⁽³⁴⁾

l _-X

For _{m m}

0,

_{we use} the _wave

8j~

~ i which has the smaller allowed range in

agreement

^with the rule

developed

in

Appendix

A For _m

<

0, et,

~

has the smaller allowed _{range, so we} take

8j~

with

8(~

~i

To calculate matrix-elements of L_ _we fornl the Hermltian

adjoint

of

L~

In

equation (35)

the _waves chosen _are

8j~

and

8(~~i

for _mm 0 and _{vice versa} for

m'<

0 In this _case the

i-

_{wave is} not associated with the shorter

branch-cut,

this agrees

with the

procedure

followed

in'part

I of this paper and leads to

interesting

but non-Hertnitian matrices.

Calculation of these matnx-elements is _a

straightforward

contour

integration.

^The contour may be

displaced

to

large xi

because the factors

'il x~)

cancel and _{one must}

siniply

extract the rdsidue of the

integrand

at

xi

- oJ.

The results _are

simplified by defining

~(*(~'~' ji

~~

fi)jm<

~

~f+I-<m+i<

^{11 +}

/fi)'~+~'

^~~~~

Then _we have

,

~

IL

+

)'. z

^{+ '}

=

~~ji '~~1~[l~ i i

>.

137)

j I

_~i, m

If

m

)/r

_{m »} °

j38)

~'~

±' _if

+ m + I r _m

< 0 where _x,

go

are defined with respect to the pairs

ii,

m

)

and

ii,

m + I

)

It _is useful to write out these matrices for

I

= 1, ²

(Tabs III, IV)

Table III Semiclassical

angular

_{momentum matrices}

for

L

=

0.000 0 719 0 000

L~

= 0.695 0 000 0 719

0.000 0 695 0 000

0.000 0 7191 0.000

L~

=

0 6951 0 000 0.7191

0.000 0 6951 0.000

000 0 000 0.000

L~ ₌ ^{0.000 0 000 0 000}

o coo o.o00 1.000

N 2 SEMICLASSICAL MATRIX-MECHANICS II 107

Table IV. Semiclassical

angular

_{momentum matrices}

for

L

=

2

o coo 1.022 o coo o coo o coo

0 979 0 000 _.1 228 o 000 o.oo0

L~

₌ ^{0 coo 1.221} 0.000 228 o 000

0 000

0.00i.

_{221 o 000 022}

0 000 0.000 o 000 0 979 o coo

.000

^{0221 0.000 o.000 0.000}

o.9791 o 000 1.2281 o.000 0.000

L~

=

0.000 221t o.000 228t o.000

0.o00 o coo 1.221 0 oo0 1.0221

0 000 coo 0 oo0 o 979t 0 000

-

^{2 000 o coo o coo o coo o coo}

o coo o00 o coo o coo o coo

L= ₌ o 000 0.000 o 000 0 000 o coo

0.000 0 000 0.000 1.000 0.000

0.000 0 000 0.000 0 000 2.000

The most important ^{feature of these} matnces is their overall accuracy each entry is within 2 fb of the

corresponding quantum

^{value However the} matrices

L~ L~

are

non-Hermttianj

_a

typical

result of calculations based _on

equation 15).

The matnces for

L~, L~

^{have the}

property

^{that if the}

product

is fornled between matrix-

elements _on

opposite

sides of the

diagonal,

the result is the square of the

quantum

matrix- element. This symmetry

property

holds also for the

harnloniq

oscillator

[4]

Given these sen~iclassical matrices

L~, L~, L~

an immediate question is whether

they obey algebraic

conditions which

parallel

the laws of the

Heisenberg

matrix mechanics

The matrices given in tables

III,

IV

exactly satisfy

the

following algebraic

relations _:

[L~, L~]

=

tL~, [L~, L=]

=

iL~, [L=,L~]

=

tL~ (39)

L~

₌ ^ma _~~.

140)

L~

=

Lj

₊

L(

₊

Lj

=

iii

₊

_{1) 3~~> 141)}

These _are the

algebraic

relations

obeyed by

the

quantum angular

momentum matrices.

Equation (41)

_is

especially

_surpnsing in view of equation

(11)

Thus the

contour-integral

semiclassical

theory reproduces

the _main results of the quantum

treatment. The matrix-elements of x=cos@

obey

the quantum selection-rules. The

semiclassical matrices L have the correct

eigenvalues, exactly obey

the commutation relations but _are non-Hernlitian and inexact

by

_a few percent.

Since the situation

closely parallels

that found for the harmonic oscillator _one

might

_suspect

there _{is a}

general proof

of these

algebraic relations, perhaps

based _on the classical Poisson brackets We have not found such _a

proof

and _m fact there _{is a} sort of counter

example provided by

the radial matrix-elements for the Coulomb

potential,

which

only approximately satisfy

the desired

algebraic

relations

Perhaps

the most

general

conclusion _is that semiclassical

results _are

good

approximations containing most features of the quantum picture.

108 JOURNAL DE

PHYSIQUE

II N 2

5.

HaW-integer spin.

The wave-function defined

by

_equation

(23)

_{is a}

single-valued

function of _x when

I

_and

m are

integers

^Thls may be tested

by following

the

changes

_m

8f~

when _{x varies} around _{a contour} that just surrounds the allowed _range

i

_xo,

xJ)

Surprisingly, fJf~

_is also

single-valued ^when'i =1/2,

_m

=

±1/2

Thls _means that the

functions

fJf~

can be used to calculate matrix-elements for _a

spin-1/2

_system

For this calculation _we substitute _xJ

=

go

₌

@

in equations

(36)-(38).

The results _are

~ l

jo

^I

~~ 2 0

L~

₌ ^~

(42)

~ l

j-

^{I o}

~~2

_o

Despite

this nice result the calculation does not give an intuitive picture or mechanical model for spin

phenomena

That _{is more}

likely

to be found

by

the method of _section 6.

6.

Superposition

of classical

paths.

This ~section examines the meaning of WKB wave-functions and the _connection between classical

paths

and quantum eigenstates ^{The discussion assembles} a number of

elementary

_or

known results

[9, 13-15]

and

interprets-

^them in terms of the

superposition

of classical

paths [16]

The

emphasis

here _{is on} the construction of semiclassical wave-functions rather than

eigenvalue spectra

or

space-time

Green's functions

II 7, 18, 19]

The construction

incorporates

the EBK quantization ^rule

[14, 15]

There _are three ways ^to obtain the three-dimensional semiclassical wave-function

first,

it is a

product

of one-dimensional WKB functions for the

separate

coordinates

ir,

_@,

4 )

second,

it _is the result of

rotating

_a

single

^classical orbit to

generate

a

space-filling family

of copies of the

orbit,

with each _copy

assigned

_a

phase

deternlined

by

the rotation

angles

and third it _is constructed

by

invariant vector-integration ^{of the} momentum field

pir

defined

by

the

family

of orbits.

For _an electron moving m a

sphencally symmetric

atomic

potential Vir)

with _energy

E~i,

the

product waye-function

_is

~nfm i~ )

"

I _R>

i~) 8tm iC°S

~ 1°

m

i ) 143)

The radial wave-function

R~i

^is

~

a~f ^±i p~1(r') dr'/h _±iw/4

~~

_~~~

fi

^{~ ~~~~}

The radial momentum

p~iir)

= r

pi (r

is

p~iir)

=

fi jE~i

⁺

^evir)

₂^~~

^{~~ (45)}

mr

N 2 SEMICLASSICAL MATRIX-MECHANICS II 109

The

angular

wave-function _is taken from

equation (28)

of section 3 The

angular

momentum quantum ^numbers are

L

=

(I+ (46)

2

L~

₌ m

(47)

The

phase

of the

product

wave-function in equation

(43)

_is

(S"(lpnfl~)d~+ lLf~Lz~)+Lz4 148)

The two tennis in

parenthesis

_come from

equation (22)

for the

phase

of

8(~. Analysis

of the

classical orbits will _now make clear the meaning of equation

(48)

Until the work of

Feynman [16]

it _was

widely

believed that _one could not understand quantum ^states in tennis of classical

orbits, ispecially

states of small

quantum

numbers such _as the

Is, 2s, 2p

states

Indeed,

there _are several important differences between classical motion

along

_an orbit and the

quantum

state

having

the _{same energy} and

angular

momentum. These differences include _some of the _more difficult _or unique features of the quantum

theory

I)

The classical _{motion is} localized

along

_a trajectory, r = ro

it ), leading

to _a

time-averaged probability density

of the fornl

p

jr

_oc 3

jr

r

oit)

^dt

149)

Thls function _{is nonzero}

along~a

_{curve or} at most a

planar

surface If _p

ir)

_were

regarded

_as

the square of _some

wave-function,

_{m an} effort to reconcile classical and quantum

descnptions,

that wave-function would have

large (or infinite)

derivatives

perpendicular

to the classical

trajectory,

^{and therefore} a

large (or infinite)

kinetic _energy

computed by

the quantum formula This _is quite ^{unlike the} quantum state, ^which gives a

smoothly

_varying

probability density

and _a kinetic _energy close to the classical value

ii)

The classical electron _is

accelerated, especially

^when it passes close _to the nucleus, and this leads to _a

fluctuating

or

time-dependent

electric current However the current vector

calculated from _a quantum energy

eigenstate

is

time-independent

iii)

While the classical Orbit has

sharply

defined values for all three

components

^{of the}

angular

momentum vector

L,

the

quantum

state does not,

typically only L~

_and

L~

have

sharply

defined values

These

important

differences _are reconciled if the

quantum

state is

compared

to a

superposition ^of classical

particles

_{on a}

family

of classical

paths

The

family

F consists of allowed orbits such that _one orbit traverses each point r m the region

permitted by

the conservation

laws,

and the

superposition

includes copies of the electron distributed

along

each orbit

(see Fig 4).

i)

Because orbits of the

family

_pass

through

each point m the

permitted volume,

there is _a

probability

for _a copy of the electron to be at any position ^{and hence} a smooth

probab1llty density

As will be _seen

below,

the quantum ^{and classical kinetic} energies are

nearly equal.

ii)

The current vector associated with the

superposition

^{of classical}

particles

is time-

independent

because the

copies

of the electron flow _{m a} smooth and laminar fashion around their orbits. As _{one copy moves away} from the

nucleus,

another

approaches

and there _is

only

a

steady

_average current

jir).

l10 JOURNAL DE PHYSIQUE II M 2

'

, _I

~

Fig 2 An allowed classical orbit located _m the _x-y

plane

The point of largest'radius _is located by the

angle

_a The

angle

_y

it)

locates the current point on the orbit The orbit _is taken to be _an

ellipse (corresponding

to the Coulomb potential) for

simplicity

but the results apply to any central-force

motion

,

z

'i

Fig

angle p

M 2 SEMICLASSICAL MATRIX-MECHANICS II ii1

z

X

Fig

4 A one-parameter

family

of orbits defined by _{varying a} for constant go and fl The

complete

two-parameter

family

is constructed by allowing fl to vary Orbits of the family pass

through

each point

m the

spatial

_regJon bounded

by

two

spheres

and _{a cone} around the z-axis Thls

family

of orbits generates the senudassical

eigenfunction

of equation (43)

iii)

The

family F(E,

L _~,

L~)

fills space, i-e-, one orbit of the

family

traverses each

point

r in

the

permitted

_region. However _one cannot not construct a

space-filling

superposition of

pa~hs

without

allowing

two

components

of L to vary

(see

below _near

Eq. (76))

The

possibility

of

constructing

a precise picture ^{of this}

type

is best _seen

by working through

some

examples

We

thereby

obtain

simple

semiclassical pictures ^for various known

solution>

of

Schroedinger's

equation,

together

with intuitive denvations of many formulas of quantum

theory

In this paper we consider the bound states of _an electron _{m a}

sphencally syrnmetnc potential.

First _we construct a

family F(E, L~, L~)

^of classical orbits all

having

the _same

energy

E~i(< o),

total

angular

momentum L

=

IL (,

^and z-component

L~.

The

family

_is not

uniquely

determined

by

the classical mechanics _in this _case, because of the

degeneracy

of the

sphencally

_{symrnetnc system}

The discussion and

figures

refer to

elliptic

orbits _{in a} Coulomb

potential (Figs 2, 3)

Thls

case is chosen for

simplicity

of presentation ^{but the} method _is not limited

to'that casel However,

^the Coulomb system has additional

degeneracy (or symmetry)

which

plays

no

esiential

_role

m the

following

discussion

l12 JOURNAL DE

PHYSIQUE

II N 2

We

begin

with _an orbit _m the x-y

plane,

with

angular

momentum vector

along

_z, and rotated in the x-y

plane

_so that the point ^of

largest

radius r~~~ occurs at _an

angle

_a from the _x-

ax1s The

angle

_y

it)

locates the current point on the

orbit,

relative to the _x-ax~s The orbit _is given

by (Fig. 2)

x ₌ r y a cos y

y = r

iy

_{a sin y}

z =

o

(50)

The function

riy

_{a is} obtained

by

solution of the classical equations ^of motion, as is the

time-dependence

of _y

it)

For the Coulomb

potential,

r =

i

l~llijl~~

n

isi)

(~

=

~~~ (52)

t mr

Here _r~ and

e are constant for orbits of the

family

F

Equation (52) applies

for _any central- force

potential.

We rotate the orbit

by

_an

angle

_Ho around the _x-axis, lo that the

angular

momentum vector becomes

L

=

10,

^L _sin

o, L _cos

o

(53)

and then rotate

by

an

angle p

around the _{z-axis, giving} the final

angular

momentum vector

L

=

iL

_sin Ho sin

p,

L sin o cos

p,

L _cos

o) (54)

The

angle

_{Ho is} constant,

r ₌

rojE,

L _~,

Lz,

_a,

p,

_y

it)) j56)

The set of orbits defined

by

_varying

a and

p

define _a two parameter

family

of orbits

,

together

with the

time-dependence

of

yit)

this

family

_covers the

permitted

_region of three- dimensional space. Tl~is _is venfied

by forming

the Jacobian

derivative,

~ii~iii~~

=

Ii t

x

II

=

r2t

cos y sin

~o 157)

This Jacobian _{is nonzero}

except

at isolated

points Equation (56)

_can be written _in

spherical polar

coordinates

ir,

_@,

4

with the result

r ₌

rjy

_a

j58)

cos = sin y sin

Ho (59)

sin

e*~~

=

icos

_{y ± i} sin _{y cos}

Ho) ^e*'~ (60)

The momentum p is

~

dro PIE,

L

,

Lz,

a,

fl,

_Y

It))

= m

~. (61)