Semiclassical matrix mechanics. I. The harmonic oscillator

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Semiclassical matrix mechanics. I. The harmonic

oscillator

R.M. More

To cite this version:

Semiclassical matrix mechanics

I. The harmonic oscillator

R. M. More

Lawrence Livermore National

_Laboratory,

P.O. Box

_{808, Livermore, California, 94550,}

U.S.A.

(Reçu

le Il

_juillet

1989,

accepté

le 25

_{septembre 1989)}

Résumé. 2014 Les matrices des variables

_dynamiques

d’un oscillateur

_harmonique

à 1 dimension

sont _{calculées par} une nouvelle méthode

_{semi-classique}

formulée par More et Warren. Les éléments de matrice

_{semi-classiques}

obtenus

_{Xn, m, Pn, m :}

_i)

obéissent exactement aux

_règles

de sélections pour les éléments de matrice nuls ;

ii)

sont très

_précis

_(environ

1

_%)

_{pour les éléments} de matrice non nuls ;

iii)

donnent un Hamiltonien sous forme

_{diagonale ;}

_iv)

satisfont exactement

les relations de commutation de

_Heisenberg

et les

_équations

d’évolution. Ces

_propriétés

sont

vérifiées pour les éléments de matrice

Fn, m

avec m ~ n. Les résultats montrent

_{qu’une grande}

partie

de la

_{Physique Quantique}

est contenue dans cette théorie

_{semi-classique}

étendue. Abstract. 2014 Matrices of

dynamical

variables of the one-dimensional harmonic oscillator are

calculated

_by

a new semiclassical method formulated

_by

More and Warren. The results are

semiclassical matrices

_{Xn, m, Pn, m,}

etc., which

i)

exactly obey

selection rules for zero matrix-elements,

ii)

are _{very accurate, circa 1} %, for nonzero matrix-elements,

iii)

which

_diagonalize

the Hamiltonian H, and

_iv)

which

_exactly

_satisfy

the

_Heisenberg

commutation relations and

equations

of motion. These statements are true for matrix-elements

_{Fn, m}

with m ~ n. The results

show that most of the _quantum

_physics

is contained in the extended semiclassical

_theory.

Classification

Physics

Abstracts 03.65S

1. Introduction.

Although

quantum

mechanics is a

_{comprehensive}

self-contained

_theory

for atomic events,

many

physicists

will agree there remains

something mysterious

about the

_{correspondence}

between classical and

_quantum

theories : how does the classical behavior of

_{large objects}

emerge out of the

_dynamics

of a

_quantum

microworld ? Interest in this fundamental

_question

combined with a search for better

_{computational techniques}

for

_applied

atomic

_physics

has drawn our attention to the

_possibility

of

_improving

the semiclassical

_theory.

To this end we have

_explored

the calculation of

_quantities

which arise in the

_Heisenberg

and

_Feynman

formulations of

_quantum

mechanics

_using

the WKB

_{approximation}

to the

Schroedinger

wave-functions. _{In this paper} we consider a

_{simple specific}

_system,

the

one-dimensional harmonic

_oscillator,

and form the semiclassical matrix-elements

_following

a new

calculational scheme based on a

_{contour-integral}

inner

_product

of WKB wave-functions

_[1-4].

The results are

_surprising.

The matrix-elements we calculate with WKB wave-functions are in excellent

_{( ~}

1

_%)

or

exact

_agreement

with

quantum

mechanics for

_half

of the matrices

_{representing dynamical}

variables such as

X, P,

X2, P 2,

and the Hamiltonian. The

_agreement

is

_especially

visible

because many matrix-elements are

_exactly

zero in both WKB and exact theories.

Matrix-elements on the

_opposite

side of the

_diagonal

can be

_badly

_incorrect,

and so the matrices are

not Hermitian. It is as if the semiclassical

_theory

of

_absorption

_{is very close} to

_quantum

mechanics,

including

all rates and selection

_rules,

but the results for emission are incorrect. Of

course, it is easy to correct the situation with the

_principle

of detailed balance.

We find that the semiclassical matrices of X and P

_{obey algebraic}

relations similar to those of the

_Heisenberg

_{quantum mechanics ;}

_they

have the correct commutator

_{[P, X],}

_they

exactly diagonalize

the Hamiltonian and

_obey

the

_{Heisenberg equations}

of motion for the

good

half of these

matrices,

associated with

_absorption

transitions.

After

_seeing

these

results,

which show that most of the

_quantum

_physics

is

_already

_present

in the semiclassical

_theory,

the reader _{may wonder where is the}

_boundary

between

_classical,

quantum

and semiclassical

_descriptions

of the world

_{[4, 5].}

We do not _{propose any}

_rigid

definition or

_{terminology ;}

our _{paper deals}

_mainly

in

_equations

which will

_speak

for themselves. Nevertheless we can

_identify

three hallmarks of a

semiclassical

_{theory :}

i)

semiclassical

_equations

should contain classical

_ingredients,

such as

_{position, velocity,}

action, etc. ;

ii)

however,

these

_ingredients

are combined with the use of

_quantum

ideas of

_wavelength,

interference,

superposition,

etc. For

_example,

the

_{deBroglie wavelength}

and Einstein-Bohr

frequency

relation are

_part

of the semiclassical

theory ;

iii)

the

_equations

are at least

_partially

derived from

_quantum

mechanics

_{by taking}

the formal limit in which Planck’s constant becomes small.

To obtain a coherent structure at the semiclassical level we add another

_ingredient,

which will be made

_precise

in sections 2 and 3 :

iv)

the semiclassical wave-functions are

_analytic

functions of the coordinates and every

singularity

has a

_{physical interpretation.}

The matrix-elements are contour

_integrals

of these

functions.

The form of these contour

_integrals

is

_{given by}

a

_specific

semiclassical rule of

calculation,

which we call the restricted

interference approximation,

derived from the

_quantum

formulas

by

systematically neglecting

interference contributions which oscillate

_rapidly

in the limit of

_large

quantum

numbers.

_(See

_Eqs.

_{(17), (18) below).}

For electrons in a Coulomb

_potential,

More and Warren

_[1]

find

_reasonably

accurate

matrix-elements for

_dipole

and

_quadrupole

transitions,

including

photoelectric

processes and

even

_including

line transitions between states with low

_{quantum-numbers,}

such as ls -->

2p,

using

WKB wave-functions and the restricted interference

_{approximation.}

The same method

also succeeds for other

_{potentials including}

the

_{Debye-screened}

Coulomb

_potential

and the self-consistent fields

_produced

_by

bound-electron

_screening

in

_{multiply-charged}

ions

_{[2, 3].}

In references

_[1-4]

the matrix-elements are evaluated

_{by saddle-point integration}

and this is a

_simple,

convenient

_{approximation}

with a

_pleasing

intuitive

_{interpretation.}

However for the

harmonic oscillator the matrix-elements can be calculated

_by

contour deformation and the results are obtained in closed form with no

_ambiguity

from numerical

_{approximations.}

These

closed-form results reveal that the semiclassical matrices are not

_merely

_good

_{approximations}

but in fact

_{exactly satisfy}

the

_expected

commutation rules

_(for

half the

_matrix,

_always).

The results obtained have a

_bearing

on the

_question

of

_{correspondence}

_identities,

the

systems.

For the

oscillator,

the semiclassical average

(X2)

is exact, while

_(X4)

is not ; we will see _{this fact emerge from the} structure of the semiclassical matrix for x itself.

These identities are limited to _{the oscillator which has many}

_{unique properties}

not to be found in more

_general

mechanical

_systems,

and the fundamental

_question

is to determine

which statements are valid for more

_{general potentials.}

This

_question

is best addressed

_by

detailed

_study

of further

_specific

cases.

What is the

_practical

value of this sort of

_{theory ?}

We do not have an

_{impressive application}

for the harmonic oscillator because the

_quantum

_theory

itself is so

_{straightforward.}

_However,

insofar as the method works for more

_complicated

_{quantum systems,}

it is clear that one can

develop

a new

computational technique

which is

_{substantially simpler}

and more convenient

than the full

_quantum

_theory,

at the

_price

of a minor

_{(10 %)}

_{loss of accuracy. Therefore} we

expect

to see a _{wide range of}

_applications

in

atomic, molecular,

plasma

and solid-state

physics.

2. WKB wave-functions.

In this section we

_develop

the

_{analytic properties}

of the WKB wave-functions. It may be assumed that the reader is familiar with the

quantum

solution as

_{given by}

Schiff

_[6]

or Landau and Lifshitz

_[7].

The

_physical

_system

is taken to be an electron

_moving

in a one-dimensional

potential

with oscillation

_{frequency w}

=

Vk/m,

zero-point scale-length a

=

Vh/mw

and energy

eigenvalue En =

(n

+

1/2 )

hw. The classical

_{turning-points}

are at the solutions of

_{En =}

The WKB wave-vector is

_m/h

times the classical

_velocity,

This is considered as a function of the

_complex

coordinate z = x +

_{i y.}

We _{define the square}

root to have a branch-cut

_{from - xn}

to + _{xn with}

_positive

real values

_just

above the cut. Such a

branch-cut is

_required

_by

the

_{physical possibility}

of + or - velocities at _any

_{point x}

in the

classically

allowed range. With this

definition,

and

where the

_{expression given}

is

_interpreted

as a contour

_integral

which defines

_0,,(z)

for all

complex

z.

Analytic properties

of

cf>n(z)

can be established

_directly

from

_equation

_{(6) (see}

the

Appendix),

but also the

_integral

can be evaluated in closed

_form,

The

_{phase integral}

is

_1/1t

times the action function of classical mechanics. The WKB

wave-functions will be normalized with a coefficient

_Nn,

The WKB

_travelling

wave is

In our work we use these functions without modification

_despite

their weak

_singularity

(~

[x + xn ]-1l4 )

at the

_{turning points.}

In

_particular

we do not _{introduce any}

_smoothing

or

connection formula across the

_{turning-points.}

In the calculation of matrix-elements we will

simply displace

the contour of

_integration

to avoid the

_singularity.

The role of a connection formula is

_{played by}

_{equation (15)}

below.

The WKB functions

_{11, n (+ )}

are

_analytic

functions of z

_except

for the branch-cut on _{the range}

(- Xn, xn)

and for the

_point

at

_infinity.

The

_{analytic properties}

are

_{readily developed}

either from the

_{defining integral, equation}

_(6),

or from the

_explicit

evaluation in

_equation

_(7).

One

might

be concerned

_{whether cP n (z)}

and/or

_{P n (± )(z)}

are

_{single-valued}

because the contour of

integration

in

_equation

₍₆₎

can reach a

_{given point}

(e.g.,

real x >

_xn)

in two _ways.

_However,

by

virtue of the

_quantization

condition and the

_properties

of the

_prefactor

1/ V q (z ),

the

functions

_{’’n± )(z)}

are

_{single-valued}

in the

_z-plane

cut _{from - Xn} to _{xn. The derivation of this}

important

technical result is sketched in the

_appendix.

Just as for

_{q (x ),}

the branch-cut on _{the allowed range}

_(-

_Xn,

_{xn )}

results

_physically

from the

two

_signs

of the

_velocity.

The

_point

at

_infinity

is a

_{singular point}

of the

_potential.

P n (± )

has no other

_singularity.

The

_asymptotic

form of

_{1/1’ J:t )}

is

Along

the real axis

_wm(+)

is

a

_{growing exponential,}

which would make for an

_{unsatisfactory}

wave-function,

but its coefficient

_{Dm is purely imaginary.}

In

fact,

Re

_{[ IP n (+}

_)(x)]

is zero for all real x

_having

_[x

₁

> xm. On the other

hand,

W)n(- )

is real and decreases for

large

(real)

z.

Because of this the WKB

_{approximation}

to the

_quantum

_{wave-function may be taken} to be

This is well-behaved at

_{large x}

and turns out to be a _very

_{good approximation}

to the

_quantum

wave-function

_everywhere

_{away from the}

_{turning points.}

A

_symmetry

formula

can be verified

_{by combining}

_properties

of

llq-(Z-)

and

_{0,, (z)}

_given

in the

_appendix.

Equation (16)

is valid

_only

in the allowed range

lx

1

x,,.

3. Semiclassical matrix elements.

If the wave-function of

_{equation (15)}

is

_directly

used to calculate matrix-elements

_by

the

quantum

formulas the results are

_{disappointing, probably}

because the matrix-elements are

integrals

of

_oscillatory

functions so that even small errors

_(near

the

_{tuming-points)}

have a

large

effect on the result.

The matrix-elements calculated

_by

a modified method are much better. In this paper we

discuss

_only

the harmonic oscillator and in this case it is the

_{extraordinary}

_{accuracy of the}

resulting

matrices which

_gives

evidence for our

_{computational}

scheme. However the modified

semiclassical rules also succeed for many other

systems

[1-3].

We use a modified inner

_product

of wave-functions defined

_by

the contour

_integral

and we calculate the matrix-element of the

_{quantum operator}

_f(P,x)

_by

the

_analogous

expression

For the

_{momentum p}

we substitute the usual differential

_operator

(see

_Eq.

(30) below).

The

integration

contour C is taken to encircle the allowed

_region

_(for

both states n,

_m)

once in the clockwise sense. Because the wave functions are

_analytic

functions of z the contour can be

deformed if needed.

In this paper we have included a factor 1/2 in the wave-function and for this reason

equations

(17),

(18)

do not need the factor 1/4 used in references

_[1-4].

The normalization

Nn

of

_equation

₍₈₎

_{has the consequence that}

_Unn

= 1 for all n.

Equations

(17)

and

₍₁₈₎

are semiclassical in the sense that the WKB functions

IF n (, )

are

_explicitly

constructed from the classical

_velocity

and action functions without ever

solving

a

_{Schroedinger equation.}

It is

_possible

to

_give

a

_rough

derivation of

_equations

(17), (18)

from the

_quantum

theory

by

inserting

the wave function of

_equation

₍₁₅₎

into the usual

_quantum

_{orthogonality}

and matrix-element

_integrals

and then

_arguing

that terms

’w’n(+) Wm(+),

Wn(-) Wm(- )

can be

neglected,

at

least for

_high

_quantum

_numbers,

because

_they

oscillate

_rapidly,

while the terms

wn(+) wm(-) ,

Similar

_reasoning

can be found in the

_literature,

without

_(evidently)

_having

been tested

critically

(e.g.,

Ref.

_[7]).

The flaw in the derivation is that the method will be

_applied

for low

_{quantum numbers,}

and the

_{integrals involving}

P («± )

P (± )

are

not small in that case. We therefore take the

_point

of view that

_equations

_{(17), (18)}

constitute basic

_assumptions

of a new semiclassical

_theory,

and

we will examine the consequences of these

_assumptions.

The

_{physical interpretation}

of

equations

(17), (18)

is

_developed

in

_greater

detail in references

_[1-4].

4. Matrix-elements for the harmonic oscillator.

The

_{surprises begin}

when we evaluate the inner

_product

of WKB functions defined

_by

equation

(17).

The matrix U is

Such matrix-elements can be

_explicitly

calculated

_{by deforming}

the

_integration

contour to

large

z where the WKB wave-functions take their

_asymptotic

forms :

where C’ is a

_large

circle at oo. The

_integral

is

_obviously

zero for all m> n. For m = _n,

Unn

is

unity

_as the result of _a

simple

_contour

integration.

For _m = n -

_1,

n - 3 and n -

_5,

_Unm

is also zero. For m = n -

_2,

it is zero because

_Bn

+

_{Ln _ 2}

= 0. Thus we

have,

in

_agreement

with

_equation

_(19).

The WKB wave-functions are

_exactly

orthonormal in the sense of the restricted interference

approximation,

a

_striking

result.

Equation

(21)

is a

_surprise

in mathematical terms because the real-axis version of the

integral

in

_equation

₍₁₇₎

has four

_{singularities}

at

_{tuming points ;}

but

_despite

the

_singular

integrand,

the result is

_simple.

_{Equation (21)}

is remarkable in

_physical

terms because a

fundamental

_{quantum property}

has

_emerged

from

_integrals

over classical velocities and action

functions.

For m = n -

_4,

_Un.

is not zero, and somewhat

simplifies

due to the

_{equation :}

Thus the matrix

_Unm

is in exact

_agreement

with the

_quantum

_theory

for all m > n -

4,

but incorrect for m n 2013 4. In

_{particular, Unm}

is not Hermitian. We have Lim

_{Un, n -}

4 =

0,

_so

evidently

_at

large n

the

region

_near the

diagonal

becomes more and more

n->oo

correct.

If U is

_{symmetrized by forming}

_Ùnm

=

Ùmn

=

Unm

Umn,

the result agrees

exactly

with the

quantum

theory,

Le.,

Ùnm

=

8 nm.

The matrix-element of the coordinate x is

We evaluate this matrix and factor out the scale

_length

Mathematically equation

(24)

is the same as

_equation

(20)

_except

that the

_large-circle

integral

has an extra _{power of} z.

_Xnm

is

_exactly

zero for all

_{m> n + 1,’}

zero for m = _n, and

Xn, n -1

1 is nonzero ; we advance the index

by

one unit to write

Also,

we have

_{Xn,n - 2}

= 0. However

Xn,n - 3

is not zero ; the formula

_simplifies

due to

which

_gives

We observe that the main values

_{Xn, n + 1, Xn + 1 n}

are _{very close} to the

_quantum

values

(= a [(n +"’1 )/2]112).

The fractional

error

is

== O.’02/n2.

The semiclassical matrix is not

Hermitian. The

_symmetrized

matrix

_{Xn, m}

=

.Km, n

=

V Xn, m Xm, n

_agrees

exactly

with the

the

_diagonal,

but

_again

decrease as n - oo while the

_nearby

«

good

» values are

_increasing.

The extreme lower-left corner of the matrix is

_badly

incorrect.

For the matrix

_{of p}

= - i Il

_{d /dx}

we

_adopt

the semiclassical formula

This

_equation

remains semiclassical because we still have not solved a

_Schroedinger

equation ;

P n (--’: )

are semiclassical wave-functions.

_Equation

(30)

is

_directly

evaluated to

_{give :}

In this case

_everything

above m = n - 3 is

good,

either exact or within 2 % of exact.

_Again

the matrix is non-Hermitian with

_growing

incorrect values in the lower-left corner. For the

good

matrix-element we have

just

as in the

_quantum

_theory.

The

_subsequent

rows

_obey

The coefficient 1.267 = 57/45 _can be verified

by

a little

_algebra.

Next we examine the commutator C =

[P, X]/ili = (PX -

XP )li h.

The result obtained

by matrix-multiplication

is

The most evident

_properties

of C are

i)

all matrix-elements

_{Cn,m are}

_exactly

correct for all

m > n - 2 and even for all m > n - 4 for n >

_{0 ;}

_ii)

_except

_{for n} =

0,

the

off-diagonal

values

This means that C = -

U2

for all matrix elements n > 0. If the matrix

Cn, m

is

_symmetrized

_by

forming

_{C n, m}

=

C m, n

=

N/èn, m Cm, n

this matrix is

again

_exact,

C n, m = - ’6 n,

m-A most

_interesting

_aspect

of

_{equation (36)}

is that the incorrect values

_{Xn, n - 3}

and

P n, n - 3

participate

in

_obtaining

_{Cn, n - 2}

= 0 for n > 0.

A matrix for

X2

can be constructed in two _ways, first

_by

_{matrix-multiplication}

from the

result in

_equation

_(25),

.

and

second,

by

contour

_integration

of

_equation

₍₁₈₎

with

_f

=

x2.

This second method

gives

Evidently

these matrices are

_i)

exact

_along

the main

_diagonal

_{and very} accurate for all

m > n -

2,

ii)

_exactly

_equal

for m > n -

_2,

so the matrix

_{multiplication}

is

_valid,

and

iii)

asymptotically

correct at

_{large n}

for the

_diagonal

line with m = n - 2.

Next we form a Hamiltonian matrix

Again

the matrix elements are exact for m > n -

_2,

and the incorrect

_subdiagonal

terms

Hn, n - 2

approach

zero as n - oo.

_Again

the

_symmetrized

matrix is exact.

Finally,

we examine the

_{Heisenberg equations}

of

_{motion, Le.,}

the commutators

[H, X ]

and

_{[H, P ].}

These

_quantities

should

_{be - i hP /m}

and

_{ihmw 2X,}

_{respectively.}

To compare with

equation

(25)

we form

and to _{compare with}

_equation

_(31),

we form

In this case all matrix-elements

_{having m > n -}

1 are in exact

_agreement

with those in

equations

(25), (31).

Evidently

a

_simple

rule is followed : the incorrect matrix-elements

_begin

at m =

n - 4 for

U,

m = n - 3

for X and P, at m

= n - 2 for

X2,

C and

_H,

and then

(evidently)

will occur at m = n - 1 for

X3

and on the

_diagonal

for

X4.

The rule is a _consequence of

matrix-multiplication

for matrices

_having

zeroes in the

_upper-right

corner. The result is also

_exactly

in

_agreement

with what one

_already

knew about the

_{correspondence}

identities for the

oscillator ;

(X2)

is exact but

_(x4)

is not. However as the incorrect values rise toward the

diagonal,

the errors become

_{comparatively}

small.

, 5.

Summary

discussion.

In _{this paper} we have

_explored

the semiclassical matrices formed

_by

the restricted interference

_{approximation,}

_{equations (17), (18),}

in order to see which matrix-elements are

correct and what

_{algebraic properties}

the semiclassical matrices have. We find that matrix-elements calculated

_according

to

_equations

_{(17), (18)}

are

_remarkably

accurate for all transitions with m > n for the harmonic oscillator. This is the most

_important

result because it appears to remain valid for other

_potentials.

We find the

_especially

_provocative

result that the

_product

or commutator of semiclassical

matrices is

_exactly

as

_expected,

_i.e.,

_{exactly obeys}

the

_{Heisenberg matrix-algebra,}

even while

This means

_that,

_subject

to the restriction to m > n -

N,

where N

depends

on the matrix

under

consideration,

the semiclassical matrices form a second

_inequivalent

_solution

of the

Heisenberg

axioms for matrix

mechanics

except

for the

_requirement

that observables be

represented by

Hermitian matrices.

The semiclassical matrices are accurate on one side of the

_diagonal,

but we must ask whether it is

_really

correct to

_identify

the

_good

matrix-elements with

_absorption

_(as

distinct from

_{emission) ;}

does that identification read too much into

_{equations (17), (18)}

which could be written with the

_opposite

convention ? Do the results

_imply

a basic difference between

quantum

mechanics and semiclassical mechanics with

respect

to time-reversal

_{symmetry ?}

Why

is the

_Hermiticity

condition the rule which must be relaxed to obtain the semiclassical

theory ?

Where does Planck’s constant enter as an

_{expansion parameter ?}

Does the

_higher

order WKB

_theory

not

_only

_get

more accurate

matrix-elements,

but also more

_{diagonal lines}

correct ? How many of these remarkable results are due to the

_{algebraic pecularities}

of the harmonic oscillator ? These

_questions

must be addressed in future work.

While this paper has concentrated on

_developing

exact

_properties

of the semiclassical

matrices,

we want to conclude with a reminder of the most

_{important practical}

consideration. The WKB functions are

_{simple explicit}

functions of the

_potential

and therefore

_equations

(17),

(18)

give

the matrix-elements of any

separable

system

by

a

_quadrature.

As far as the results of

this paper are a

_guide,

it can be

_expected

that matrix-elements obtained

_by

this

_quadrature

will be accurate.

Acknowledgment.

A

_part

of this work was

_performed

while the author visited the LULI

_(GRECO-ILM)

Laboratory

of the Ecole

_{Polytechnique,}

_{and the author is very}

_grateful

for the

_hospitality

of that

_Laboratory.

Numerical calculations related to _{the results in this paper} were

_performed

with the

_help

of K. H. Warren.

Work

_performed

under the

_auspices

of the U.S.

_Department

of

_{Energy by}

the Lawrence

Livermore National

_Laboratory

under contract number W-7405-ENG-48.

Appendix.

, In this

appendix

we

supply

a few more details

concerning

the

analytic

properties

of the WKB

wave-functions. The

_prefactor

in the WKB

_{wave-function, 1 q (z ),}

has an extra branch-cut

which we take to run

_{from xn}

to oo. With this

choice,

B/?(z)

is

i)

positive

real for z = x + i e

with

1 x

_[

_Xn

ii)

positive

real for z

_{= i y,}

_{y > 0}

iii)

positive

imaginary

for z = x - i e

_with

_{1 x 1}

_Xn

iv) positive

imaginary

for z = -

i y,

_{y >} 0

Comparing (vi)

and

_(vii),

we find a

_{sign change}

across the extra branch-cut. -Z

Analytic properties

of çb,,(z)

=

q (z’ )

dz’ can be determined from the

_integral

and/or

-

xn

is

_negative

real for

)

is

_{negative imaginary}

In

_(vi),

_(vii),

the

_imaginary

_part

is

_{negative ;}

in fact as z - _{oo ,}

The subdominant term has a branch cut _{from x = xn} to 0o across

_{which 0,, jumps}

_by

The

exponential

can

_readily

be

_{analyzed by}

means of these results. We are

_specially

interested in the limit as z - 00. A

_{straightforward expansion gives}

Now a small additional effort will convince the reader that

_{P n (± )}

is

_{analytic everywhere}

_except

at z = _oo

and along

the branch-cut

linking -

Xn to _{xn. In}

_particular,

the extra branch-cut from xn to 0o in

0

_{and 0 n (z)}

_exactly

cancels in both

_{P n (+ )}

and

_{1/1’ J- ).}

_Equation

₍₁₆₎

is verified

from

_properties

_{(vi, vii)}

of

.J q (z)

_and

On(Z)-References

[1]

MORE R. M. and WARREN K. H., J.

_Phys.

France 50

₍₁₉₈₉₎

35.

[2]

MORE R. M. and WARREN K. H., J.

_Phys.

_Colloq.

France 49

₍₁₉₈₈₎

C7-59 ; MORE R. and WARREN K. H., to _appear.

[3]

MORE R. M., to _{appear in Nucl. Instrum. Methods.}

[4]

MORE R. M., to _{appear in} « Atomic

_Physics

of

_{Highly-Charged}

Atoms », Ed. R. Marrus

(Plenum

Press).

[5]

MILLER W. H., Adv. Chem.

_Phys.

25

₍₁₉₇₄₎

345.

[6]

SCHIFF L. I., Quantum Mechanics, 3rd Ed.

_{(McGraw-Hill)}

1975.