Table Of ContentCase Studies in
Atomic Physics III
edited by
E. W. McDANIEL
Georgia Institute of Technology, Atlanta, Georgia, U.S.A.
M. R. C. MCDOWELL
Royal Holloway College, University of London, London, England
1974
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
© NORTH-HOLLAND PUBLISHING COMPANY - 1974
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording,
or otherwise without the prior permission of the copyright owner
Library of Congress Catalog Card Number 79-91451
North-Holland I.S.B.N. 0 7204 0292 1
American Elsevier I.S.B.N. 0 444 10598 0
Publishers:
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
NORTH-HOLLAND PUBLISHING COMPANY, LTD.-LONDON
Sole distributors for the U.S.A. and Canada:
AMERICAN ELSEVIER PUBLISHING COMPANY, INC.
52 VANDERBILT AVENUE
NEW YORK, N.Y. 10017
PRINTED IN THE NETHERLANDS
PREFACE
This volume collects the five Case Studies so far published in the new
journal format. It will be clear that as promised we now intend to cover
the whole field of atomic and molecular physics. A full statement of the
scope of our coverage is printed in each issue and in this volume.
We intend to continue publication approximately bi-monthly, and we
have at this moment five further issues in press, and twelve more in
various stages of preparation.
Scientists working in the field who feel they may wish to contribute
are invited to contact one of the Editors or Associate Editors whose names:
and professional addressses are given in the preliminaries to this volume.
We shall be interested in considering articles for inclusion in volume
seven (with planned publication dates in 1976—77) and later volumes.
M.R.C. McDowell
E.W. McDaniel
SCOPE OF THE JOURNAL
CASE STUDIES IN ATOMIC PHYSICS
Case Studies in Atomic Physics will accept studies written in English in the
following fields:
1. Atomic collisions (including collisions between electrons, atoms,
molecules, ions, and photons, in the energy range where the
phenomena are unaffected by nuclear forces).
2. Transport properties of electrons, ions, molecules, and photons,
normally in the gas phase.
3. Interaction potentials.
4. Surface phenomena of an impact or adsorption type.
5. Theoretical studies of the structure of atomic and molecular systems.
6. Spectroscopy (especially rf, double resonance and level crossing
spectroscopy).
7. Mesic atoms and positronium.
8. Line broadening.
9. Atomic and molecular beams.
lO.Physics of free electrons.
11 .Applications to gas lasers, optical pumping, astrophysics, atmospheric
physics, plasmas and other fields.
Emphasis will be on the fundamental properties of atoms, molecules, ions,
and electrons and on theoretical and experimental methods of studying
them. Where applications are considered, the emphasis is on the knowledge
of atomic properties required, rather than on the application per se.
Each study will present a view of the author or authors on the topic con
sidered, in as much details as may required, but will not in general attempt
a comprehensive review of the field, stressing rather those points which
the author believes are important for understanding the present position,
and for the direction of growth of the subject.
Each study will be published under the authors' names, and should be
referred to in the literature by the authors' names.
DETAILED BALANCING IN THE
TIME-DEPENDENT IMPACT
PARAMETER METHOD
E.GERJUOY
Department of Physics, University of Pittsburgh,
Pittsburgh, Pennsylvania 15213, USA
Contents:
1. Introduction 1
2. Time-reversal in the impact parameters method 4
2.1 Time-reversal with a time-independent Hamiltonian 4
2.2 Time-reversal with prescribed relative motion 9
2.3 Detailed balance in the impact parameter method 13
2.4 Symmetry under reflection 15
2.5 Moving eigenfunctions 20
3. The coupled state approximation 27
3.1 Expansion in time-independent functions 28
3.2 Microreversibility relations involving identical intervals 34
3.3 Microreversibility proof for moving expansion functions 39
Acknowledgment 45
References 45
CHAPTER 1
DETAILED BALANCING IN THE TIME-DEPENDENT
IMPACT PARAMETER METHOD
E. GERJUOY
Department of Physics, University of Pittsburgh,
Pittsburgh, Pennsylvania 15213, USA
Abstract: For collisions of the form a + b-*· c + d it is shown that detailed balance will be form
y
ally satisfied in the impact parameter coupled state approximation (a sit is commonly employed
in atomic collisions), and will have a believable physical interpretation, provided the coupled
state expansion functions are properly chosen. The connections between microreversibility rela
tions in the coupled state approximation, in the impact parameter method without the coupled
state approximation, and in the original pre-impact-parameter description of the collision by a
time-independent Hamiltonian, are elucidated. The discussion is valid for systems composed of
spinless and/or spin xh particles, distinguishable or indistinguishable, and takes into accoun tthe
transformation properties of the spin functions under the time-reversal and reflection operations
relevant to microreversibility in the impact parameter method; straight line constant speed
motion of a relative to b is not assumed. The relationship - of the present more general treat
ment - to a proof of detailed balancing in the coupled state approximation previously given by
Green, but valid only for spinless particles and relatively simply transforming expansion func
tions is clarified.
1. Introduction
Consider the collision
a + b-*c+d (1)
where c, d may be a rearrangement of the particles comprising the possibly compo
site quantum mechanical systems a, b. Although there are reasons to doubt time-
reversal invariance in certain fundamental particle reactions (Casella [1969]), in
general one supposes the Hamiltonian governing (1) is time-reversal invariant, im
plying as is well known (Messiah [1962]) that the reaction (1) must obey detailed
balance (also often termed microreversibility). More specifically, let
Τ(μ > ub>va>vb~*uc> ud> vc>vd)
α
denote the amplitude for producing c, d in bound states u , u moving with labora
c d
tory velocities v, v when a, b are incident in bound states u , u with laboratory
c d a b
velocities v , v \ then
a b
Γ(Κ^,/^,-ι> ,^+/^ (2)
0
2 E. Gerjuoy, Detailed balancing in the time-dependent impact parameter method
where K is the time-reversal operator. For systems of spinless particles
K = K (3a)
0
where K denotes the operation of complex conjugation. For systems composed of
0
n spin Y particles along with other spinless particles,
s
K = U [io(j))K ^in^yK (3b)
y 0 0
7=1
where σ (/) is the conventional Pauli spin operator σ for the/th particle, and the
product in (3b) runs over the set of spin j particles only; the quantity
y = Π [σ (/)] (3c)
0
7=1
defined by (3b) obviously is a Hermitian operator whose square is the unit operator
in the spin space of all the particles. It can be argued (Newton [1966] ) that for
systems of spinless and/or spin \ particles the definitions (3) are consistent with
Ku(J, M) = ei7r</+M>u(J, -M) (4)
where u(J,M) are a degenerate set of wave functions of total angular momentum/
and total magnetic quantum number M, constructed from one-particle space and
spin functions via standardly defined Clebsch-Gordan coefficients. Other consistent
choices of the phases of u{JM), thereby (ultimately inconsequentially) altering the
t
phase factors in (4), are possible of course (Biedenharn and Rose [1953] ). With the
choice (4), however, the reciprocity relation (2) takes the form
T(J , -M, J , -M , -v, -v -+/ , -M , J , -M , -v , -v ) =
C c d d c d a a b b a b
= exp{-m(J+M+J +M -J -M -J -M )} X
c c d d a a b b
X T(J ,M , J ,M , v , v ^J,M,J ,M , v, v ) (5)
a a b b a b c c d d c d
where, e.g, a, b in (1) are atoms in their ground states and c, d are excited states of
these atoms. Because J+M in (4) always is an integer, the signs o fthe combinations
(J +M ), etc., in the phase factor on the right side of (5) may be altered at will.
a a
In the impact parameter method of treating reactions (1) (especially as this
method is employed in atomic collisions) the relative motion of a, b is postulated to
follow a classical trajectory, whose dependence on time, energy and impact param
eter is prescribed. As a result, the originally time-independent Hamiltonian governing
the collision (1) becomes time-dependent, so that the usual proofs of detailed
balance (Messiah [1962], Newton [1966] ) are not immediately applicable to re
sults obtained from the impact parameter method, especially in the so-called coupled
E. Gerjuoy, Detailed balancing in the time-dependent impact parameter method 3
state approximation; in this approximation the wave function is projected on a
finite (therefore incomplete) non-orthogonal (except at times t = ±°°) time-dependent
set of eigenfunctions Φ,·(/), therewith replacing the partial differential time-dependent
Schrodinger equation by a finite set of coupled total differential time-dependent
equations for the projection amplitudes (Wilets and Gallaher [1966], Flannery
[1970] ). This gap in the theory largely has been filled by Green [1965], who has
given a proof of detailed balancing which is applicable to the impact parameter
coupled state approximation. However Green's proof is deficient in the following
respects *: i(i)' The time-reversal properties of spin eigenfunctions — responsible for
the io factors in (3b) - are ignored, i.e., in effect the particles are all assumed spin-
y
less. (ii) Each of the coupled state expansion eigenfunctions is assumed to transform
into itself under time reversal, a property not possessed by the set u(J,M) of equa
tion (4) even in the purely spinless case, (iii) Rather insufficient attention is paid to
the complications - in the impact parameter method and the proof of detailed
balance therein — induced by particle indistinguishability. (iv) For collisions such
as (1), it is assumed that a, b move with constant velocity, (v) The connection be
tween Green's proof of detailed balance and the more usual proofs (Messiah [1962],
Newton [1966] ) holding for time-independent Hamiltonians is concealed.
For the foregoing reasons, and because verification of detailed balance recently
(Tai [1970] ) has been found to be a useful check on the accuracy of impact param
eter coupled state calculations, this paper presents a proof of detailed balance ap
plicable to the impact parameter coupled state approximation and free from the
aforementioned deficiencies. The relation expressing detailed balance in this approx
imation is not quite identical with (2), however, as will be seen. To keep the discus
sion from becoming too diffuse, it is assumed that the reaction (1) is an atomic
(and/or molecular) collision, these being the types of collisions wherein the impact
parameter coupled state approximation primarily is employed **. Thus it can be
assumed that the colliding systems a, b are composed of spinless and spin y particles
only, i.e., it can be assumed that K of (3b) indeed is the time-reversal operator
appropriate to the original Hamiltonian, before the impact parameter method has
been introduced via prescription of the relative motion of a, b. This assumption ob
viously presumes the spin functions of the atomic nuclei (composed of spin \ particles)
only) are chosen consistent with (4), as always is possible. Of course, when the spins
of the atomic nuclei involved in the collision are wholly irrelevant (the usual situa
tion in atomic collisions), the nuclei can be treated as if they were spinless; the same
remark pertains to the spin y electrons involved in the collision. In general the elec-
* A proof seemingly along the same lines as Green [1965], but considerably less detailed and
with the same deficiencies, also has been given by Browne [ 1965].
** For reasons detailed below (subsection 2.2) the atomic systems a, b, c, d in (1) each must
contain atomic nuclei, i.e., they really must be atoms or molecules in the usual sense, and
cannot be, e.g., single electrons or positronium atoms; otherwise the results of this paper
do not apply.
4 E. Gerjuoy, Detailed balancing in the time-dependent impact parameter method
trons cannot be considered spinless, however, even when the Hamiltonian is assumed
spin-independent, because in general the reaction (1) leads to final bound states
u u whose spin-dependences differ from the spin-dependences of the initial states
c> d
Ua>ub'
2. Time-reversal in the impact parameter method
In this section I shall examine the time-reversal properties of the Hamiltonian and
wave function in the impact parameter method as primarily formulated, without the
further imposition of the coupled state approximation. It is useful to be able to
compare the results and procedures for the cases of time-dependent and time-inde
pendent Hamiltonians. With this objective in mind, in subsection 2.1 immediately
below I review certain time-reversal properties of the original time-independent
collision (1), before its Hamiltonian has been made time-dependent by prescribing
the relative motion of a, b. The corresponding properties of the same collision in the
time-dependent impact parameter description then are examined in subsections
2.2—2.5.1 stress that the contents of subsection 2.1 mainly are well known (Messiah
[1962], Newton [1966] ) and are displayed here solely for convenience.
2.1. Time-reversal with a time-independent Hamiltonian
With a time-independent Hamiltonian i/, a wave function Ψ at time t = 0 evolves
0
to
Ψ(τ) = βχρ{-ΐ#τ/«}* (6a)
0
by the time r. The wave function ΚΨ(τ) then evolves to
Φ(2τ) = exp {-\Ηφ } Κ*(τ) = exp {-ίΗτ/Λ } K exp {-ϊΗτ/h }* (6b)
0
by time 2r, where K is given by (3b). The invariance of Hinder time reversal is
expressed by the postulate that
oyH*=Hoy (7a)
where of course generally H* ΦΗ for particles with spin. The relation (7a) is equiv
alent to
KH = HK (7b)
with the understanding that for any wave function Ψ
K(HV) = i'V//*** = i"sHoy** = H(KV) . (7c)
Employing (7) in (6b)
Φ(2τ) = exp {-\Ηφ }exp{iHT/h } ΚΨ = K* . (8)
0 0
Eq. (8) expresses the time-reversed property of a solution Ψ(τ) to the time-dependent
