Lectures in Scattering Theory

271 Pages·1971·2.961 MB·English

by  A. G. Sitenko and P. J. Shepherd (Auth.)

OTHER TITLES IN THE SERIES IN NATURAL PHILOSOPHY
Vol.1. DAVYDOV—Quantum Mechanics
Vol. 2. FOKKER—Time and Space, Weight and Inertia
Vol. 3. KAPLAN—Interstellar Gas Dynamics
Vol.4. ABRIKOSOV, GOR'KOV and DZYALOSHINSKII—Quantum Field Theoretical
Methods in Statistical Physics
Vol. 5. OKUN'—Weak Intraction of Elementary Particles
Vol. 6. SHKLOVSKII—Physics of the Solar Corona
Vol. 7. AKHIEZER et al.—Collective Oscillations in a Plasma
Vol. 8. KIRZHNITS—Field Theoretical Methods in Many-body Systems
Vol. 9. KLIMONTOVICH—Statistical Theory of Non-equilibrium Processes in a Plasma
Vol. 10. KURTH—Introduction to Stellar Statistics
Vol. 11. CHALMERS—Atmospheric Electricity (2nd Edition)
Vol. 12. RENNER—Current Algebras and their Applications
Vol. 13. FAIN and KHANIN—Quantum Electronics, Volume 1—Basic Theory
Vol. 14. FAIN and KHANIN—Quantum Electronics, Volume 2—Maser Amplifiers and
Oscillators
Vol. 15. MARCH—Liquid Metals
Vol. 16. HORI—Spectral Properties of Disordered Chains and Lattices
Vol. 17. SAINT JAMES, THOMAS and SARMA—Type II Superconductivity
Vol. 18. MARGENAU and KESTNER—Theory of Intermolecular Forces
Vol. 19. JANCEL—Foundations of Classical and Quantum Statistical Mechanics
Vol. 20. TAKAHASHI—An Introduction to Field Quantization
Vol. 21. YVON—Correlations and Entropy in Classical Statistical Mechanics
Vol. 22. PENROSE—Foundations of Statistical Mechanics
Vol. 23. VISCONTI—Quantum Field Theory, Volume 1
Vol. 24. FURTH—Fundamental Principles of Theoretical Physics
Vol. 25. ZHELEZNYAKOV—Radioemission of the Sun and Planets
Vol. 26. GRINDLAY—An Introduction to the Phenomenological Theory of Ferroelec-
tricity
Vol. 27. UNGER—Introduction to Quantum Electronics
Vol. 28. KOGA—Introduction to KineticTheory Stochastic Processes in Gaseous System s
Vol. 29. GALASIEWICZ—Superconductivity and Quantum Fluids
Vol. 30. CONSTANTINESCU and MAGYARI—Problems in Quantum Mechanics
Vol. 31. KOTKIN and SERBO—Collection of Problems in Classical Mechanics
Vol. 32. PANCHEV—Random Functions and Turbulence
Vol. 33 TALPE—Thedry of Experiments in Paramagnetic Resonance
Vol. 34 TER HAAR—Elements of Hamiltonian Mechanics 2nd Edition
Vol. 35 CLARKE & GRAINGER—Polarized Light and Optical Measurement
Vol. 36 HAUG—Theoretical Solid State Physics Volume 1
Vol. 37 JORDAN & BEER—The Expanding Earth
Vol. 38 TODOROV—Analytic Properties of Feyman Diagram in Quantum Field Theory
LECTURES IN
SCATTERING THEORY
BY
A.G. SITENKO
Institute of Theoretical Physics,
Academy of Sciences of the Ukrainian SSR
TRANSLATED AND EDITED BY
P. J. SHEPHERD
Worcester College, Oxford
PERGAMON PRESS
OXFORD NEW YORK TORONTO
SYDNEY BRAUNSCHWEIG
Pergamon Press Ltd., Headington Hill Hall, Oxford
Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford,
New York 10523
Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1
Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street,
Rushcutters Bay, N.S.W. 2011, Australia
Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig
Copyright © 1971 Pergamon Press Ltd.
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
Pergamon Press Ltd.
First edition 1971
Library of Congress Catalog Card No. 74-150692
Printed in Germany
08 016574 5
Preface
THE present book is an expanded version of a course of lectures on
supplementary problems in quantum mechanics given by the author to
students of Kiev State University specializing in theoretical nuclear
physics. In the lectures an account is given of the principles of the non-
relativistic theory of potential scattering. The discussion is based on the
introduction of the concept of the scattering matrix. The properties of
the scattering matrix and its connection with physically observable quan­
tities are treated in detail. A stationary formulation of the scattering
problem is given and the wave functions of a particle in an external field
are examined. The optical theorem is formulated and time reversal and
the reciprocity theorem are discussed. The analytic properties of the
scattering matrix, dispersion relations and complex angular momenta
are investigated in detail, and the separable representation of the scatter­
ing amplitude is also treated. Scattering and bound states in a three-
particle system are examined. In the last chapter an account is given of
the theory of scattering of particles with spin, and the polarization phe­
nomena arising in the scattering are investigated. The lectures fill the
existing gap between a university quantum mechanics course and current
original papers on scattering theory, and can serve as an introduction to
the theory of nuclear reactions and elementary particles. At the end of
the book a list is given of further books, in which the reader will be able
to find not only a more detailed discussion of various problems but also
references to original papers and review articles.
The author expresses his sincere thanks to Dr. D. ter Haar for his
interest in the lectures, and to Dr. P. J. Shepherd for translating the lec­
tures into English and for useful comments.
la LST IX
Chapter 1
Quantum-mechanical Description and
Representations
1.1. Quantum-mechanical Description of Physical Systems
In quantum mechanics any physical quantity (dynamical variable) can
be represented by an operator. Associated with each operator is a linear
equation, soluble only for certain eigenvalues of the operator. The
corresponding solutions of the linear equation are called eigenfunctions.
In quantum mechanics we usually consider Hermitean operators. A linear
self-conjugate operator is said to be Hermitean; the eigenvalues of such
an operator are real.
The eigenvalues of a Hermitean operator define the possible values of
the physical quantity and are characterized by definite quantum num­
bers. The corresponding eigenfunctions define the possible states of the
physical system. Certain conditions are usually imposed on the eigen­
functions (that they should be finite, single-valued and continuous). The
eigenfunctions of Hermitean operators also satisfy orthonormality and
completeness conditions.
In the general case a physical system may be characterized by a num­
ber of dynamical variables. The state of the system is described by a wave
function (state vector) ψ (χ), where <x is the state index, i.e. a set of eigen­
Λ
values of the physical quantities or a set of the corresponding quantum
numbers which determine the state of the system, and x is the representa­
tion index, i.e. the aggregate of the variables on which the wave function
depends. The square of the modulus of the wave function ψ (χ) deter­
Λ
mines directly the probability of finding the system at a certain point x
for the given state oc.
We expand the wave function ψ (χ) in terms of the eigenfunctions of
Λ
some operator Q :
V>«(x) = Σ€*«Ψ«(Χ) (1.1)
1
Lectures ïn Scattering Theory
(q is an eigenvalue of the operator Q). It is assumed that the functions
y> (x) form a complete orthonormal set of functions, satisfying the condi-
q
tl0n μχψί(χ)ψ (χ) = δ .. (1.2)
9 ββ
The square of the modulus of the expansion coefficient c£ characterizes
the probability that the quantity Q has the value q in the state a. There­
fore we may consider the totality of expansion coefficients c\ as the wave
function of the state oc in the ^-representation. This becomes particularly
clear if we make use of the Dirac notation :
Ψ<Χ(Χ) = <*!*>> V>q(x) = <*k> and c\ = <?k>.
With this, the equality (1.1), which describes the transformation from
the ^-representation to the x-representation, can be rewritten in the form
<*μ> = Σ<χ|?χ?Ι<%>. (ΐ·3)
Q
It follows from (1.3) that the eigenfunction <Λ:| q) = ip (x) of the opera­
q
tor Q in the ^-representation is the transformation function from the
^-representation to the ^-representation.
Writing the transformation function in the form (x \ q} emphasizes the
symmetry between the representation index x and the state index q. It
is not difficult to see that the function effecting the reverse transformation,
<<7|x>, coincides with the complex conjugate <JC|#>* of the direct trans­
formation function. We multiply the equality (1.3) by <x|#>* and inte­
grate over x. Then, by virtue of the orthonormality (1.2) of the functions,
we obtain
f
whence it follows from the definition of the transformation function that
<q\x> = <x\q>*· (1.4)
The transformation of matrices of operators from one representation
to another follows directly from (1.3),
<*'| O \x} = Σ <x'\q'> W\ O \q> <q\x>, (1.5)
q.Q'
and is effected by the same transformation functions.
2
Quantum-mechanical Description and Representations
The eigenfunction of the momentum operator in the coordinate re­
presentation may be cited as an example of a transformation function :
%(r) = <r\F> = euimp"\ (1.6)
This function effects the transformation from the momentum representa­
tion to the coordinate representation. The eigenfunction of the angular
momentum operator in the momentum representation affords another
eXamplC Ψ,Μ = <«|/m> = Y (Θ, φ) (1.7)
lm
(n is the unit vector in the direction of the momentum p). The func­
tion (1.7) effects the transformation from the representation given by
the values of the angular momentum to the representation given by the
direction of motion of the particles.
The change from one representation to another (i.e. the change from
certain independent variables to others) is called a canonical transforma­
tion. Canonical transformations are carried out by unitary operators.
The physical properties of a system are invariant with respect to canonical
transformations.
In quantum mechanics, apart from the unitary canonical transforma­
tions corresponding to a change from one set of independent variables
to another, we also consider unitary transformations describing the
variation of states of physical systems in time. In contrast to canonical
transformations, the unitary operators in this case depend on time ; the
change of the state of the system with time is represented as the result of
the action of a unitary operator on the wave function. Various methods
of describing the time behaviour of physical systems in such a way are
possible and are usually called representations. These representations,
which characterize the time behaviour of physical systems, must not be
confused with the representations introduced above of physical quan­
tities and states, which are defined by a set of independent variables.
1.2. The Schrödinger Representation
In the Schrödinger representation the operators do not depend explicitly
on time and the change of the state of the system in time is determined
by change of the wave function. The time dependence of the wave func-
3
Lectures in Scattering Theory
tion is defined by the Schrödinger equation
ihd-^l = Htp(t), (1.8)
dt
where H is the Hamiltonian of the system.
The time dependence of the wave function may be described by means
of the time-shift operator R (t, 0),
(t) = R(t, 0)ψ(0), (1.9)
W
where ψ(0) is the value of the wave function at the initial moment of
time t = 0. From the condition that the probability is conserved in time
it follows that R is a unitary operator
R*R = 1. (1.10)
From the Schrödinger equation (1.8) it is not difficult to obtain the fol­
lowing operator equation for R (t, 0),
^ 0)
/ A =
dt
with the initial condition R (0, 0) = 1. The formal solution of the opera­
tor equation (1.11) can be written in the form
R(t,0) = e'iiimH\ (1.12)
1.3. The Heisenberg Representation
An alternative description of a physical system is possible, when the
wave function does not change in time, but the operators corresponding
to the physical quantités do change in the course of time.
We shall define the wave functions in the Heisenberg representation
in such a way that they coincide with the wave functions of the Schrö­
dinger representation at a certain moment of time, for example t = 0:
ΨΗ = Ws(0). (1.13)
4
Quantum-mechanical Description and Representations
According to (1.9), the transition from the Schrödinger representation
to the Heisenberg representation can be carried out by means of the
unitary transformation
^ = ^ • ^ 0 ) ^ ). (1.14)
Operators in the Heisenberg representation will be expressed in terms of
operators in the Schrödinger representation by means of the relation
Q (t) = R-Ht,0)QR(t,0). (1.15)
H s
Differentiating (1.15) with respect to time and taking (1.11) into account,
we obtain the Heisenberg equation of motion which determines the
change of an operator in the Heisenberg representation with time:
iï^rQH(t) = [QH(t\H]. (1.16)
dt
Operators which commute with H are independent of time in the
Heisenberg representation. In particular the Hamiltonian of the system H
remains unchanged in going from the Schrödinger representation to the
Heisenberg representation and vice versa.
1.4. The Interaction Representation
We shall consider a system consisting of several particles interacting
with each other. In this case it is convenient to decompose the Hamil­
tonian of the system into two parts
H=H +V, (1.17)
0
where H is the Hamiltonian of the system without interaction of the
0
particles and Fis the interaction. In such systems it is convenient to use
the interaction representation to describe change of the state with time.
We introduce a new wave function ψ^ί) such that
(î) = Rô1(t,0) (t) (1.18)
Wl Ws 9
where the shift operator R (t, 0) is a solution of the equation
0
3ÄoM , ) (1.19)
/Ä = jffoÄo(i 0
dt
5
Lectures in Scattering Theory
with the initial condition R (0, 0) = 1. The formal solution of eqn. (1.19)
0
has the form
R (t,0) = e~(lin)Hot. (1.20)
o
Differentiating eqn. (1.18) with respect to time and using the Schrödinger
equation (1.8), we find the equation of motion in the interaction repre­
sentation
dJ^l = ν^θψΜ, (1.21)
m
dt
where Vj(t) is the interaction operator in the interaction representation
Vj(t) = Rô\t 0) VR (t, 0) = e(il*)Hot V 'it,mHot. (1.22)
9 s 0 s e
The time dependence QÎ any operator in the interaction representation
is defined similarly:
(t) = eiilh)HQt Qe-(il«)Hot. (1.23)
Ql s
Differentiating this relation, we find the variation with time of an opera­
tor in the interaction representation
ift^Q(t) = [Q(t),H ]. (1.24)
I I 0
dt
The interaction representation is intermediate between the Heisenberg
and Schrödinger representations. Operators in the interaction represen­
tation depend on time in the same way as do the Heisenberg operators
of the system in the absence of the interaction, and the change with time
of the wave functions is caused completely by the interaction. Below we
shall omit the index / on quantities in the interaction representation.
PROBLEMS
1.1. Find the eigenfunctions of an operator # in its own representa­
tion.
Let q be an eigenvalue of the operator #. The equation for the eigen­
0
functions of an operator # in any representation has the form
4ψ*ο = SW*o· (1-25)