Irreducible Tensor Methods - An Introduction for Chemists [1 ed.]

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Irreducible Tensor Methods AN INTRODUCTION

FOR

CHEMISTS

Brian L. Silver Department of Chemistry Technion—Israel Institute of Technology Haifa, Israel

ACADEMIC P R E S S A Subsidiary

of Harcourt

New York

Brace Jovanovich,

San Francisco Publishers

London

1976

COPYRIGHT © 1 9 7 6 , BY A C A D E M I C PRESS, I N C . ALL RIGHTS RESERVED. N O PART O F THIS P U B L I C A T I O N M A Y B E R E P R O D U C E D OR T R A N S M I T T E D I N A N Y F O R M OR B Y A N Y M E A N S , E L E C T R O N I C OR M E C H A N I C A L , I N C L U D I N G P H O T O C O P Y , RECORDING, OR ANY I N F O R M A T I O N STORAGE A N D RETRIEVAL S Y S T E M , W I T H O U T PERMISSION IN WRITING FROM THE PUBLISHER.

A C A D E M I C PRESS, INC. I l l Fifth A v e n u e , N e w York, N e w Y o r k 10003

United

Kingdom

Edition

published

A C A D E M I C PRESS, INC. 2 4 / 2 8 Oval R o a d , L o n d o n N W 1

by

(LONDON)

LTD.

Library of Congress Cataloging in Publication Data Silver, Brian L Irreducible tensor methods. (Physical chemistry, a series of monographs ; 3 6 ) Bibliography: p. 1. Symmetry (Physics) 2. Quantum theory. 3. Calculus of tensors. I. Title. II. Series. QC174.17.S9S54 515'.63 75-36655 ISBN 0 - 1 2 - 6 4 3 6 5 0 - 9

P R I N T E D I N T H E U N I T E D STATES O F AMERICA

For Danny To my Mother and t h e m e m o r y of m y

Father

Preface

In this b o o k I have a t t e m p t e d t o give an i n t r o d u c t i o n t o t h e t h e o r y and use of irreducible tensor o p e r a t o r s . T h e fact t h a t t h e t e x t is aimed at chemistry graduates has controlled b o t h t h e m a t h e m a t i c a l level and t h e choice of illustrative e x a m p l e s . T h a t part of the t h e o r y devoted t o systems having spherical s y m m e t r y c o m e s w i t h i n t h e range of w h a t physicists w o u l d call t h e t h e o r y of angular m o m e n t u m , a n d has b e e n summarized a u t h o r i t a t i v e l y by F a n o and R a c a h ( 1 9 5 9 ) . T h e t h e o r y as applied t o molecules is t h e subject of a m o n o g r a p h b y Griffith ( 1 9 6 2 ) . Since t h e p u b l i c a t i o n of these classic treatises there have been advances in t e c h n i q u e m o s t of w h i c h have n o t been i n c o r p o r a t e d i n t o t e x t b o o k s or treatises. T h e range of applications, especially in chemical physics, has increased rapidly in t h e last few y e a r s . I have included t h e d e v e l o p m e n t s in t e c h n i q u e , in particular t h e use of second q u a n t i z a t i o n , real Cartesian tensors, off-diagonal o p e r a t o r equivalents, and coupling and recoupling coefficients for spinor g r o u p s . In a d d i t i o n I have given an i n t r o d u c t i o n t o vector spherical h a r m o n i c s and s u b s e q u e n t l y described t h e m u l t i p o l e e x p a n s i o n of light waves and t h e expression of t h e c o h e r e n c y m a t r i x in t e r m s of irreducible t e n s o r s . This material is used in t h e s u m m a r y of t h e m o d e r n formalism for light scattering w h i c h c o n c l u d e s t h e b o o k . T h e close relationship of the theories for spherical and lower s y m m e t r y has b e e n stressed t h r o u g h o u t the t e x t . T h e a m o u n t of m a t h e m a t i c s required of t h e reader is m o d e s t . A little e l e m e n t a r y group t h e o r y goes a long w a y . Physicists, m a t h e m a t i c i a n s , or theoretical chemists m a y feel t h a t d e p t h has b e e n sacrificed. N o t as a concession t o t h e m , b u t as a p o i n t e r to t h e average chemist, there are, scattered in t h e t e x t , brief—almost vanishing—paragraphs hinting at the delights of Lie g r o u p s , c o n t r a g r e d i e n c e , time reversal, a n d o t h e r imaginary beings (Borges, 1 9 7 4 ) .

xiii

xiv

Preface

T h e examples are almost all t a k e n from t h e recent research literature a n d have b e e n w o r k e d t h r o u g h in considerable detail. I believe all the e x a m p l e s t o be of interest in their o w n right, b u t their m a i n p u r p o s e is t o illustrate t h e practical application of t h e theoretical t e c h n i q u e s . T h e godfather of this b o o k is Professor R u b e n P a u n c z , an invaluable colleague t o have just d o w n the corridor. My t h a n k s are d u e t o critical s t u d e n t s at t h e T e c h n i o n and t h e Weizmann I n s t i t u t e for forcing m e , against the d e m a n d s of e n t r o p y , t o order my thoughts. T h e m a n u s c r i p t was e x p e r t l y and intelligently t y p e d b y Mrs. N o r m a J a c o b . I t h a n k those at A c a d e m i c Press involved in the p r o d u c t i o n of this b o o k . Their c o o p e r a t i o n , care, and professionalism m a d e it a pleasure t o w o r k w i t h t h e m .

Introduction

"Then the king's colour changed, and his thoughts alarmed him; his limbs gave way, and his knees knocked together. The king cried aloud to bring in the enchanters, the Chaldeans, and the astrologers. The king said to the wise men of Babylon, * Whoever reads this writing, and shows me its interpretation, shall be clothed with purple, and have a chain of gold about his neck, and shall be third ruler in the kingdom.' " Daniel 5 : 6 , 7

T h e use of s y m m e t r y t o simplify p r o b l e m s in a t o m i c and molecular q u a n t u m mechanics is familiar t o those c h e m i s t r y or physics u n d e r g r a d u a t e s w h o have t a k e n a course in e l e m e n t a r y g r o u p t h e o r y and learned to classify wave functions and operators according t o g r o u p r e p r e s e n t a t i o n s . T h e almost trivial derivation of selection rules from this classification is t h e best k n o w n d e m o n s t r a t i o n of the p o w e r of g r o u p t h e o r y t o reveal generalizations w i t h o u t t h e n e e d for c o m p u t a t i o n . R o u g h l y speaking t h e irreducible tensor m e t h o d is designed t o derive all t h o s e p r o p e r t i e s of a system w h i c h are consequences of its spatial s y m m e t r y . T h e m e t h o d m a y be regarded as a b r a n c h of group t h e o r y , or, in its application t o a t o m i c systems, as a part of t h e t h e o r y of angular m o m e n t u m . T h e c o n c e p t of irreducible tensors and t h e d e v e l o p m e n t of a formalism for m a k i n g practical use of t h e m are due mainly t o Guilio R a c a h w h o s e four classic p a p e r s ( R a c a h , xv

xvi

Introduction

1 9 4 2 a , b , 1 9 4 3 , 1 9 4 9 ) are recognized as a b r e a k t h r o u g h in a t o m i c s p e c t r o s c o p y . R a c a h ' s m e t h o d s for handling m u l t i e l e c t r o n systems were soon seized u p o n b y nuclear physicists, w h o realized the close m a t h e m a t i c a l relationship b e t w e e n t w o central field m o d e l s : t h e electrons in an a t o m and the n u c l e o n s in t h e nuclear shell t h e o r y (cf. de-Shalit and Talmi, 1 9 6 3 ) . R a c a h ' s w o r k was n o t created in a v a c u u m , b u t was partially based o n , and developed c o n c u r r e n t l y w i t h advances in t h e t h e o r y of angular m o m e n t u m m a d e b y Eugene Wigner. T w o a u t h o r i t a t i v e t e x t s summarize their w o r k : " I r r e d u c i b l e Tensorial S e t s " b y F a n o a n d R a c a h ( 1 9 5 9 ) and " G r o u p T h e o r y and Its Application t o t h e Q u a n t u m Mechanics of A t o m i c S p e c t r a " b y Wigner ( 1 9 5 9 ) . In Part I of this b o o k there is an a c c o u n t of t h o s e parts of t h e t h e o r y of angular m o m e n t u m t h a t are n e e d e d for an understanding of the irreducible tensor m e t h o d . At the e n d of Part I, in C h a p t e r 9 , we derive t h e few basic expressions n e e d e d for a wide range of applications. Part II consists mainly of a variety of applications. Wigner's w o r k o n angular m o m e n t u m was largely c o u c h e d in the language of g r o u p t h e o r y , a n d he p o i n t e d o u t t h a t m u c h of t h e t h e o r y was directly applicable t o p o i n t groups (Wigner, 1 9 4 0 ) . This aspect was left u n t o u c h e d until t h e 1 9 5 0 s when t h e t h e o r y of irreducible tensors for p o i n t groups was developed, p r e d o m i n a n t l y b y Y u k i t o T a n a b e in J a p a n and s u b s e q u e n t l y b y J o h n Griffith in England w h o s e m o n o graph " T h e Irreducible Tensor M e t h o d for Molecular S y m m e t r y G r o u p s " (Griffith, 1 9 6 2 ) is t h e m o s t comprehensive s u m m a r y of t h e t h e o r y . Part III is p a r t l y a presentation of Griffith's w o r k . In a d d i t i o n t h e r e is an a c c o u n t of m o r e recent w o r k o n spinor ( d o u b l e ) groups. T h e t e c h n i q u e s are again based o n a correct classification of states and o p e r a t o r s , b u t this time in t e r m s of t h e t r a n s f o r m a t i o n s i n d u c e d b y a usually finite group of s y m m e t r y o p e r a t i o n s . T h e similarities b e t w e e n the m e t h o d s for h a n d i n g p o i n t g r o u p a n d spherical s y m m e t r i e s are m a n y and pleasing a n d will be c o m m e n t e d on frequently. Part IV includes applications of the t h e o r y given in Part III and also a short a c c o u n t of t h e use of second q u a n t i z a t i o n . In evaluating m a t r i x e l e m e n t s in m u l t i e l e c t r o n states it is convenient t o m a k e use of coefficients of fractional parentage (cfp). These coefficeints allow us t o build u p a n t i s y m m e t r i c states of m u l t i e l e c t r o n systems from states containing fewer electrons. T h e advantages of this p r o c e d u r e will, it is h o p e d , b e c o m e a p p a r e n t in C h a p t e r 2 1 . T h e use of cfp was developed almost entirely b y R a c a h w h o was c o n c e r n e d w i t h p r o b l e m s of a t o m i c s p e c t r o s c o p y . T h e idea spread i n t o nuclear shell t h e o r y and t h e n i n t o molecular electronic s t r u c t u r e . T h e basic and related c o n c e p t s of s y m m e t r y and t r a n s f o r m a t i o n a l p r o p e r t i e s can be applied t o vector fields. Multipole expansions of fields have b e e n used in physics for m a n y years, for e x a m p l e , in c o n n e c t i o n w i t h nuclear s p e c t r o s c o p y ( R o s e , 1 9 5 5 ) . T h e physical properties of molecules (e.g., polarizability, electric charge d i s t r i b u t i o n ) are frequently described b y m u l t i p o l e e x p a n s i o n s . In dealing w i t h t h e i n t e r a c t i o n of static fields or radiation w i t h molecules it seems n a t u r a l t o use m u l t i p o l e expansions of the relevant fields. In Part V an analysis is given of t h e t r a n s f o r m a t i o n s of vector fields under coordinate rotations.

Introduction

xvii

T h e reader should n o t be discouraged b y t h e occasionally forbidding formalism. T h e m a t h e m a t i c a l b a c k g r o u n d of the average chemist is m o r e t h a n sufficient t o o v e r c o m e any difficulties. S o m e sections are m a r k e d ( n ) t o indicate t h a t t h e y are digressions a n d are t e m p o r a r i l y or p e r m a n e n t l y dispensable. T h e form t h a t these asides t a k e is often t h a t of a h i n t at t h e wider or deeper aspects of some part of t h e t e x t , and in such cases t h e reader should n o t e x p e c t t o find a substitute for serious discussion. T h e t e x t is a set of variations on a single simple theme—the behavior u n d e r coordinate t r a n s f o r m a t i o n s of scalar, spinor, and vector fields. We have limited ourselves t o c o o r d i n a t e rotations and inversions. Even within these b o u n d a r i e s there are far m o r e sophisticated d e v e l o p m e n t s w h i c h we d o n o t t o u c h , b u t w h i c h are primarily of importance t o a t o m i c and nuclear spectroscopists (cf. J u d d , 1 9 6 3 , 1 9 6 7 ) . F u r t h e r still, b e y o n d these h o r i z o n s , are the wide plains of general t r a n s f o r m a t i o n t h e o r y , w h e r e r o a m the h e r d s of e x o t i c g r o u p s and t r a n s f o r m a t i o n s t a m e d b y the physicists. T h e tensor m e t h o d s w h i c h are o u r c o n c e r n represent b u t a small d o m a i n w i t h i n the empire of s y m m e t r y . A l t h o u g h those p o i n t i n g o u t t h e inevitable errors in the t e x t will n o t be a p p o i n t e d third ruler in t h e k i n g d o m , t h e y will gain t h e a u t h o r ' s g r a t i t u d e .

PART

Chapter

I

1

The Rotation Operator

1.1

COORDINATE

ROTATIONS

O u r initial object is t o find a c o m p a c t formalism for describing t h e effect p r o d u c e d on an arbitrary function of a given set of c o o r d i n a t e s w h e n t h a t set is subject t o a r o t a t i o n a b o u t t h e origin. This p r o b l e m is central t o t h e irreducible tensor t e c h n i q u e . It is easiest t o visualize t h e p r o b l e m in t e r m s of a physical field in space, for example t h e two-dimensional scalar field formed b y t h e t e m p e r a t u r e at every p o i n t o n this page. T o describe this t e m p e r a t u r e distribution we can choose t w o m u t u a l l y perpendicular c o o r d i n a t e axes, labeling t h e m , s a y , x and >>, a n d t h e n m a k e a list of the t e m p e r a t u r e s at a n u m b e r of p o i n t s on t h e page. A c o m p l e t e description of t h e field w o u l d a p p a r e n t l y require an infinite list of p o i n t s w i t h their associated t e m p e r a t u r e s . However, m o s t scalar (and vector) fields t h a t arise in physics and chemistry can be described b y a function of t h e c o o r d i n a t e s . T h u s a possible, if i m p r o b a b l e , t e m p e r a t u r e distribution on this page might b e given b y t h e function t e m p ( x , y) = xy, in certain u n i t s . This function is of course only applicable if t h e c o o r d i n a t e s (x, y) of a given p o i n t are referred t o t h e axes we have c h o s e n . S u p p o s e we n o w r o t a t e o u r c o o r d i n a t e system. T h e new axes X, Y might be o b t a i n e d , for e x a m p l e , b y a r o t a t i o n of t h e old axes x, y b y 4 5 ° anticlockwise a b o u t t h e origin. N o w we wish t o describe the scalar ( t e m p e r a t u r e ) field in t e r m s of X and Y. We obviously c a n n o t use t h e same function t e m p ( x , y) a n d replace x b y X a n d y b y Y. T h u s in t h e old system = ut m te n n we t e m p ( l , 1) 1> b system the t e m p e r a t u r e at t h e p o i n t ( 1 , 1 ) is z e r o . We are going t o n e e d a different function of X a n d Y t o describe t h e scalar field. T h e 2 2 function we n e e d is TEMP(X, Y) = \{X - Y ), w h i c h can b e c h e c k e d for a few p o i n t s a n d w h i c h w e will later establish r a t h e r m o r e elegantly. T h e q u e s t i o n we n o w ask is " C a n I find TEMP(X, Y) if I k n o w b o t h t h e function t e m p ( x , y) and t h e relative 1

2

1

The Rotation

Operator

orientations of t h e (X, Y) a n d (x, y) axis s y s t e m s ? " In o t h e r w o r d s , surely it c a n n o t be necessary t o find t h e form of t h e function T E M P ( Z , Y) b y going b a c k t o the raw physical data. It is n o t . In this c h a p t e r we will be c o n c e r n e d w i t h t h e " t r a n s f o r m a t i o n of functions u n d e r c o o r d i n a t e r o t a t i o n s , " w h i c h is t h e general n a m e for the p r o b l e m typified b y discovering h o w t o derive TEMP(X, Y) from t e m p ( x , y). T h e first step is t o agree o n an u n a m b i g u o u s and convenient m e t h o d for describing c o o r d i n a t e r o t a t i o n s , b u t before proceeding it is i m p o r t a n t t o realize t h a t for the m o m e n t we are going t o limit ourselves t o the t r a n s f o r m a t i o n of scalar functions. The t r a n s f o r m a t i o n s of a vector field, such as a magnetic field, are n o t so simple because each p o i n t of space is associated w i t h a q u a n t i t y w h i c h h a s , in a d d i t i o n t o a m a g n i t u d e , a direction. T h e t r a n s f o r m a t i o n s of vector fields are dealt w i t h in C h a p t e r 2 8 .

yn 43

(X-I.Y-I)

/

2 /

\

FIGURE

THE EULER

/

- ( x - i , y-i)

1

1

1.2

/

/

2

3

4

x

1

ANGLES

There are t w o obvious w a y s of describing c o o r d i n a t e rotations—we can specify either the operations p e r f o r m e d or t h e final result. T h u s in our opening e x a m p l e we could say (i) t h e x, y axis system was r o t a t e d anticlockwise b y 4 5 ° a b o u t t h e z axis, or (ii) t h e X axis is inclined at 4 5 ° t o b o t h t h e x a n d y axes. T h e Y axis is inclined at 4 5 ° t o t h e ; ; axis and 135° t o t h e x axis. T h e difference b e t w e e n these t w o a p p r o a c h e s m a y seem a c a d e m i c , b u t we will see in C h a p t e r 2 t h a t in certain cases t h e effect of a r o t a t i o n of, say, 2n is n o t t h e same as t h e effect of a r o t a t i o n of 4rr, despite t h e fact t h a t for t h e m a n in t h e street t h e relative o r i e n t a t i o n of t h e original a n d final c o o r d i n a t e systems is t h e same for b o t h r o t a t i o n s . F o r this reason we describe c o o r d i n a t e r o t a t i o n s in t e r m s of o p e r a t i o n s .

1.2

The Euler

3

Angles Z A

X FIGURE

We will w o r k , t h r o u g h o u t

2

this b o o k , w i t h a right-handed

coordinate

system

(Fig. 2 ) . This is an arbitrary choice b u t it is i m p o r t a n t t o realize t h a t if we choose a left-handed s y s t e m , there will be a change of sign in some of t h e expressions derived later. A positive rotation is defined t o be such t h a t it w o u l d carry a right-handed screw a forward along the relevant axis, away from t h e origin. We symbolize b y Rn(0) r o t a t i o n b y a positive angle 6 a b o u t an axis n. Three s t a n d a r d r o t a t i o n s are n o w defined (Fig. 3). Starting with t h e original c o o r d i n a t e system x, y, z, we (i)

r o t a t e t h e axes b y a positive angle a ( 0 < a < 2n) a b o u t t h e z axis (this first

r o t a t i o n gives us a n e w c o o r d i n a t e system x yy, original z axis);

FIGURE

3

z

in w h i c h z

is identical t o t h e

1

4 r o t a t e b y a positive angle ] 3 ( 0 < j 3 < 7 r )

(ii)

The Rota tion Opera tor

about the y

axis, giving a n e w

c o o r d i n a t e system x", y \ z" in w h i c h y " - y \ (hi) ttt

x

rotate ttt

,y

,z

ttt .

b y 7 ( 0 < 7 < 2n) ttt

i - i

in w h i c h z

j •

,.

about

the z "

axis

t o give

t h e axis

system

tt

1 ,

is identical t o z .

T h e angles a , 0 , a n d 7 are t h e f a m o u s Euler angles, w h i c h are widely used in describing r o t a t i o n s of c o o r d i n a t e s y s t e m s , functions, or solid b o d i e s a b o u t a fixed p o i n t . A t least o n e practical advantage of their use is t h e fact t h a t t h e angles a a n d ]3 are t h e same as t h e polar angles of t h e final a x i s z " w i t h respect t o t h e original a x i s z . is limited t o an angle j3 of 7: or less. T h e reader can

Note that the rotation about y

convince himself w i t h a m o d e l t h a t t h e c o o r d i n a t e axes can b e r o t a t e d t o a n y final position desired, if any t w o of t h e angles a, j3, a n d 7 are allowed t o range from 0 t o 2ir a n d t h e t h i r d is limited t o t h e range

0-n.

This sequence of three r o t a t i o n s can b e w r i t t e n Rz» (y)Ry'(P)Rz((x) order implies t h a t w e operate

w i t h Rz(a)

first a n d Rz"(y)

p e r f o r m i n g these t h r e e r o t a t i o n s a b o u t t h e axes z, y\

where t h e

last. T h e final result of

a n d z" can be s h o w n t o be

equivalent t o t h r e e r o t a t i o n s of t h e c o o r d i n a t e system a b o u t t h e fixed

(original) axes

x, y, a n d z. In fact

(i-i)

RAi)Ryimz(cc)=RMRy(P)Rz(i)

N o t e t h e reversal of t h e o r d e r of a , j3, a n d 7. T h e s t a n d a r d p r o o f relies on t h e fact t h a t we c a n , for e x a m p l e , write

Ry'(fi =

Rz(a)Ry(p)Rz(r")

This says t h a t t h e result of a r o t a t i o n of /3 a b o u t y can b e o b t a i n e d b y first moving t h e y

axis b a c k t o y w i t h a r o t a t i o n of —a a b o u t z ; t h e n r o t a t i n g t h e system a b o u t y b y

t h e angle j3; a n d finally moving t h e y axis b a c k to>>' w i t h a r o t a t i o n ofRz(a).

By t h e

same k i n d of reasoning, r e m e m b e r i n g t h a t z = z ,

Rz"(y)=Ry'(ff)RAi)Ry'(rP)

= [Rz(a)Ry(fiRz(-a)]

Rz(y)

[Rz(a)Ry(-P)Rz(r*)]

N o w r o t a t i o n s a b o u t t h e same axis c o m m u t e , so t h a t w e can reverse t h e order of t h e f o u r t h a n d fifth o p e r a t o r s in this expression a n d t h e n , since Rz (—a)Rz(oc)

Rz 0 . We can o b t a i n a n o t h e r form for Rz(86) b y using t h e i d e n t i t y lz = -i(x

d/dy-y

d/dx)

w h i c h , from ( 1 . 5 ) , results in Rz(86)=

\+i86lz

(1.7)

N o t e t h a t we w o r k t h r o u g h o u t in u n i t s of h f o r angular m o m e n t u m . In t h e language of t h e physicist we say t h a t the o p e r a t o r lz is t h e " g e n e r a t o r of infinitesimal r o t a t i o n s a b o u t t h e z a x i s . " T o b e p e d a n t i c w e might a d d " i n t h e case of scalar fields." There are o t h e r infinitesimal t r a n s f o r m a t i o n s , for e x a m p l e translations, w h i c h can b e expressed in t h e form 1 + iaS, w h e r e S is an o p e r a t o r a n d is said t o " g e n e r a t e " t h e t r a n s f o r m a t i o n and a is an infinitesimal scalar d i s p l a c e m e n t of some k i n d (Messiah, 1 9 6 2 , C h a p t e r X V ) . 1.4

TRANSFORMED

FUNCTIONS

There is often confusion as t o t h e detailed m e c h a n i s m b y w h i c h we t r a n s f o r m functions from one frame t o a n o t h e r . In this section we a t t e m p t t o clarify some c o m m o n m i s u n d e r s t a n d i n g s . E q u a t i o n ( 1 . 3 ) can b e w r i t t e n generally as

where / a n d F are t h e functions in t h e t w o frames and r and r' are t h e c o o r d i n a t e s of

1

8

The Rotation

Operator

in physical space. Of course t h e t w o sets of c o o r d i n a t e s r a n d r' refer t o

t h e same point

different frames a n d are therefore different

sets of numbers.

We can write E q . ( 1 . 6 )

m o r e generally a n d m o r e pedantically as

(1.8)

F(r') = Rf(r') w h e r e R is t h e o p e r a t o r c o r r e s p o n d i n g t o any arbitrary c o o r d i n a t e

transformation.

N o t e very carefully t h a t t h e c o o r d i n a t e s r' on b o t h sides of this e q u a t i o n are identical. This is i m p o r t a n t . If w e have o b t a i n e d t h e explicit form of t h e o p e r a t o r R w h i c h satisfies E q . ( 1 . 8 ) a n d w e wish t o o b t a i n t h e function F ( r ' ) , w e m u s t o p e r a t e w i t h / ? o n t h e function / ( r ' ) , not o n t h e f u n c t i o n / ( r ) . R m u s t b e an o p e r a t o r expressed in t e r m s of t h e c o o r d i n a t e s r' if it is t o o p e r a t e on / ( r ' ) . To obtain

the transformed

o p e r a t o r R(r)

function

o n the function f(r),

F(r)

we must operate with

t h e original

function of t h e

the new

coordinates. This p o i n t is b r o u g h t o u t specifically in E q . ( 1 . 5 ) . O n t h e left-hand side F(x, y, z) is a function exactly

of t h e c o o r d i n a t e s in t h e t r a n s f o r m e d appear

in

the

operator

frame. But t h e same

(1 + 89 [x d/dy - y d/dx])

and

in

coordinates

the

function

y, z) o n t h e right-hand side.

fix,

If we h a d used a left-handed c o o r d i n a t e s y s t e m , i.e., i n t e r c h a n g e d x andj> in Figs. 4 and 5, t h e f o r m of Rz(86)

w o u l d b e 1 - / 86 lz as can b e verified b y rederiving ( 1 . 5 )

and n o t i n g t h a t w e m u s t n o w p u t CP2 = y\

1.5

86 a n d CP\ = x x

86.

THE ROTATION OPERATOR FOR ONE AXIS A r m e d w i t h t h e o p e r a t o r for infinitesimal r o t a t i o n s a b o u t t h e z axis w e can n o w

derive t h e o p e r a t o r for finite r o t a t i o n s a b o u t this axis, w h i c h will enable us t o d e d u c e t h e effect of a finite c o o r d i n a t e r o t a t i o n on a f u n c t i o n . Consider t h e o p e r a t o r Rz

(6)

c o r r e s p o n d i n g t o a positive finite r o t a t i o n b y 6 a b o u t t h e z axis. A n y finite r o t a t i o n b y 6 can b e regarded as t h e s u m of a series of n successive r o t a t i o n s t h r o u g h an angle of 6/n.

If we let n t e n d t o infinity, t h e n 6/n

infinitesimal Rz(6)

rotation

t e n d s t o z e r o , and we can define an We n o w express t h e finite

angle 86 = \imn^oo(6/n).

rotation

as a succession of n infinitesimal r o t a t i o n s : Rz(6)

n

= lim [Rz(6/n)] = n-*oo

using expression ( 1 . 7 ) for Rz(86).

lim [Rz(86)f

de

60-K)

= Urn

6d

[l+i86lzf

80-K)

A t this p o i n t we use o n e of t h e expressions for t h e y x

e x p o n e n t i a l f u n c t i o n , n a m e l y l i m x_ > 0 (1 + x) ^ Rz(6)

= exp(i6lz)

= e x p j>, a n d o b t a i n (1.9)

This is a very i m p o r t a n t e q u a l i t y ; from it w e can o b t a i n t h e o p e r a t o r for any r o t a t i o n of t h e c o o r d i n a t e s y s t e m b y n o t i n g t h a t t h e r e m u s t b e analogous expressions for finite r o t a t i o n s a b o u t t h e x and y axes.

1.5

The Rotation

Operator

9

for One Axis

We l o o k at a very simple ( a n d s t a n d a r d ) e x a m p l e of t h e use of t h e o p e r a t o r Rz(6). Suppose w e have a scalar field / ( 0 ) such t h a t at each p o i n t P there is associated a n u m b e r exp(/m0) where m is an integer a n d 0 measures t h e angle of inclination of t h e line OA' t o t h e c o o r d i n a t e axis OA, where OA' passes t h r o u g h P and t h e origin (Fig. 6 ) . 614 is o u r reference line in t h e plane a n d defines t h e c o o r d i n a t e system we are using. Every p o i n t o n OA' will have t h e same n u m b e r exp(/ra0) associated w i t h it since t h e scalar field / ( 0 ) is defined t o d e p e n d only o n 0. We define a z axis passing t h r o u g h O, perpendicular t o t h e plane of t h e page, a n d p o i n t i n g u p w a r d . L e t us n o w r o t a t e t h e

FIGURE

6

c o o r d i n a t e axes a b o u t z b y a positive angle a. so t h a t OA is carried t o OB. What function n o w describes t h e scalar field? It will b e a function F ( 0 ' ) = Rz(a)f( -x, w h i c h , s u b s t i t u t e d i n t o this expression, gives Lx = -i(z d/dy - y 3 / 3 z ) . T h e right-hand side of this expression is equal t o -lx, w h i c h defines t h e s y m b o l l.x. T h u s u n d e r this c o o r d i n a t e r o t a t i o n lz w h i c h m e a n s t h a t w e are asking for t h e same physical q u a n t i t y b u t expressing it in a different c o o r d i n a t e system. On t h e o t h e r h a n d , if we rotate the operator lz, we u n d e r s t a n d this t o m e a n t h a t we are asking for a different physical q u a n t i t y . If w e ' ' r o t a t e lz " b y 9 0 ° a b o u t t h e y axis a n d s u b s e q u e n t l y o p e r a t e on a state, we will o b t a i n t h e e x p e c t a t i o n value of lx.

1.13

COMMENTS ON T H E ROTATION

GROUP

It was s h o w n b y Euler t h a t a n y t w o r o t a t i o n s can b e expressed as a single r o t a t i o n a b o u t a suitably chosen axis, from w h i c h it follows t h a t any series of r o t a t i o n s can be reduced t o a single r o t a t i o n a b o u t s o m e axis n , say. It is easy t o believe t h a t t h e r o t a t i o n o p e r a t o r for r o t a t i o n b y an angle 0 a b o u t an axis n is exp(/0n • J ) , w h i c h means t h a t t h e three p a r a m e t e r s we n e e d t o define a r o t a t i o n are n o w t h e angle 0 and t h e t w o polar angles of n . If n is equivalent t o x, y, or z, w e recover t h e simple r o t a t i o n operators. We will n o t have occasion t o use t h e o p e r a t o r exp(/0n • J ) , b u t t h e fact t h a t any t w o r o t a t i o n s are equivalent t o a n o t h e r r o t a t i o n m e a n s t h a t w e can write D(ap'y')D(a"p"y")

= D(oLPy)

(1.20)

1

14

The Rotation

Operator

for any choice of t h e t w o sets (afi'y) a n d (a"fi"y"). This is o n e of t h e properties o f a g r o u p , a n d t h e r o t a t i o n o p e r a t o r s d o in fact form a g r o u p : there is a unit e l e m e n t , Z>(000); an inverse, D~\a^y)\ associativity holds,D(l)[D(2)D(3)] = [D(l)D(2)]Z)(3); and closure is d e m o n s t r a t e d b y ( 1 . 2 0 ) . The r o t a t i o n g r o u p is usually called R3. The representations of R3 are infinite in n u m b e r a n d are labeled w h e r e t h e dimension of t h e r e p r e s e n t a t i o n is 2/ + 1 a n d / can t a k e o n all positive integer a n d half-integer values 0, j , 1, § , . . . . These facts should b e familiar t o t h e reader. In t h e n e x t c h a p t e r we will e x a m i n e these r e p r e s e n t a t i o n s m o r e closely b u t in passing we d r a w t h e reader's a t t e n t i o n t o a certain looseness in t h e preceding s e n t e n c e . The representations with / half-odd-integer are often t e r m e d two-valued representations of R3, although t h e y are in fact representations of a larger g r o u p U2. This p o i n t is enlarged o n in Section 2 . 6 .

n 1.14

COMMENTS ON LIE GROUPS

The g r o u p R 3 is an e x a m p l e of a continuous group, since t h e o p e r a t o r s of t h e group D(a$y) are functions of c o n t i n u o u s l y variable p a r a m e t e r s . Starting w i t h a definite choice of Euler angles we can arrive at any o t h e r set ( a , )3, 7 ) b y c o n t i n u o u s l y varying the angles. R3 is also of course an infinite group since t h e p a r a m e t e r s a, ]3, 7 can be chosen in an infinite n u m b e r of w a y s . It is because R3 is c o n t i n u o u s t h a t we could deal w i t h infinitesimal t r a n s f o r m a t i o n s , w h i c h have n o meaning for a finite g r o u p . A set of infinitesimal t r a n s f o r m a t i o n s Ot derived from a c o n t i n u o u s g r o u p can b e w r i t t e n in t h e general form

Ot=

l+ZStfyP/ /

w h e r e da; is an infinitesimal change in a relevant variable and t h e Pj are o p e r a t o r s k n o w n as generators [cf. Eq. ( 1 . 7 ) ] . Lie d e m o n s t r a t e d t h a t t h e Ot themselves form a group a n d t h a t t h e generators Pj of such Lie groups o b e y c o m m u t a t i o n relations of t h e general t y p e [P;,Pk]

= lcU,k,t)Pi

(1.21)

i.e., t h e c o m m u t a t o r s are linear c o m b i n a t i o n s of the o p e r a t o r s . The elements Pj of a Lie g r o u p and their c o m m u t a t i o n relationships define w h a t m a t h e m a t i c i a n s call an algebra, in this case a Lie algebra, a n d the c o m m u t a t i o n relations d e t e r m i n e t h e p r o p e r t i e s of t h e Lie g r o u p and t h e c o r r e s p o n d i n g group of finite o p e r a t i o n s . T h e Jx, Jy, and Jz of t h e infinitesimal r o t a t i o n o p e r a t o r s , 1 + 60,-//, satisfy t h e c o m m u t a t i o n relations ( 1 . 2 1 ) a n d therefore define a Lie algebra, from w h i c h on one h a n d t h e w h o l e structure of t h e t h e o r y of angular m o m e n t u m m a y b e derived, and on t h e o t h e r t h e c o m p l e t e p r o p e r t i e s of t h e g r o u p R3. This is b u t one example of the deep c o n n e c t i o n b e t w e e n s y m m e t r y a n d physical observables. O t h e r o p e r a t o r s are used t o generate c o n t i n u o u s groups in the advanced t h e o r y of a t o m i c s t r u c t u r e . This subject is outside o u r range b u t is dealt w i t h b y J u d d ( 1 9 6 3 ) .

1.15

1.15

Conventions

15

CONVENTIONS

T h e o p e r a t o r D(a$y) m e a n s different things t o different investigators. S o m e a u t h o r s define it t o operate o n functions a n d s o m e t o operate on c o o r d i n a t e s . T o a d d t o t h e confusion, some use it t o o p e r a t e o n functions defined in a fixed initial frame and some t o operate o n functions defined in t h e final frame, w h i c h is t h e c o n v e n t i o n we use. There are m a n y incorrect forms in t h e literature, even in s t a n d a r d t e x t s . The following a u t h o r s use D(a^y) t o give t h e effect on a function of rotating t h e coordinate system b y positive angles, for w h i c h t h e correct form of t h e o p e r a t o r is (1'.15): Rose ( 1 9 5 5 ) . E d m o n d s ( 1 9 5 7 ; 2 n d ed., 1 9 6 0 ) : As p o i n t e d o u t b y Wolf ( 1 9 6 9 ) E q . 4 . 1 . 8 has t h e e x p o n e n t i a l o p e r a t o r s in t h e w r o n g order. In 4 . 1 . 1 2 t h e order is correct, b u t this e q u a t i o n does n o t follow from t h e previous t e x t . E d m o n d s ( 1 9 6 0 , revised printing 1 9 6 8 ) : E q . 4 . 1 . 8 remains i n c o r r e c t ; E q . 4 . 1 . 1 2 is n o w c o r r e c t e d , w h i c h makes it inconsistent w i t h 4 . 1 . 8 . Wigner ( 1 9 5 9 ) : Those w h o read t h e original G e r m a n edition ( 1 9 3 1 ) should realize t h a t a left-handed c o o r d i n a t e system is used, w h i c h is u n c o m m o n ; t h e 1 9 5 9 English t e x t uses a right-handed system. Heine ( 1 9 6 0 ) . The following a u t h o r s write D(aj5y) in t h e form e x p ( - / a / z ) exp(-iPjy) e x p ( - r y / z) and use it t o r o t a t e functions b y t h e positive angles y, ]3, a n d a w i t h respect t o a fixed c o o r d i n a t e system [cf. E q . ( 1 . 1 7 ) a n d n o t e t h e different ordering of t h e a n g l e s ] : Rose ( 1 9 5 7 ) , Messiah ( 1 9 6 2 ) , T i n k h a m ( 1 9 6 4 ) . A very full discussion of t h e r o t a t i o n of spherical h a r m o n i c s is given in a review b y Steinborn and Ruedenberg (1973).

Chapter

2

The Wigner Rotation Matrices

2.1

THE ROTATION MATRICES

The angular m o m e n t u m eigenfunctions |/ra>have a p r o p e r t y of great significance f o r u s . The2j + l states\jm)offixed jspan an irreducible representation ofR3, the infinite rotation group (Wigner, 1 9 5 9 ; Heine, 1 9 6 0 ; T i n k h a m , 1 9 6 4 ) . P u t a n o t h e r w a y : u n d e r an arbitrary coordinate r o t a t i o n D(a@y) t h e state \jm) transforms i n t o a linear c o m b i n a t i o n of t h e c o m p l e t e set of 2 / + 1 states \jm) of the same /': /

D(aPy)\jm)

=

I

(2.1)

\jm') 9$J*ti)

m =-/

where ^JJm(°fiy)is

a

c o e f f i c i e n t d e p e n d i n g o n / , m9 a n d m a n d o n t h e angles a , j3, and

7. More fully we can write D(apy)(\ff\

| / f / - 1 > , . . . \ j m ) . . . I / , - / + 1>, | / - / »

m

= (!//>, lA / - 1 > , • • • l / ^ > . . . I/, - / + 1>, 1 / - / »

Notice t h e ordering of t h e indices m a n d m, a n d t h a t if instead of ( 2 . 1 ) w e write t h e states as a c o l u m n vector a n d t h e t r a n s f o r m a t i o n as D(apy)\fm)=

t

,

^^(a^7)l/^ >

(2.2)

m = -/

the m a t r i x ^mm'iP^Pi) is t h e transpose of t h e m a t r i x < ^ ^ m( a | 3 7 ) . We will from n o w o n use E q . ( 2 . 1 ) in defining t h e expansion coefficients a n d warn t h e reader t h a t 16

2.1

The Rotation

Matrices

17

confusion can arise if it is n o t clear w h i c h of t h e e q u a t i o n s ( 2 . 1 ) or ( 2 . 2 ) is being used. T h e p r o p e r t y defined in ( 2 . 1 ) is a familiar p r o p e r t y of t h e spherical h a r m o n i c s l Ym(0, |; 0 0 ) = 0 | 0 0 > , we see t h a t &°\a(ly)

is i n d e p e n d e n t of

t h e angles of r o t a t i o n a n d equal t o 1. It describes t h e trivial t r a n s f o r m a t i o n u n d e r coordinate harmonic

rotations

of

the

spherically s y m m e t r i c function

|00>. T h e

spherical

F 0 ° , w h i c h is t h e angular p a r t of t h e h y d r o g e n i c s orbitals is such a

function, a n d is obviously invariant u n d e r c o o r d i n a t e r o t a t i o n s : D(ofiy)Y0°

= 1 • Y0°

(for any a, j3, a n d 7 ) . If y o u sit on t h e n u c l e u s , an s orbital looks exactly t h e same whichever way y o u arrange y o u r c o o r d i n a t e system. Before we l o o k at t h e f o r m of t h e Wigner matrices for / > 0 we m u s t go over an i m p o r t a n t p o i n t c o n c e r n i n g phases.

2.2

QUESTIONS O F PHASE l

Consider t h e explicit form of t h e functions Y m . x

Yx

=-(3/87r)

V=(3/4T0

(x + z»/r

1 / 2

(Z//-)

=+(3/87r) These functions span

In m a n y standard t e x t s we find

1 / 2

(2.8)

1 / 2

(x-z»/r

i.e., t h e y form an invariant subspace u n d e r t h e o p e r a t i o n s

of R 3. N o w let us r o t a t e t h e c o o r d i n a t e system b y + 9 0 ° a b o u t y inducing t h e changes x -> z, y -> y, and z

-x.

Making t h e c o r r e s p o n d i n g s u b s t i t u t i o n s in ( 2 . 8 ) w e find r

1

r

1

I i -^il i -2l

Y0 ^

1 / 2

+

y

1

r

0

+il -i

U2 l

1/2

1 / 2

1 +

1

l

2- Yl~2- Y_l

1

r_1 -> + ^r

1 1

+2-

Fo

5^-i

1

so t h a t t h e t r a n s f o r m a t i o n m a t r i x is given b y / 1

i

1

D(0,90,0)(Y1 ,Yo,Y-1 )

1

= (Yl ,

Y 0\

Y.^

1

5

-2" h

0-1/2

1

0

2"

2

1 2/

_ 2- l / 2

5 1 2 /

2.2

Questions

of Phase

19

The m a t r i x is identical t o t h a t o b t a i n e d b y p u t t i n g (afiy) =(0, 90,0) in t h e explicit form of [Eq. (2.12)]. However, if we h a d t a k e n positive phases for t h e entire set (2.8), i.e., changed t h e sign of > V , t h e r o t a t i o n m a t r i x w o u l d be different:

0(0,90,0) =

It is clear t h a t for t h e states \\m) at least, t h e form of t h e Wigner m a t r i x d e p e n d s on l the relative phases of t h e c o m p o n e n t s of t h e set Y m . More generally, t h e sets \jm) m u s t have k n o w n relative phases if w e are t o c o n s t r u c t t h e matrices $ £ ) m. T h e r e is n o universally accepted convention for phases, b u t in m o s t of t h e chemical literature, and a good part of t h e physical, t h e c o n v e n t i o n used is t h a t of C o n d o n a n d S h o r t l e y (1951) w h i c h we will use a n d on which we n o w c o m m e n t . If t h e reader is puzzled b y t h e a p p a r e n t l y arbitrary m a n n e r in w h i c h phases are defined, it m a y help t o p o i n t o u t t h a t t h e r o o t of t h e trouble lies way back in t h e c o m m u t a t i o n relationships [fx ,fy]=

ijz,

[jy ,Jz]

= ijx,

[j'z,Jx ] = IJy

w h i c h are t h e basis o f t h e t h e o r y o f angular m o m e n t u m . T h e c o m m u t a t o r s eventually lead t o t h e e q u a t i o n s fz\jm)

2

= m\jm),

J \jm)

= /(/

+

l)\jm) 1/2

UI im) = e x p ( / 0 ) [ / ( / + 1) - m(m + 1 ) ] 1 / , m + 1> jjfm>

1/2

= e x p ( - / 0 ) [ / ( / ' + 1) - m(m-1)] 1/,

m-l)

where 0 is arbitrary: T h e c o m m u t a t i o n relationships d o n o t fix t h e m a g n i t u d e of 0. The a c t i o n of, say, /+ on a state \jm) gives a n o t h e r state \j,m + l) multiplied b y a numerical factor, b u t w i t h c o m p l e t e l y arbitrary phase with respect t o t h e first s t a t e . 2 P u t differently, t h e set of 2/ + 1 states \jm) has definite eigenvalues for j a n d j z b u t n o phase relations are implied b y t h e symbols / a n d m. C o n d o n a n d Shortley take 0 = 0, which fixes t h e relative phases of t h e 2/+ 1 states \jm) of different m. O t h e r choices of 0 m a y be more or less convenient b u t are equally acceptable, m a t h e matically or physically. There is n o physical significance a t t a c h e d t o t h e phase of a wave function a n d t h e calculated value of a physical observable c a n n o t d e p e n d on t h e phase c o n v e n t i o n . However, it is i m p o r t a n t t o be consistent—we m u s t choose a c o n v e n t i o n a n d stick t o it. O n e consequence of C o n d o n a n d S h o r t l e y ' s choice of phase is t h a t / m / ^-m (^0) = (-l) rm (0,0)* (2.9) x

which is satisfied, for e x a m p l e , b y Yx

and F ^ i n

(2.8). More generally,

m

(2.10)

\j -m) = (-\) \jm)* The spherical h a r m o n i c s defined as above are called standard this b o o k , a n d t h e Wigner matrices are consistent w i t h t h e m .

spherical

harmonics

in

20

2

The Wigner Rota tion Ma trices

In C h a p t e r 16 we will w o r k w i t h sets of functions o b t a i n e d b y performing u n i t a r y ( 1) b u t t h e explicit t r a n s f o r m a t i o n s o n t h e sets | l m > . T h e transformed sets still span ® l form of t h e r o t a t i o n matrices will b e different. T h u s for t h e set Ym a coordinate r o t a t i o n of 9 0 ° a b o u t t h e z axis gives t h e r o t a t i o n m a t r i x

l

The set (x, y, z) is related t o Ym

by the unitary matrix

.1/2

a n d for this set a r o t a t i o n of 9 0 ° a b o u t z is described b y t h e m a t r i x

b u t n o t in t h e s t a n d a r d form given b y ^ ^ ? m . It

We say t h a t t h e set x, y, z spans

is a familiar result of m a t r i x algebra t h a t a u n i t a r y t r a n s f o r m a t i o n ^ o f t h e basis set for a given r e p r e s e n t a t i o n induces a corresponding change in t h e matrices 0t of t h e r e p r e s e n t a t i o n ; t h e original a n d final matrices are said t o b e equivalent, a n d are related x

b y & -+Sf0lSf~ .

2.3

T h e reader can check t h a t for o u r e x a m p l e Jt = ^ ^ ( 9 0 , 0 , O ) ^

THE FORMS OF ®

( 1 2)

'

AND ®

- 1

.

( 1)

We n o w c o m e d o w n t o e a r t h a n d consider t h e explicit forms of t h e r o t a t i o n matrices for j = \ a n d 1. T h e elements of OfiM

are given b y

^^W7)=0

exp(-z'7r) • 0 \ (a\

_ /-l

exp(-i7r)-l/\jJ/"\

0 \ / a \ _ l-a

0

\

~ \-0 /

The spinors a a n d j3 suffer a well-known change of phase w h e n t h e c o o r d i n a t e system is r o t a t e d by 2tt a b o u t t h e z axis. A r o t a t i o n of 4tt leaves t h e phases u n c h a n g e d , justifying our earlier warning t h a t c o o r d i n a t e r o t a t i o n s m u s t b e t r e a t e d as o p e r a t i o n s . Notice also t h a t r o t a t i o n s of +7r a n d -it a b o u t a given axis are n o t equivalent, in t h e =

sense t h a t , e.g.,Z)(7r00)lH> *'l4£>. butZ>(-7r00)lH> - - * l l i > . Using t h e n o m e n c l a t u r e \m) w e can write t h e s t a n d a r d basis states spanning 9)^ |1>, |0>, a n d | - 1 > . In this basis set has t h e form

|0>

l+l> l+l>

I0> -1>

exp0*7>7(l + cosj3) expO'7) -exp(/a)2"

1 2/

sin /3

e x p ( - / 7 > | ( l - cos 0) exp(za)

exp(/7)2"

|-1>

1 ?/

sin/3

e x p ( / 7 > | ( l - cos0) e x p ( - / a )

cos|3

exp(-/a)2"

-exp(/7)2"

as

1 2/

1 2/

sin |3

e x p ( - / 7 > ^ ( l + cos /?) e x p ( - / a )

sinj3

(2.12)

As a simple e x a m p l e of t h e use of this m a t r i x we consider t h e effect of a c o o r d i n a t e l r o t a t i o n of —90° a b o u t t h e y axis on a pz orbital, Y0f(r). T h e system before a n d after r o t a t i o n is s h o w n in Fig. 7. In t h e n e w c o o r d i n a t e system t h e original function n o w l o o k s like a px orbital. L e t us s h o w t h a t this is t h e result we get b y using t h e l m a t r i x @( \apy). T h e angles a , j3, a n d y occurring in the Wigner matrices are of course those b y which we r o t a t e t h e c o o r d i n a t e system. In our case w e have o Rz(Q°)Ry(-90 )Rz(0°l giving a = y = 0° a n d (3 = - 9 0 ° . Using these angles in ( 2 . 1 2 ) we o b t a i n the corresponding r o t a t i o n m a t r i x : 4 ^ ( 0 , - 9 0 , 0 ) ^ 1

2, 1

21 /2

xn

h

0

2-

9-1/2

1

1 /2

)

(2.13)

2

22

FIGURE

The Wigner Rotation

Matrices

7

N o w formally w e have l

£

D{a&*,)pz=DWi)Yo m=f{f)

Ym^{\m'\D{a&i)\\Q)

m'=+l

m'=+l I

I

1

)

(

I

)

= / ( r ) [ r , M o ( « U 7 ) + r 0 ^ o o ( ^ 7 ) + r-i'^-VoCa^)] w h i c h , using ( 2 . 1 3 ) ,

=/(r>[-iV • =/(r)(-2=-2-

1 / 2

2 - 1 / 2

I / 2

[(-2"

+ iV • o + r., 1

• 2-

1 / 2

]

I

)(r1 -y.1 ) 1 / 2

1

1/2

) f e +ipy)-2- (Px

-iPy)]

=

Px

Expressing this in w o r d s , we can say t h a t t h e effect of a c o o r d i n a t e rotationD(ocfiy) o n t h e function pz is t o change it i n t o t h e function px in t h e n e w c o o r d i n a t e system. N o t e t h e i m p o r t a n c e , stressed above, of maintaining o u r sign conventions in these l m a n i p u l a t i o n s . T h u s w e m u s t n o t forget t h e m i n u s sign in t h e i d e n t i t y f(r)Yi = l/2 —2~ (px +ipy). Of course we arrive at t h e same final relationship of c o o r d i n a t e axes t o function if we r o t a t e t h e pz orbital b y + 9 0 ° a b o u t t h e y axis a n d leave t h e axes alone. 2.4

PROPERTIES O F T H E ROTATION MATRICES

Having seen t h e explicit form of t w o of t h e r o t a t i o n matrices, w e n o w consider t h e ; p r o p e r t i e s of t h e general m a t r i x ^ ^ m ( a 0 7 ) .

2.4

Properties (i)

23

of the Rota tion Ma trices

If Sn is a Hermitian o p e r a t o r , t h e n t h e o p e r a t o r C- exp(iSn)

is t h e o p e r a t o r exp(JSa)

exp(iSb)

c o m m u t e . It follows t h a t D(etfiy) e x p ( / a / z) a n d t h e o p e r a t o r s j

z

is u n i t a r y since it h a s t h e form e x p ( r y / z)

and j

y

1

= ^

m

Sn

exp(ifijy)

are H e r m i t i a n . T h e Wigner r o t a t i o n matrices are

therefore u n i t a r y matrices and it follows t h a t t h e inverse of WflmWy)]-

is u n i t a r y a n d so

. . . w h e t h e r or n o t t h e o p e r a t o r s

exp(iSc)

( - 7 - ^ - a ) =

t ^ w( « P 7 ) ]

equals t h e adjoint t

/ )

= ^ w w' ( « j 8 7 ) *

(2.14)

w h e r e t h e first equality follows from ( 1 . 1 6 ) . (ii)

T h e o r t h o g o n a l i t y c o n d i t i o n s w h i c h apply t o t h e rows a n d c o l u m n s of a

u n i t a r y m a t r i x a p p l y t o t h e matrices @(aPy): I £ ^ U « < W $ m( a / } 7 ) = S m


0 7 )

< £ a( « 0 7 ) d n = M P / . + 0 ]

h

h h

where d£l =

da

dp> sin [

(2.17)

dy

This states t h a t t h e elements of t h e r o t a t i o n m a t r i c e s , considered as functions of a , j3, a n d 7 , are o r t h o g o n a l t o each o t h e r in t h e d o m a i n covered b y the allowed range of t h e Euler angles. T h e reader should realize t h e close analogy b e t w e e n this result a n d the great o r t h o g o n a l i t y t h e o r e m which occurs in t h e t h e o r y of finite g r o u p s : I

^u%SR)^u%SR)-

W//)8

5 ,

M l U 2m

mj

5,,

(2.18)

This relationship can b e seen "visually" from Fig. 8 in which w e show t h e matrices of t w o r e p r e s e n t a t i o n s / and / of a finite g r o u p . T h e s u m over R i n ( 2 . 1 8 ) is just t h e s u m of t h e p r o d u c t s of m a t r i x e l e m e n t s c o n n e c t e d b y t h e a r r o w s , r e m e m b e r i n g t o take t h e c o m p l e x conjugate of t h e e l e m e n t s of T ^ . T h e t h e o r e m says t h a t this s u m always vanishes unless t h e p r o d u c t s are b e t w e e n a given m a t r i x element and itself. N o w t h e set of all c o o r d i n a t e r o t a t i o n s a b o u t a p o i n t forms t h e g r o u p The sets of 2 / + 1 functions \jm) form basis functions for irreducible representations of this g r o u p . T h u s t h e five functions for / = 2 form t h e basis for a five-dimensional r e p r e s e n t a t i o n o f R3. Each e l e m e n t R of t h e group ^ 3 will b e represented in this representation b y a m a t r i x giving t h e t r a n s f o r m a t i o n of t h e set |22), |21>, |20>, |2 —1>, |2 —2) u n d e r t h e o p e r a t i o n of t h e e l e m e n t R. But t h e e l e m e n t s of R3 are t h e set of all c o o r d i n a t e r o t a t i o n s a b o u t a p o i n t , a n d are just t h e o p e r a t o r s D(afiy). The matrix

24

2

FIGURE

representing a certain D(aPy)

Matrices

8

in this set of functions is m e r e l y t h e Wigner r o t a t i o n

2

matrix The

The Wigner Rotation

& \apy). set

of

matrices

J

for all a , 0,

& \aPy)

r e p r e s e n t a t i o n of d i m e n s i o n ( 2 / + l )

2

of t h e g r o u p R3.

and y

forms

an

irreducible

From now on the symbol

w i t h o u t t h e Euler angles, will b e used t o label t h e ( 2 / + l ) - d i m e n s i o n a l r e p r e s e n t a t i o n ofR3. We might a n t i c i p a t e t h a t there will b e an o r t h o g o n a l i t y relationship b e t w e e n t h e e l e m e n t s of t h e matrices &i\aPy)

of t h e t y p e found for finite g r o u p s . T h e r e is, a n d

it is c o n t a i n e d in E q . ( 2 . 1 7 ) . T h e great o r t h o g o n a l i t y t h e o r e m , as s t a t e d in E q . ( 2 . 1 8 ) , is applicable only t o finite g r o u p s , b u t t h e g r o u p R3 is an infinite, c o n t i n u o u s g r o u p . In E q . ( 2 . 1 7 ) t h e place o f t h e d i s c o n t i n u o u s label R in E q . ( 2 . 1 8 ) is t a k e n b y t h e values of t h e t h r e e Euler angles a , j3, a n d 7. Since these are c o n t i n u o u s l y variable w e have t o integrate r a t h e r t h a n s u m . T h e t h e o r y of c o n t i n u o u s groups is n o t quite as simple as this discussion might i m p l y , b u t at least t h e reader should b e aware of t h e group-theoretical u n d e r t o n e s of E q . ( 2 . 1 7 ) . A simple illustration of o r t h o g o n a l i t y is provided b y t h e e l e m e n t s

&tfl

l9

of

Eq.(2.12). (

i

)

| ^ o r i( a j 3 7 ) M o ( a i 3 7 ) ^ = J V

1

/2

= -£

sin/3exp(/a)exp(/7)2"

sm^dn

3

sin j3tfj3

exp(iot)do(. Jo

1 2/

Jo

exp(-/7)^7 Jo

= - i [Ox!xO]=0 (iv)

T w o useful identities are ®mo(°07) (

=

^ Om(, t h e c o m p o n e n t s of u in t h e n e w c o o r d i n a t e s y s t e m .

FIGURE

9

2.6

Another

Look

at9

T h e matrices

{ x ,) 2

27

are n o t designed t o handle Cartesian vectors, so w e p e r f o r m a

u n i t a r y t r a n s f o r m a t i o n t o t h e suitable set of c o m p o n e n t s 7V(r) = - 2 -

1 / 2

(W

c;

i

+ iuy),

T0(t)

= uz,

T.x\x)

v

= 2- \ux-iuy)

(2.27)

C o m p a r i s o n with ( 2 . 8 ) will show w h y we c h o o s e these c o m b i n a t i o n s . Using ( 2 . 2 6 ) w e can write for t h e z c o m p o n e n t of u , for e x a m p l e , l

uz = 7 V (r)9hi (0/50) + T0 (t)9®W0)

l

+ T.x

(r^-iCOjSO)

(2.28)

w h i c h , using ( 2 . 1 2 ) and ( 2 . 2 7 ) , leads t o t h e readily verifiable result = ux s i n 0 + uz cos]3

uz

T r a n s f o r m a t i o n s of H a m i l t o n i a n s are t r e a t e d b y B u c k m a s t e r et al. ( 1 9 7 2 ) . F o r o t h e r applications of t h e r o t a t i o n matrices see McConnell et al. ( 1 9 6 0 ) a n d R e u v e n i et al. ( 1 9 7 3 ) .

n 2.6

ANOTHER LOOK AT

^

T h e following n o t e s on idea of double groups.

( 1 /) 2

e

We can write t h e m a t r i x 9)^^ cos 0 / 2 = a a n d e x p [ - z ( a - y)/2] expressed as

^

i n ta e rn d e d t o i n t r o d u c e , in a simple m a n n e r , t h e

in t w o different w a y s . If w e define e x p [ z ( a + y)/2] ( 1 /) 2 sin 0 / 2 = b, we see from ( 2 . 1 1 ) t h a t 5 > can b e

)

s

C

.

I)

-

2 9)

In t h e language of t h e m a t h e m a t i c i a n a a n d b are t h e Cayley-Klein p a r a m e t e r s of t h e r o t a t i o n D(a$y). Alternatively, if we express D(a(3y) in t h e form e x p ( / 0 n • J ) a n d define a0 = cos 6/2 a n d a = s i n ( 0 / 2 ) n , w e find

^^W)^!

\-iai

— a2

) a0 — ia3 I

(2.30)

T h e equivalence of ( 2 . 2 9 ) a n d ( 2 . 3 0 ) can easily be c h e c k e d for a r o t a t i o n a b o u t t h e z axis, for w h i c h n = ( 0 0 1 ) . a0 a n d a are k n o w n as t h e h o m o g e n e o u s Euler p a r a m e t e r s . l In either of these forms 9^ ^(a^y) is seen to be a u n i t a r y m a t r i x w i t h d e t e r m i n a n t 1. It is a c o n s e q u e n c e of m a t r i x m u l t i p l i c a t i o n t h a t t h e set of all u n i t a r y u n i m o d u l a r ( t h a t m e a n s det = 1) n x n matrices forms a g r o u p . T h e infinite set of 2 x 2 matrices of this k i n d defines t h e g r o u p U2. A n y physical r o t a t i o n can b e defined b y its Euler angles, w h i c h in t u r n can be used 1 / 2 via ( 2 . 2 9 ) or ( 2 . 3 0 ) t o build a c o r r e s p o n d i n g 2 x 2 m a t r i x belonging t o 3 ^ ) , i.e., t o U2. T h u s every physical r o t a t i o n D(a@y) appears t o b e associated with a u n i q u e m a t r i x of U2.

( 2

28

2

The Wigner Rota tion Ma trices

N o w consider a r o t a t i o n of 0 a b o u t t h e z axis, for w h i c h /exp(/0/2)

2

^/ )(0OO)=(

\

If we perform the physically

0

,

\

)

exp(-/0/2)/

0

indistinguishable r o t a t i o n 0 + 27r, we find

2

^ / > ( 0 + 2tt, 0 , 0 ) = / " \

e

x

p

0

O

)0

/

2

)

° = _ -exp(-/0/2)/

( i » ( 0o o )

S

1 2

But — ^ / ^(0OO) is a m a t r i x t h a t is u n o b t a i n a b l e if we limit ourselves t o " n o r m a l " Euler angle r o t a t i o n s 0 < a < 27T, 0 < ]3 < 7r, 0 < y < 27T. This result can be generalized. For every r o t a t i o n D( | / 2 m 2 ) a n d e a c h function will b e a linear c o m b i n a t i o n of these p r o d u c t states, i.e., l

l

C^i

Z

\lm) =

mlm2

If we can find all t h e coefficients generalize

our

nomenclature

, we will have solved o u r p r o b l e m . We can m

slightly

t o give t h e states \jm)

\]\tni)\j2m2) J

(3.2)

\l2lm m1 l)\l2m2)

to

consider

the

coupling

of

the

states

a n d write t h e c o r r e s p o n d i n g coefficients as

These coefficients are usually called vector-coupling (VC) coefficients or

C m[m2m-

C l e b s c h - G o r d a n coefficients, a n d this c h a p t e r is d e v o t e d t o their p r o p e r t i e s . T h e origin of t h e label " v e c t o r c o u p l i n g " lies in t h e fact t h a t an angular m o m e n t u m m a y be regarded as a vector a n d t h e coupling of angular m o m e n t a , as exemplified in Eq. ( 3 . 2 ) , can b e regarded as a process of v e c t o r a d d i t i o n . (Recall t h e vector m o d e l of t h e a t o m . ) We p r o d u c e , o u t of a h a t , t h e coefficients for o u r e x a m p l e , a n d list all t h e states arising from t h e c o u p l i n g of t w o p e l e c t r o n s . We use t h e subscript 1 or = 3 -

( | 1 1 > 1| 1 - 1 > 2- | 1 0 > 1| 1 0 > 2 + |1 - D i l l D j )

1=1

111) = 2 -

( | l l > 1| 1 0 > 2- | 1 0 ) 1| l l > 2)

( | 1 1 > 1| 1 - 1 > 2 - | 1 - l > i l l l > 2 )

1 / 2

11 - 1 > = 2 1= 2

1 / 2

1 / 2

|10> = 2 -

( | 1 0 > 1| 1 - 1 > 2 - | 1

\22)

= | 1 1 > 1| 1 1 > 2

|21>

=2-

|20>

(3.3a)

(3.3b)

- O i l l 0 > 2)

1 / 2

=6-

|2-1> = 2-

( | l l > 1 | 1 0 > 2 + | 1 0 > 1| l l > 2)

1 / 2

( | l l > 1 | l - 1 > 2+ 2 | 1 0 > 1| 1 0 > 2+ | 1 - D i l l l > 2 )

1 / 2

( | 1 0 > 1| 1 - D 2 + I I

| 2 - 2 > = |1 - D i l l

(3.3c)

- D i l l 0 > 2)

-1>2

Later we will indicate w h e r e t o find VC coefficients. F o r t h e p r e s e n t we use these results t o illustrate s o m e general p o i n t s . (a)

If t h e t r a n s f o r m a t i o n is w r i t t e n in t h e m a t r i x form ( 3 . 4 ) , it is easy t o check

t h a t t h e r o w s a n d c o l u m n s are o r t h o n o r m a l vectors. T h e t r a n s f o r m a t i o n c o n n e c t i n g orthonormal

sets

is, in

general,

unitary

but

in

this

case t h e m a t r i x

elements

(coefficients) are all real a n d t h e m a t r i x is o r t h o g o n a l . N o t e t h a t t h e s u b m a t r i c e s , each of w h i c h c o r r e s p o n d s t o a fixed value of mx + m2,

are also o r t h o g o n a l .

32

3

The Coupling

of Two Angular

Momenta

3.2

The Vector-Coupling (b)

Coefficients

33

R e t u r n i n g t o t h e explicit forms in ( 3 . 3 ) , it is informative t o specialize t o t w o

equivalent p e l e c t r o n s , say t w o 2p e l e c t r o n s , a n d a p p l y t h e p e r m u t a t i o n o p e r a t o r P 1 2. T h e states belonging t o / = 0 a n d / = 2 are u n c h a n g e d , b u t for / = 1 t h e states change sign. The t o t a l wave f u n c t i o n m u s t be a n t i s y m m e t r i c , w h i c h m e a n s t h a t for the / = 0 and 2 states we n e e d an a n t i s y m m e t r i c t w o - e l e c t r o n spin function, a singlet

spin

function, a n d for / = 1 we require t h e s y m m e t r i c triplet f u n c t i o n . So we are led t o l

3

p r e d i c t , c o r r e c t l y , t h e t e r m s S, (c)

1

P, D for t h e configuration

2

p.

The t r a n s f o r m a t i o n c a n be inverted, allowing us t o express any of t h e simple

p r o d u c t states as a linear c o m b i n a t i o n of t h e c o u p l e d states. If we write ( 3 . 4 ) as s/ =M3,

0b=JT d. But in o u r x

we can invert t o o b t a i n

c a s e j ^ i s real a n d a real u n i t a r y

m a t r i x is an o r t h o g o n a l m a t r i x for w h i c h t h e inverse equals t h e t r a n s p o s e . T h u s it is 1

trivial t o c o n s t r u c t ^ " , a n d t h e reader m a y care t o verify t h a t , for e x a m p l e , | 1 0 > a| 1 - l > 2 = 2 (d)

1 / 2

| 2 - l > + 2-

1 / 2

|l

-1>

T h e o r d e r of t h e basis functions has b e e n deliberately chosen t o give a

t r a n s f o r m a t i o n m a t r i x divided i n t o b l o c k s . We can see t h a t e a c h b l o c k c o n n e c t s states of identical t o t a l m.

( T h u s t h e state

| 1 1 > 1| 1 1 > 2. )

|22> can only c o n t a i n

This

c o r r e s p o n d s t o t h e fact t h a t m a g n e t i c q u a n t u m n u m b e r s are a d d e d algebraically in adding angular m o m e n t a , a n d a p r o d u c t state \lxm\) c o u p l e d state \lm)

w i t h m = mi +m2.

\l2m2)

can only c o n t r i b u t e t o a

The t o t a l angular m o m e n t a lx and l2

add

vectorially a n d d o n o t provide a m e a n s of " b l o c k i n g " t h e t r a n s f o r m a t i o n m a t r i x . As

another,

coefficients

for

^(i/2)0^(i/2)

very two =

^

(

familiar, 1

spins )

+^

(

0

example

of ,

\.

)

we

The

ad n

write

coupled

ea r

te h

d o w n t h e m a t r i x of states

span

the

direct

coupling product

well-known singlet and triplet

spin

functions for t w o e l e c t r o n s : Iolo\ i i

22"

\o o

i n 1 2 /

2" -2-

1 2 /

I /2

0

0a

0

ap

(3.5)

o

w h e r e we have used t h e c o m m o n s y m b o l s for t h e states \ ^ \ ) a n d \ \ — \ ) .

3.2

THE VECTOR-COUPLING

COEFFICIENTS

Having l o o k e d at t w o simple e x a m p l e s of t h e c o u p l i n g of angular m o m e n t a we n o w formulate t h e p r o b l e m in general t e r m s . We will d o this in t w o w a y s . (i) \j\mi),

Consider a set of (2j\ + 1) states, a typical m e m b e r of the set being labeled and a n o t h e r set of ( 2 / 2 + 1) states labeled \j2m2).

( 2 / 2 + 1) p r o d u c t states \jimi)\j2m2)

We f o r m t h e (2j\ + 1)

a n d from these w e c o n s t r u c t t h e (2j\ + 1)

34

3

( 2 / 2 + 1) states \jm)

The Coupling

w h i c h are eigenfunctions of

2

\JiJ2Jtn)is

states

2

j = (j* + j 2 )

T o indicate t h e origin of t h e c o u p l e d states \jm) product

of Two Angular a n d j \ = (jlz

+

we will write t h e m \jij2jm).

will be w r i t t e n in t h e form

\j\m\)\j2m2)

Momenta

a linear c o m b i n a t i o n of t h e states \i\mx]2m2)

T h e state

\i\mxj2m2).

:

Z C / m ' m l/i^i/2^2> m; m 2

\j\J2Jrn)=

/2z). The

(3.6)

The d o u b l e s u m m a t i o n over mx a n d m2 ensures t h a t we cover all t h e possible p r o d u c t states. The coefficients C of course d e p e n d on all t h e q u a n t u m n u m b e r s as indicated b y t h e subscripts a n d superscripts. N o w t h e angular m o m e n t u m eigenfunctions form an o r t h o n o r m a l set ( 2 . 3 ) , a n d w e can t h u s find t h e coefficients simply b y multiplying b o t h sides of E q . ( 3 . 6 ) b y \j\mxj2m2) C

a n d integrating. We o b t a i n = < |. (ii)

A n o t h e r w a y of a p p r o a c h i n g t h e VC coefficients is t h e following. We saw t h a t

t h e direct p r o d u c t of t w o r e p r e s e n t a t i o n s of R3 can b e expressed as a c o m b i n a t i o n of representations

[Eq. ( 3 . 1 ) ] . N o w t h e set of p r o d u c t functions \j\mx)\j2m2)

form a basis for t h e reducible

r e p r e s e n t a t i o n 3^^®^^)

consisting of matrices of d i m e n s i o n [(2j\

must

and this r e p r e s e n t a t i o n ,

2

+ 1) (2/* 2 + 1 ) ] , can b e b r o u g h t i n t o b l o c k

form b y a u n i t a r y t r a n s f o r m a t i o n of t h e basis functions. The required t r a n s f o r m a t i o n S is of course t h a t w h i c h gives linear c o m b i n a t i o n s of t h e p r o d u c t states t r a n s f o r m i n g 1

as s t a n d a r d basis functions for t h e distinct r e p r e s e n t a t i o n s appearing

in^ ^®

T h e m a t r i x Sf is simply m a d e u p of t h e relevant VC coefficients a n d t h e " b l o c k e d " l

matrices of t h e r e d u c e d r e p r e s e n t a t i o n will have t h e general form £f~ [ i o

110)

100)

11 -1>

0

0

0

ei

0

0

e

0

0

0 te

0 e'

The e l e m e n t s of t h e m a t r i x all have unit m a g n i t u d e and are identical for all elements t en s ae m °%jm,jm °f J- T h e m a t r i x has b e e n deliberately c o n s t r u c t e d t o satisfy the following c o n d i t i o n s : (a) When °U multiplies $f9 t h e m a t r i x of t h e V C coefficients, it gives a m a t r i x QlSf which differs only in a change of phase for t h e e l e m e n t s of Sf. (b)

The

unitary

transformation

(^^)

_ 1

[^

( 1 / 2 )

®^

( 1 / 2 )

]^reduces

t h e direct

p r o d u c t m a t r i x t o b l o c k form identical t o t h a t o b t a i n e d b y t h e action of £f. It requires little w o r k t o check these s t a t e m e n t s . What d o t h e y imply? T h a t t h e m a t r i x Sf of V C coefficients is n o t t h e only m a t r i x t h a t can be used t o reduce t h e direct product matrix ® ^ 0 / 2 ) m \ n o t h e r w o r d s , it appears t h a t the r e q u i r e m e n t t h a t t h e m a t r i x of coefficients reduces t h e direct p r o d u c t m a t r i x is n o t sufficient t o fix u n a m b i g u o u s l y t h e values of t h e V C coefficients. However, from (a) we see t h a t t h e coefficients are fixed within a phase factor. We will n o t e n t e r i n t o w h y and h o w t h e c o m m o n l y used phases have b e e n c h o s e n . F o r o u r p u r p o s e s it suffices t o realize t h a t there is a certain freedom of choice in defining t h e coefficients, w h i c h in the final analysis arises from t h e physical irrelevance of t h e phase of a state. The phase c o n v e n t i o n in almost universal use is s u c h t h a t all V C coefficients are real, a l t h o u g h the restriction t o reality is n o t sufficient t o fix phases. Those interested in t h e m i n i m u m necessary restrictions can consult E d m o n d s ( 1 9 6 8 ) o r Wigner ( 1 9 5 9 ) . T h e coupling coefficients used in this b o o k are t h o s e in a p p a r e n t l y universal use.

3.4

THE EVALUATION AND PROPERTIES OF THE VC COEFFICIENTS

In accordance w i t h o u r policy of occasionally sidestepping p r o o f s we will merely state t h e p r o p e r t i e s t h a t are of relevance t o us. (i) A coefficient vanishes unless mi +m2 -m and unless j\ +j2 >/> Iji-jilThe second c o n d i t i o n , w h i c h is often called t h e triangle c o n d i t i o n and w r i t t e n

3.4

The Evaluation

and Properties

of the VC

Coefficients

37

A ( / i / 2 / ) , can be regarded as a c o n s e q u e n c e , or a proof, of the vector addition rule for angular m o m e n t u m . (ii) The coefficients are real a n d form an o r t h o g o n a l m a t r i x . The definition of a scalar p r o d u c t is such t h a t if (a\b)is real, t h e n it is equal t o (b\a). This m e a n s t h a t . The p o i n t is t h a t w e only need tabulate one of the t w o coefficients since t h e s y m m e t r y p r o p e r t y allows us t o evaluate t h e o t h e r . Relationships of this t y p e are n o t particularly easy t o r e m e m b e r , b u t Wigner h a s devised a s y m b o l , closely related t o the V C coefficients, t h a t has very simple symmetry properties. Wigner's " 3 - / s y m b o l " is defined b y (

1

2 7

m2

\mi

3 7

/ l i

2

m

)=(-l) - " - K2/3 + ir

1 / 2

m3

< / i ^ i / 2 m 2i / 1/ 2/ 3 - m 3>

(3.14)

As an example

(!

J

^=(-i)

, / 2 , / 2

-

1 2

- ° ( 2 - i + i)- ' = 6-

1 /2

The 3-/ s y m b o l has easily r e m e m b e r e d s y m m e t r y p r o p e r t i e s : (i)

A n even p e r m u t a t i o n of c o l u m n s leaves t h e value of t h e s y m b o l u n c h a n g e d :

lh

h

/,\//

\m{

m2

m31

a

(ii) + A+ n o d d p e r m u t a t i o n (_iyi /2 /3:

\m2

m2

m3I

/, \ //,

/,

m3

mxJ

m\

\m3

u\

( 3 1 5 )

m2j

of c o l u m n s multiplies t h e value of t h e s y m b o l b y

\ _ lii \mx

/,

\m2

ji

is \ _ Ih

h

h \ _ lis

h

h

mi

m3J

m3

m2

m2

mi

\mx

\m3

(3.16) There are six possible orderings of t h e s y m b o l s ] \ 9j 2 > a n d j3 in the 3-/ s y m b o l , a n d t h e value of all six s y m b o l s can b e o b t a i n e d from t h a t of a n y o n e s y m b o l b y using rules (i) and (ii). F o r e x a m p l e , k n o w i n g t h a t

3.6

Evaluation

of the 3-j Symbols /

39

1

2 3 \

(-.

. n

o

2

/

3

" "

s

)

w e can d e d u c e t h a t /

ll

1 2 3\

\-l

0 1/

A /

3

\01- 1 /

(iii)

l

h

2

1 \ _ /

2

1

\1 - 1 0 / \ 0 - 1

3

\ _ /

1 3

l) \ - l

2

\ _ /

3

Oj

1

2

l

\

(2/35)

1 2/

\1 0 - 1 /

M .(_!)/,•/,•/./ M h

h

\mi

3

m2

mJ

-m2

3

(3.17)

-m J 3

It follows from this t h a t a 3-j s y m b o l w i t h all m values e q u a l t o z e r o vanishes unless Ji

+

+

7*2 7*3 is even. We will use this fact later.

T h e o r t h o g o n a l i t y relationship ( 3 . 1 1 ) of t h e V C coefficients implies t h a t

£

//i

72 7 y/i

^wijl «tj

m

2

72 7 j

= ( 2 / + 1 )

-i

5 / 7 5 m

,

m

A ( / i /)2 /

a

i ) 8

m/\mi m ml 2

A further s u m m a t i o n over t h e i n d e x m will be of use t o us later:

E

(

A

H

mlm2m\mi

m2

(3.19) ml

w h i c h follows i m m e d i a t e l y from ( 3 . 1 8 ) since t h e r e are 2 / + 1 values of m. T h e practical advantages of t h e 3-j s y m b o l are evident from t h e simplicity of these relationships. N o t i c e from t h e definition of t h e 3-j s y m b o l [Eq. ( 3 . 1 4 ) ] t h a t since t h e VC

coefficient

m i + m

2

+ m

3

vanishes unless mi + m2 = —m3,

t h e 3-j s y m b o l vanishes unless

= 0.

T h e 3-j s y m b o l vanishes unless mi+m2

(3.20)

+ m3 = 0

The c o n d i t i o n on t h e / s is t h e same as for t h e V C coefficients, i.e.,

3.6

A(jij2j3).

E V A L U A T I O N O F T H E 3-/ S Y M B O L S Algebraic formulas for a large n u m b e r of t y p e s of 3-j coefficients are given b y

E d m o n d s ( 1 9 6 8 ) . S y m b o l s in frequent use are

I

\m

1

/

m

J = (-l) '- m[(2/+l)(/ + l)/]-

-m 0 /

1/2

(3.21)

3

40

7i

12

H\

0

0

0 J

= (-D

The Coupling

of Two A ngular Momen ta

//2

(3.22) w h e r e J=j\

+]\

This s y m b o l vanishes i f /

is o d d , a result t h a t follows from

E q . ( 3 . 1 7 ) . In passing w e n o t e t h a t any s y m b o l vanishes if it has t w o identical c o l u m n s a n d ;'i + y 2 + h is o d d , a c o n s e q u e n c e of ( 3 . 1 6 ) . F o r e x a m p l e

13

3

3

\l

1

-2

= 0

A n o t h e r very useful general formula is

Ih

h

1 mi

-m2

o Oi x

=

_ l( ) / I - m l

/ +- 1I / 2) S

( i 2

/2 i

/

62

m

j

m( 3 . 2 3 )

Thus I=

I

1-1/2

Buckmaster et al ( 1 9 7 2 ) have listed algebraic formulas for 3-/ s y m b o l s of t h e t y p e

// \M

/ -(M±m)

r

for

/ = 0(1)7

±m

i.e., for all values of / r u n n i n g from 0 t o 7 inclusive in i n c r e m e n t s of 1. These symbols frequently occur in calculations. R o t e n b e r g et al ( 1 9 5 9 ) have t a b u l a t e d algebraic forms f o r / = 0 ( ^ ) 3 . The m o s t c o n v e n i e n t collection of n u m e r i c a l values for 3-/ s y m b o l s is p r o b a b l y t h a t compiled b y R o t e n b e r g et al ( 1 9 5 9 ) .

3.7

THE CLEBSCH-GORDAN

SERIES

We n o w discuss a relationship c o n n e c t i n g elements of t h e r o t a t i o n m a t r i c e s . T h e reason we did n o t present this at t h e e n d of C h a p t e r 2 is t h a t we n e e d t h e V C coefficients, w h i c h were o n l y i n t r o d u c e d in this c h a p t e r . T h e t h e o r e m states t h a t :

\ij\m

ij^rrii I / 1 , / 2 ,

(

x^ i'

+

m

' 2>

mi

+m

j,ml+m2) M 2 7 )

(3-24)

3.7

The Clebsch-Gordan

41

Series l

This says t h a t if w e take a n y e l e m e n t of t h e m a t r i x 3^ \a^y) and m u l t i p l y it b y any 2 e l e m e n t of t h e m a t r i x 3^ \aPy) t h e result can be expressed as a linear c o m b i n a t i o n of e l e m e n t s of t h e matrices 3^\a0y) w h e r e / r u n s over t h e values j \ +/2, + O i / 2 —1)> • • •> l / i - 721- T h e range over w h i c h / runs is limited b y t h e triangle c o n d i t i o n o b t a i n i n g for t h e vector-coupling coefficients. N o t e t h a t , of course, t h e angles a , /3, a n d y m u s t b e t h e same for all m a t r i c e s . We d o n o t prove t h e C l e b s c h - G o r d a n series b u t t h e reader s h o u l d b e able t o see t h a t our previous discussion [in Section 3 . 2 ( h ) ] of t h e r e d u c t i o n of a direct p r o d u c t m a t r i x 3^^ ® 3^^ implies t h a t t h e e l e m e n t s of t h e r e d u c e d matrices are linear 2 c o m b i n a t i o n s of p r o d u c t s of e l e m e n t s of 3^^ a n d 3^ \ Conversely, p r o d u c t s of t h e cn a a 2 a type 3 ^ ( Py)®m }m2 ( Qy) b e expressed in t e r m s of t h e elements of t h e matrices 3"' c o n t a i n e d in t h e reduced direct p r o d u c t m a t r i x . T h e formal s t a t e m e n t of t h e latter relationship is t h e C l e b s c h - G o r d a n series. We c o n t e n t ourselves w i t h giving e x a m p l e s t o show h o w the t h e o r e m w o r k s in t w o specific cases a n d deriving an i m p o r t a n t relationship from t h e t h e o r e m , concerning t h e integral over t h e p r o d u c t of three spherical h a r m o n i c s . Consider t h e p r o d u c t

According t o t h e t h e o r y this p r o d u c t is equal t o a linear c o m b i n a t i o n of t h e m a t r i x elements 3 $

(ct(3y) a n d 0 ^ (a/fy) since t h e allowed / values are (£ + 2 ) a n d ( 5 — £ ) .

However, 3 $

(otPy) does n o t exist, since if y = 0 , m m u s t equal z e r o : t h e only

e l e m e n t in 3 ^

is 3 qq . T h u s t h e only t e r m in t h e e x p a n s i o n of 9

(

& ( « W < l / i ( « < l T )

is

( H H l H u x H H l H i D ^ i V = (1 x

i)9m^{oL$y)

L e t us m u l t i p l y b o t h sides of ( 3 . 2 5 ) b y angles:

a n d integrate over all t h e

. C X J T ™'. '™. ' '"'™. ' ' ™'. '». ' ''' '''* s

)

(o

r)2

(

,

jT)3

),

(

,

)

sto

- n r i w . i ) Jo J OJ 0 /

\mx

m2

mj

yrii

m2

mj

If w e use t h e o r t h o g o n a l i t y p r o p e r t i e s expressed in E q . ( 2 . 1 7 ) , w e see t h a t all t e r m s o n t h e right will vanish unless m 3 ' = m ' =

i' +

ra2m3

=m=ml

and / = ; 3 . W h e n

+m2i

these c o n d i t i o n s are c o m p l i e d w i t h t h e integral o n t h e right-hand side of t h e e q u a t i o n 2

reduces t o 87r , and w e get

Jjj \mi

J2

J3

J2

m2

m3J\mx

m2

J3

\

m3J

If we n o w use t h e relationship ( 2 . 2 0 ) , w h i c h o n l y h o l d s for / integer, w e find

JTJo ™ Y

1{ 6 y 0 )

( 6 , 0 )

( d t 0 ) s i n 6 d e d

_ r(2/1+ l)(2/2 + l ) ( 2 / 3 + l ) 4n

1/2

Ih

h

h\

\0

0

O/lm!

h

h m2

Is '

(3.26)

m3)

w h e r e we have changed t o t h e m o r e usual angles (0, 0). This is a very useful result since t h e integral over t h r e e spherical h a r m o n i c s arises f r e q u e n t l y , as we will find in later c h a p t e r s . I n using t h e e q u a t i o n b e careful t o n o t i c e t h a t it c o n t a i n s n o c o m p l e x

3

44

The Coupling

of Two A ngular

Momenta

conjugates. Also notice t h a t t h e triangle c o n d i t i o n o n t h e 3-/ s y m b o l m e a n s t h a t t h e 1

integral vanishes unless Q^ ^ ®

contains 3 ^

3

\ a result t h a t also follows from

elementary group theory.

2

As an e x a m p l e w e w o r k o u t t h e e x p e c t a t i o n value of ( 3 c o s 0 — l ) / r orbital

!

Y0f(r).

3

for t h e

This integral arises in t h e evaluation of t h e a n i s o t r o p i c m a g n e t i c

d i p o l e - d i p o l e hyperfine i n t e r a c t i o n in an a t o m . We require /=

i r / o i o °* ' l

x Y0(d, 2

0 ) / w [ (3

y i (6

N o w ( 3 c o s 0 - 1) = 4 ( T T / 5 ) integral is

1 2/

c o s 12

*

3

0- )/'* ]

0 ) / ( r ) sin 6 d$ d dr 2

Y0 .

1

N o t i n g t h a t (Y0)*

\o

0

l

a n d using ( 3 . 2 6 ) t h e

= Y0,

o/Vo

o/

0

v

'

T h e 3-/ s y m b o l s are given in closed f o r m b y ( 3 . 2 2 ) a n d w e find

F o r a 2pz

3

3

orbital (/ = 1) this gives f remains t o b e evaluated, a n d w e get n o help in this from our p r e s e n t t e c h n i q u e s since t h e y a p p l y o n l y t o angular f u n c t i o n s . T h e 3-/ s y m b o l s have b e e n s h o w n t o give a concise m e t h o d of evaluating an integral w h i c h occurs in a n u m b e r of p r o b l e m s ranging from a t o m i c physics t o terrestrial m a g n e t i c fields ( J a m e s , 1 9 7 3 ) :

= 0 ( / i / 2/ 3)

\mx

m2

m3J\0

0

0

where / 8 ( / i / a/ 3) = 4 T r [ ( 2 / 1 + l ) ( 2 / 2 + l ) ( 2 / 3 + 1 ) ]

1 2/

[(/1 + / 2 - / 3 + I ) (/1 + / 3 - / 2 + and L=l1

+ l2 +13.

+ /3-/1 + 1 ) ]

1 2/

J

3.9 h 3.9

Regge

Symmetries

45

REGGE SYMMETRIES

T h e 3-/ s y m b o l s have far m o r e s y m m e t r i e s t h a n those s u m m a r i z e d in Section 3 . 5 . Regge ( 1 9 5 8 ) has f o u n d 7 2 s y m m e t r i e s b y m a k i n g t h e following intriguing correspondence:

Ih

h

h\

U*i

m2

m3)

/ 2

+

/ 3 - / i

h-m\

/1+/W2 /1 + / 2 J2-m2

—7*3

I

J3-m3

If this square is reflected t h r o u g h its diagonals, or t h e rows or c o l u m n s cyclically p e r m u t e d , the resultant square can be associated w i t h a 3-/ s y m b o l having t h e same value as t h a t w i t h w h i c h w e started. F o r e x a m p l e , 0

2

21

1

2

1

3

0

1

Reflecting a b o u t t h e upper-left-lower-right diagonal, w e o b t a i n 0

1

3|

2

2

0|

12

1

1

w h i c h w i t h little effort can b e s h o w n t o c o r r e s p o n d t o

\o -k

hi

The equivalence of these t w o 3-/ symbols does n o t follow from t h e s y m m e t r i e s in Section 3 . 5 . Noncyclic p e r m u t a t i o n of t h e rows or c o l u m n s of the square results in a 3-/ s y m b o l equal t o +

(_l)/l /2+/3

Jl [mi

12 m2

/3

m3

w h i c h m e a n s t h a t a Regge s y m b o l w i t h t w o identical rows or c o l u m n s + + ( / 1 h 7 3 ) o d d gives a vanishing 3-/ s y m b o l . T h u s

i

6 5 4

\3 0 - 3 / vanishes, an e x a m p l e given b y J a m e s ( 1 9 7 3 ) .

and

46

3

3.10

The Coupling

of Two Angular

Momenta

THE V COEFFICIENT

In a d d i t i o n t o Wigner, o t h e r a u t h o r s have also derived symbols o f high s y m m e t r y from t h e vector-coupling coefficients. Later w e will have t o b e familiar w i t h t h e V coefficient of F a n o a n d R a c a h ( 1 9 5 9 ) w h i c h is related t o t h e coupling coefficients b y 2

i ) z 2 + z x ( x 2

T±1

2

=i2~ (x1y2-x2yll

1

= 2 " (xx ± iyx)(x2

1/2

1/2

T0 = Q) zlz2-6- (x1x2

±iy2)] ^

y 2)

y i

(4.3) k

T h e subscript k in Tq gives t h e r a n k of t h e tensor, while t h e subscript q is analogous t o a m a g n e t i c q u a n t u m n u m b e r as we will see in m o r e detail in C h a p t e r 5. There we

4.5

Irreducible

Tensor

49

Fields

present a systematic m e t h o d for c o n s t r u c t i n g irreducible spherical t e n s o r s , and t h e reason for the phases in ( 4 . 3 ) will b e c o m e a p p a r e n t . 4.4

IRREDUCIBLE CARTESIAN TENSORS

R e t u r n i n g t o first-rank Cartesian tensors we see t h a t n o t only is (x, y, z) such a t e n s o r b u t m o r e o v e r it spans t h e irreducible r e p r e s e n t a t i o n However, it does n o t 1 /2 (1/2) d o so in standard f o r m , a l t h o u g h the set [ — 2 ~ ( x + iy),z, 2 " (x - iy)] d o e s , and this set is therefore an irreducible spherical t e n s o r of r a n k 1. T h e set (x, y, z) is referred t o as an irreducible Cartesian tensor. T h e irreducible Cartesian tensors 7^ a n d l T , equivalent t o t h o s e in ( 4 . 3 ) , are 1/2

l

T° =-3- (xlx2^y1y2+zlz2l 1

Ty

= -2-

1 / 2

(x

Tx -

l

x 2) ,

Tz

l Z 2Z l

=

-2-

= -2-

l l 2

{ y , z 2- z

1 / 2

(x^

2

)

i y 2

-yxx2)

2

T h e specific forms of T are given in C h a p t e r 16 [Eq. ( 2 0 . 9 ) ] . In Chapter 16 we will discuss t h e c o n s t r u c t i o n of irreducible Cartesian t e n s o r s , and m o r e generally t h e t e c h n i q u e s involved in w o r k i n g w i t h real bases rather t h a n t h e standard c o m p l e x bases symbolized b y \jm > and typified b y ( 4 . 3 ) .

4.5

IRREDUCIBLE TENSOR FIELDS

Spherical tensors as described above are c o m b i n a t i o n s of t h e c o o r d i n a t e s of a p o i n t in space. However, a glance at ( 2 . 1 ) reveals a close similarity b e t w e e n t h a t e q u a t i o n , which describes t h e t r a n s f o r m a t i o n s of states, a n d ( 4 . 2 ) , w h i c h describes t h e t r a n s f o r m a t i o n s of spherical tensors. We define an irreducible tensor field of rank kto be a set of 2k + 1 functions transforming u n d e r c o o r d i n a t e r o t a t i o n s according t o k

D(aPy)fq (r')

= I

fq'\^}}(p$i)

(4.5)

The m o s t familiar functions satisfying this e q u a t i o n are t h e spherical h a r m o n i c s l Ym (P > 0)- N o t e t h a t , consistently w i t h t h e conclusions of Section 1.4, t h e same c o o r d i n a t e s r', referred t o t h e final frame, appear on b o t h sides of ( 4 . 5 ) . Making use of t h e relationships (Section 1.4) Rf(t')

= F(r') =

f(t)

we can write ( 4 . 5 ) in t h e form ,

//(r)= Z / / ( r ) ^ ( a / 3 7)

(4.6)

where r a n d r' belong t o t h e same p o i n t referred t o different c o o r d i n a t e systems. F o r later use we n o t e t h a t t h e inverse of ( 4 . 6 ) is / A r ' ) = I / / ( ' ) ^ P 7 )

(4.7)

50

4.6

4

Scalars, Vectors,

Tensors

SCALARS

A n rcth-rank Cartesian t e n s o r h a s 3 " c o m p o n e n t s , so a zero-rank t e n s o r h a s 3 ° , t h a t is o n e , c o m p o n e n t . I t s t r a n s f o r m a t i o n s m u s t be described b y a 1 x 1 m a t r i x , a n d therefore it does n o t m i x w i t h o t h e r q u a n t i t i e s u n d e r c o o r d i n a t e r o t a t i o n s . This unchanging n a t u r e , o r in variance, is w h a t characterizes a zero-rank tensor. T h e t e r m scalar h a s b e e n used in t w o w a y s , o n e of w h i c h is t o d e n o t e a q u a n t i t y w h i c h is invariant u n d e r c o o r d i n a t e r o t a t i o n s , i.e., a zero-rank tensor. T h e classic e x a m p l e of this usage is t h e so-called scalar p r o d u c t A • B o f t w o vectors, w h i c h is a n u m b e r . A generalization of t h e scalar p r o d u c t is m a d e in t h e n e x t c h a p t e r . A second usage is in w h i c h were given at t h e b u t it is n o t necessarily certainly n o t invariant spanning

t h e familiar phrase scalar field, or scalar function, e x a m p l e s of beginning of C h a p t e r 1. A scalar field h a s n o vector p r o p e r t i e s , 1 a scalar field, is a zero-rank t e n s o r . T h u s Y0 (6,although 1 t o c o o r d i n a t e r o t a t i o n s b u t belongs t o t h e set Ym(69 0)

It is i m p o r t a n t t o realize t h a t a vector field m a y be invariant t o r o t a t i o n s . S u p p o s e we choose a c o o r d i n a t e system a n d t h e n at every p o i n t w e place a vector directed away from t h e origin a n d of length equal t o t h e distance from t h e origin t o t h e p o i n t . We have c o n s t r u c t e d t h e vector field r, w h i c h h a s spherical s y m m e t r y a n d is invariant u n d e r c o o r d i n a t e r o t a t i o n s , b u t is definitely n o t a scalar field. We have used t h e t r a n s f o r m a t i o n a l behavior of sets of n u m b e r s or fields as a m e a n s of classifying t h e m u n d e r such t e r m s as " s c a l a r , " "rtth-rank spherical t e n s o r , " or "rtth-rank Cartesian t e n s o r . " In t h e n e x t chapter we will e x a m i n e t h e properties of sets of operators a n d again it will be their t r a n s f o r m a t i o n s t h a t will provide t h e basis for their classification.

Chapter

5

Irreducible Tensor Operators

. . in this case the spherical harmonics play the role of operators and not of eigenfunctions, and it appears convenient to consider in a general way the algebra of such operators." G . R a c a h , Phys.

5.1

DEFINITION OF IRREDUCIBLE TENSOR

Rev.

62, 441 (1942)

OPERATORS

U p t o n o w w e have b e e n largely c o n c e r n e d w i t h t h e t r a n s f o r m a t i o n p r o p e r t i e s of spatial f u n c t i o n s o r spinors. In particular, we have b e e n c o n c e r n e d w i t h t h e infinite g r o u p of c o o r d i n a t e r o t a t i o n s comprising t h e g r o u p R3.

We f o u n d t h a t t h e angular

m o m e n t u m eigenkets \jm) behave simply u n d e r such r o t a t i o n s and in fact f o r m basis sets for t h e irreducible r e p r e s e n t a t i o n s 9 ^

of R3.

F r o m t h e l a t t e r p o i n t of view the

q u a n t u m n u m b e r s / and m can b e regarded as g r o u p - t h e o r e t i c a l labels. T h e n e x t s t e p , w h i c h brings us t o t h e c o n c e p t of irreducible t e n s o r o p e r a t o r s , is t o consider

in detail

the

transformation

p r o p e r t i e s o f operators

under

t r a n s f o r m a t i o n s , and in particular u n d e r t h e g r o u p of r o t a t i o n s R3,

coordinate

a subject w e

t o u c h e d on briefly in S e c t i o n 1.12. We f o u n d previously t h a t certain t y p e s of f u n c t i o n s behave c o m p a r a t i v e l y simply under c o o r d i n a t e r o t a t i o n s in t h a t t h e y are t r a n s f o r m e d i n t o linear c o m b i n a t i o n s of a closed set of f u n c t i o n s D(0LPy)fi

= Icjtfj

for

51

/=1,...,7V

52

5

Irreducible

Tensor

Operators

i.e., t h e functions fx, . . . , fN f o r m a basis for a n irreducible invariant subspace u n d e r R$. T h e same idea can b e applied t o operators and w e define a n irreducible tensor o p e r a t o r ( I T O ) o f r a n k k t o b e a set o f 2k + 1 o p e r a t o r s Tq w h i c h u n d e r coordinate r o t a t i o n s t r a n s f o r m according t o t h e relation k

l

D(aPi)Tq D- (afiy)

= T,Tq'

k

(5.1)

9 $ (afii)

w h e r e q a n d q can t a k e o n 2k + 1 values running f r o m -k t o k. This says t h a t u n d e r a coordinate r o t a t i o n D(afiy) t h e o p e r a t o r Tq is t r a n s f o r m e d i n t o a linear c o m b i n a t i o n ,k of 2k + 1 o p e r a t o r s Tq , t h e e x p a n s i o n coefficients being t h e e l e m e n t s o f t h e Wigner matrices. T h e definition concerns itself only w i t h t h e t r a n s f o r m a t i o n a l properties of o p e r a t o r s , n o t their detailed f o r m . S u b s e q u e n t l y , w e will l o o k a t m a n y specific o p e r a t o r s b u t a t this stage i t m a y h e l p t o examine some samples o f ( 5 . 1 ) . F o r k = 0 w e have 2A: + 1 = 1 o p e r a t o r r0°. Its t r a n s f o r m a t i o n p r o p e r t i e s are given by 1

(aPi) = T0°9$

D(apy)T0°D-

= r 0° x 1 = T0°

(5.2)

i.e., T0° is u n c h a n g e d b y c o o r d i n a t e r o t a t i o n s . S u c h a zero-rank o p e r a t o r is often called a scalar operator. The simplest scalar o p e r a t o r is just a n u m b e r . The " o p e r a t o r " 2 is obviously a scalar o p e r a t o r , a n d so are all t h e t e r m s t h a t appear in t h e H a m i l t o n i a n , w h i c h m e a n s t h a t scalar o p e r a t o r s are going t o o c c u p y m u c h o f our t i m e . l

1

F o r k= 1 w e have three o p e r a t o r s Ti\ T0, a n d T^ w h i c h form a first-rank irreducible t e n s o r o p e r a t o r a n d u n d e r r o t a t i o n s t r a n s f o r m i n t o linear c o m b i n a t i o n s o f each o t h e r , e.g., missing o u t t h e angles (afiy): l

DT0D~

l

l

= r . / ^ V o + T09®

+ TS&ft

(5.3)

If w e have t h r e e functions t h a t span w e say t h a t u n d e r c o o r d i n a t e r o t a t i o n s t h e y behave like t h e c o m p o n e n t s o f a vector. By analogy w e can u s e t h e t e r m vector 1 l operator t o describe t h e set o f t h r e e o p e r a t o r s T\ , To , and T h e simplest e x a m p l e of a vector o p e r a t o r is t h e position o p e r a t o r r, for w h i c h w e c a n choose t h r e e c o m p o n e n t s rx, ry, and rz. N o w these certainly span t h e r e p r e s e n t a t i o n but they d o n o t satisfy o u r definition of an irreducible t e n s o r o p e r a t o r , i.e., t h e y d o n o t t r a n s f o r m u n d e r r o t a t i o n according t o E q . ( 5 . 1 ) . T h e reason, a n d the r e m e d y , should be obvious: rx, ry, a n d rz d o n o t span 3 ^ in standard f o r m . T h e t r a n s f o r m a t i o n coefficients in ( 5 . 1 ) are t h e elements of t h e Wigner r o t a t i o n matrices, w h i c h are defined in t e r m s of t h e t r a n s f o r m a t i o n s o f a standard set o f basis functions. Clearly, if we m a k e a suitable o r t h o g o n a l t r a n s f o r m a t i o n o f t h e set {rx, ry, rz} , w e c a n o b t a i n a n o t h e r set t h a t will span 9 ^ in standard f o r m . A correct set is -2-

1 / 2

( x + z » ss 7 V ,

l

z = T09

l/2

2- (x-iy)

l

= T.t k

(5.4)

The identification o f each o p e r a t o r w i t h its corresponding s y m b o l Tq can b e arrived at explicitly b y considering t h e t r a n s f o r m a t i o n properties of t h e set o f o p e r a t o r s .

5.1

Definition

of Irreducible

Tensor

Operators

53

However, t h e c o r r e s p o n d e n c e should b e intuitively clear after looking at ( 2 . 8 ) . We see that l

the

Ym(Q,

operators

defined

above

are

proportional

to

the

spherical

harmonics

0). This p r o p o r t i o n a l i t y is of course n o t accidental, b u t is a direct c o n s e q u e n c e

of o u r definition ( 5 . 1 ) of an irreducible t e n s o r o p e r a t o r . It is easy t o check t h a t t h e spherical h a r m o n i c s themselves t r a n s f o r m like irreducible tensor l

l

£

D(a^)Ym(6,(t>)D- (a^)=

operators (5.5)

Ym\B9fi9%m(a&i)

m'=-l l

There is n o p a r a d o x in our use of t h e Ym

b o t h as functions and as o p e r a t o r s . T h e

same duality can b e seen t o a p p l y t o t h e functions x, y, and z, w h i c h can also b e used as o p e r a t o r s . F o r e x a m p l e , if w e w a n t t o k n o w t h e average value of the position vector rx for a certain function \p, we m u s t evaluate f\p*rx\pdr

= f\p*x\pdr.Heve

x plays t h e

part of an o p e r a t o r . Similarly, a l t h o u g h the spherical h a r m o n i c s are usually considered as functions t h e r e are m a n y p r o b l e m s in molecular physics in w h i c h t h e y appear as o p e r a t o r s , p e r h a p s t h e best k n o w n being t h e m u l t i p o l e e x p a n s i o n of an electrostatic potential. O t h e r e x a m p l e s of first-rank irreducible t e n s o r o p e r a t o r s are 1 a n d s, b o t h of w h i c h have t h r e e c o m p o n e n t s . The t h r e e c o m p o n e n t s o f 1 will a p p e a r often in later c h a p t e r s : -2'

l i

\l

x

+ ily) = - 2 "

Notice t h e phases; t h e set 2 "

1 / 2

/ +,

1 /2

/+, lz,

lz = lZi 2~

1 /2

1,2

2~ (lx

-ily)

= 2"

1 / 2

/_

(5.6)

L d o e s not t r a n s f o r m c o r r e c t l y . It is

c o m m o n usage t o define t h e o p e r a t o r s / +1 = - 2 "

1 / 2

/ +,

/ 0 = / 2,

/_! = 2 "

1 / 2

/_

(5.7)

N o t e t h e subscripts carefully; d o n o t confuse / +1 w i t h /+, for e x a m p l e , in later applications. 2

2

2

2

2

For k-2 t h e r e are five o p e r a t o r s : T2 , T , T0 , T_ , T_ . The m o s t obvious 2 e x a m p l e is t h e set o f spherical h a r m o n i c s Ym (0, 0). We will later e x a m i n e o t h e r second-rank o p e r a t o r s . O p e r a t o r s of r a n k k > 2 are n o t c o m m o n in chemical physics, a n o t a b l e e x c e p t i o n being t h e spherical h a r m o n i c s for / > 2 , w h i c h arise, for e x a m p l e , in t h e m u l t i p o l e expansion of electrostatic p o t e n t i a l s . E x a m p l e s of third- and f o u r t h - r a n k t e n s o r s appear in C h a p t e r 16. T h e definition of irreducible t e n s o r o p e r a t o r s d o e s n o t formally preclude t h e possibility of k being half-integer, b u t in fact there are n o o p e r a t o r s of half-integer l/2 rank. If t h e r e w e r e , w e could have m a t r i x e l e m e n t s of t h e kind k\m \ T \ \m) w h i c h represents a transition b e t w e e n a particle of spin \ and a particle of spin 1. N e i t h e r this n o r o t h e r half-integer transitions have b e e n observed e x p e r i m e n t a l l y . F e r m i o n s and b o s o n s are n o t interconvertible.

5

54

n 5.2

Irreducible

Tensor

Operators

AN EXAMPLE

In order t o illustrate some p o i n t s w e will w o r k o u t in detail t h e t r a n s f o r m a t i o n of 1 Tq given in ( 5 . 3 ) . We will write o u t t h e r o t a t i o n o p e r a t o r s a n d t h e tensor o p e r a t o r Tq l l as matrices. Since T0 is o n e of t h e set of t h r e e o p e r a t o r s T m , w e use a three-dimensional space and t h e relevant r o t a t i o n o p e r a t o r s are 9 ^ (a/fy) and 1 9 ^ (a/37)" . N o w t h e matrices 9^ are c o n s t r u c t e d in t h e basis set | / r a > a n d w e m u s t l construct t h e o p e r a t o r s Tm in t h e same basis. A n easy w a y t o d o this is t o choose as our set t h e o p e r a t o r s y + 1, / 0 , a n d / _ l 5 w h i c h leads t o t h e following forms if t h e operators are evaluated in t h e basis set |11>, |10>, |1 — 1>:

7*-!

TV

\o

0

1

fO

0

0\

=I i

0

0

vo

1

0/

0/

l

T0=[

'1

0

o

0

0

0

0

-1

\0

1

(5.8)

We have

Dropping t h e Euler angles, w h i c h should appear in every e l e m e n t of t h e r o t a t i o n matrices, w e need t o evaluate

f^n

0io

®01

0

w h i c h gives k

k

k

JnTq -Tq Jn=% Putting=/

z

q

Tq> (kq'\Jn\kq)

we get one of the e q u a t i o n s ( 5 . 9 ) k

k

[Jz,Tq ]=qTq

If we p u t Jn = / + or / _ , we o b t a i n the o t h e r c o m m u t a t o r s in ( 5 . 9 ) . A simple illustration of these c o m m u t a t i o n relations is provided b y t h e set of 1 2/ 1 /2 /_, angular m o m e n t u m o p e r a t o r s J+1, J0, and / _ t w h i c h are — 2 " / + , Jz, and 2 " respectively, a n d form an irreducible tensor o p e r a t o r of r a n k 1. These o p e r a t o r s o b e y Racah's rules as can b e s h o w n from t h e familiar c o m m u t a t i o n rules for angular momenta.

5.4

SCALAR AND VECTOR

OPERATORS

Starting from Racah's definition it is p e r h a p s easier t o define t h e exact m e a n i n g of the t e r m s "scalar o p e r a t o r " and " v e c t o r o p e r a t o r " w h i c h frequently appear in the literature. The c o m m u t a t i o n relationships have b e e n seen t o follow directly from the k behavior u n d e r r o t a t i o n s of t h e o p e r a t o r s Tq , and any o p e r a t o r obeying E q . ( 5 . 1 ) will satisfy R a c a h ' s definition of an irreducible tensor o p e r a t o r . If the r o t a t i o n o p e r a t o r

5

56

Irreducible

Tensor

Operators

D(afiy) is in t h e general form ( 1 . 1 5 ) , t h e n a n y set of suitably normalized vector c o m p o n e n t s Vi, V0, a n d V.x transforms according t o ( 5 . 1 ) . A vector playing t h e role of an o p e r a t o r behaves as an o p e r a t o r of r a n k 1, and t h e c o m m u t a t i o n relations ( 5 . 9 ) become [J+,V0]

=2 =2

[Jz,V0]

1 / 2

F 1,

1 / 2

1 / 2

F_1

1/2

K 0,

=0,

= 2

[J.9V0]

[J-,V1]=2 V0

[Jz,V±1]

= V±l

Any o p e r a t o r obeying these c o m m u t a t i o n rules is referred t o as a vector o p e r a t o r with respect to the angular momentum J. T h e last p a r t of t h e definition is often forgotten b u t it is essential. It might be argued t h a t since J and L are b o t h angular m o m e n t u m operators, t h e y will have t h e same c o m m u t a t i o n relations w i t h V\, V0, a n d V.x. This is n o t necessarily s o , and there are o p e r a t o r s w h i c h are vector o p e r a t o r s with respect to J b u t n o t t o L. We illustrate w i t h s o m e i m p o r t a n t e x a m p l e s . The spin o p e r a t o r s S i , S 0 , a n d S_x form a n irreducible tensor o p e r a t o r of r a n k 1. Their c o m m u t a t i o n relations w i t h J and L are

=

[Jz.So]

= 0, = 0

U2

2 Sl,

[J-,S0]

[Jz,S±1]

U2

=

2 S.

=

2 > S0

=

±S±1

1

i

for all / a n d /

All c o m m u t a t o r s b e t w e e n t h e c o m p o n e n t s of S a n d L vanish since t h e o p e r a t o r s w o r k on spin a n d space, respectively. T h u s S behaves as a vector o p e r a t o r w i t h respect t o J b u t n o t w i t h respect t o L. An irreducible tensor o p e r a t o r w h i c h c o m m u t e s w i t h J is called a scalar o p e r a t o r with respect to J. We can see from t h e c o m m u t a t o r s ( 5 . 9 ) t h a t o n l y a zero-rank o p e r a t o r (for w h i c h k = q = 0 ) can b e a scalar o p e r a t o r w i t h respect t o J. T h e vanishing of t h e m u t u a l c o m m u t a t o r s of S w i t h L m e a n s t h a t S is a scalar operator w i t h respect t o L. S t a t e d in t h e language of t r a n s f o r m a t i o n t h e o r y , u n d e r a r o t a t i o n of spatial coordinates, spin c o o r d i n a t e s are left u n c h a n g e d . I n t h e language of t h e chemist, if we have an a electron in a py orbital a n d perform a spatial c o o r d i n a t e r o t a t i o n of 9 0 ° about z, we o b t a i n an a electron in a px orbital. The space function is changed b y operators of t h e t y p e e x p ( / a / „ ) b u t n o t t h e spin function. T h e generator of spatial and spin r o t a t i o n s is J , a n d S is a vector o p e r a t o r w i t h respect t o J because e x p ( / a / „ ) acts on spin as well as space. It should n o w be clear t h a t t h e o p e r a t o r L • S is a scalar o p e r a t o r w i t h respect t o J. In J space b o t h L a n d S behave as vectors a n d L • S, their scalar p r o d u c t , is of course invariant u n d e r r o t a t i o n s . N o t i c e t h a t L • S is not a scalar o p e r a t o r w i t h respect either t o L or t o S alone. Intuitively we should realize t h a t this is because L • S consists of the scalar p r o d u c t of t w o vectors lying in separate subspaces. If t h e c o o r d i n a t e axes of only one of these subspaces are r o t a t e d , this p r o d u c t will n o t b e invariant. T h e c o m m u t a t o r s of L * S w i t h J vanish, b u t t h e c o m m u t a t o r s of L • S w i t h L o r S d o n o t .

5.6

The Construction

n 5.5

A LIE

of Compound

Irreducible

Tensor

57

Operators

GROUP

In this section w e t o u c h briefly on s o m e of t h e wider i m p l i c a t i o n s of R a c a h ' s c o m m u t a t i o n rules a n d their equivalent, t h e t r a n s f o r m a t i o n rules given in E q . ( 5 . 1 ) . It is clear from t h e l a t t e r t h a t t h e t r a n s f o r m a t i o n b e h a v i o r t h a t w e d e m a n d of irreducible tensor o p e r a t o r s ensures t h a t t h e o p e r a t o r s t r a n s f o r m as bases for r e p r e s e n t a t i o n s of the full r o t a t i o n g r o u p . Evidently t h e c o m m u t a t i o n rules m u s t also i m p o s e t h e same c o n d i t i o n s , i.e., t h e y d e t e r m i n e t h e t r a n s f o r m a t i o n p r o p e r t i e s of t h e o p e r a t o r s u n d e r r o t a t i o n s . N o w let us c h o o s e t h e t e n s o r o p e r a t o r s t o b e v e c t o r o p e r a t o r s , a n d even m o r e specifically t o b e t h e o p e r a t o r s J+l,

J0,

and J_x. T h e n we have

[/±,/±i] = 0 , U,J0]

= 2

1 / 2

/ + 1,

[J.,J0]

=0

[Jo,Jo] =2

1 / 2

/.

[ / ±, /

l 5

T 1]

=2

1 / 2

/

0

w h i c h are j u s t t h e familiar c o m m u t a t i o n rules for angular m o m e n t u m o p e r a t o r s . T h u s t h e commutators

under

rotations.

But w e n o w recall t h a t / is t h e g e n e r a t o r of t h e r o t a t i o n o p e r a t o r s D(olPj)

of the angular momentum

determine

their behavior

w h i c h in

their e n t i r e t y c o n s t i t u t e t h e o p e r a t o r s of t h e full r o t a t i o n g r o u p . We arrive at the conclusion t h a t t h e b e h a v i o r of t h e r o t a t i o n o p e r a t o r s u n d e r c o o r d i n a t e r o t a t i o n s m u s t be d e t e r m i n e d c o m p l e t e l y b y t h e c o m m u t a t i o n rules of t h e i r g e n e r a t o r s . But t h e specification of t h e i r b e h a v i o r u n d e r r o t a t i o n s is equivalent t o specifying their p r o p e r t i e s , i.e., h o w t h e y c o m b i n e w i t h e a c h o t h e r . F o r finite

group

groups the group

m u l t i p l i c a t i o n table d e t e r m i n e s all t h e p r o p e r t i e s of t h e g r o u p , for continuous

groups

it is t h e c o m m u t a t i o n rules of t h e g r o u p generators t h a t define t h e g r o u p p r o p e r t i e s . L o o k i n g b a c k at S e c t i o n 1.14 we see t h a t w e have r e t u r n e d t o t h e c o n c e p t of Lie groups. R a c a h ' s definitions j u s t reduce t o t h e Lie algebra of t h e r o t a t i o n g r o u p w h e n k

t h e Tq

5.6

are t a k e n t o b e t h e angular m o m e n t u m o p e r a t o r s .

THE CONSTRUCTION OF COMPOUND TENSOR

IRREDUCIBLE

OPERATORS

M a n y of t h e o p e r a t o r s t h a t o c c u r in c h e m i s t r y a n d physics are c o m p o u n d in t h e sense t h a t t h e y are built u p from p r o d u c t s of c o m p o n e n t s of simpler o p e r a t o r s . A familiar e x a m p l e is t h e s p i n - o r b i t o p e r a t o r 1 • s. In this section w e will discuss t h e s y s t e m a t i c c o n s t r u c t i o n of c o m p o u n d t e n s o r o p e r a t o r s f r o m simple o p e r a t o r s . O u r object in doing this is t o provide ourselves w i t h a m e a n s of classifying as fully as possible

the

behavior

of

any

operator

under

coordinate

rotations

since

this

classification is an essential p a r t of t h e irreducible t e n s o r m e t h o d . Let us s u m m a r i z e o u r k n o w l e d g e at this stage: A set of 2k + 1 t e n s o r o p e r a t o r s spans an invariant subspace u n d e r t h e o p e r a t i o n s of t h e g r o u p R3.

F u r t h e r m o r e , since

the o n l y irreducible r e p r e s e n t a t i o n s of R3 can b e s h o w n t o be the r e p r e s e n t a t i o n s 3 ^ of order 21 + 1, it follows t h a t an irreducible t e n s o r o p e r a t o r of o r d e r k forms a basis k

for t h e r e p r e s e n t a t i o n 3^ \

F r o m E q . ( 5 . 1 ) we see t h a t t h e t r a n s f o r m a t i o n coefficients

5

58

Irreducible

Tensor

Operators

k

are t h e e l e m e n t s 9q q(aPy) and it follows t h a t the t e n s o r o p e r a t o r forms a standard basis as defined in Section 2 . 2 . We are already familiar w i t h t h e idea t h a t t h e direct p r o d u c t of t w o sets of basis functions separately spanning representations r,- and r ; of a g r o u p G, will span the direct p r o d u c t representation T = r,- ® Ty of G. In general V is reducible a n d we can find linear c o m b i n a t i o n s of t h e direct p r o d u c t functions such t h a t t h e y t r a n s f o r m as m e m b e r s of t h e basis set for an irreducible r e p r e s e n t a t i o n c o n t a i n e d in T. N o w there is n o t h i n g in this p r o c e d u r e t h a t d e m a n d s t h e use of functions t o form our basis sets. We can j u s t as well apply t h e w h o l e a p p a r a t u s t o operators. F o r m a l l y , if we k l \ and have t w o irreducible tensor o p e r a t o r s of rank kx and k2 spanning & k & ( * \ w e can define a direct p r o d u c t consisting of all t h e (2ki + 1)(2A; 2 + 1) possible 2 p r o d u c t s Tq^ Tq 2. We can n o w find linear c o m b i n a t i o n s of these p r o d u c t s t h a t will span t h e irreducible representations c o n t a i n e d in t h e direct p r o d u c t r e p r e s e n t a t i o n k gi( i) ® £j}(k2) 0f ^ 3 ^ simple and classic example is the direct p r o d u c t of t w o first-rank t e n s o r o p e r a t o r s . We t a k e t h e t w o sets of operators -2-

1 / a

-2"

(*,+(Vi)

1 / 2

0 c 2 + r > 2) ^

f TJW, 2-

1 / 2

( * 1- / > I) J

z2

(5.12)

>Tq\2)

u

2- \x2-fyt)

,

The o p e r a t o r s are supposed t o act on t h e c o o r d i n a t e s of t w o separate particles 1 and 2 . T h u s all t h e c o m p o n e n t s of a given o p e r a t o r c o m m u t e w i t h all those of the o t h e r o p e r a t o r . Later we will consider t h e possibility t h a t b o t h o p e r a t o r s refer to the same particle. T h e direct p r o d u c t of t h e sets ( 5 . 1 2 ) will span the representation ® 3 ^ = QjO) + + £$(o) Evidently we form one second-rank tensor o p e r a t o r spanning one first-rank tensor o p e r a t o r spanning and a scalar tensor ( 0 )

1

1

2

1

. Symbolically we might write T ® T = T + T + T ° . What o p e r a t o r spanning ^ 1 2 are the correct linear c o m b i n a t i o n s of the p r o d u c t Tg Tg t h a t we n e e d ? Obviously t h e y are given b y t h e vector-coupling coefficients w h i c h in o u r particular example are given b y ( 3 . 4 ) . If the reader t h i n k s t h a t this is n o t obvious, p e r h a p s t h e following reasoning will convince h i m . T h e vector-coupling coefficients as we originally defined t h e m were designed t o give t h e correct linear c o m b i n a t i o n s of t h e states \j\m\)\i2m2)\ required t o give a state | jiJ2Jm >. In fact, the values of the coefficients d e p e n d only on the values of t h e angular m o m e n t u m eigenvalues (the / s and ms) and n o t at all on t h e detailed form of the functions \fatna). We only required t h a t these functions b e normalized and t h a t an (arbitrary) standard choice of phase be m a d e . N o w the k o p e r a t o r s Tq , as we have seen, behave u n d e r r o t a t i o n s j u s t as t h e spherical h a r m o n i c s k Y and therefore j u s t as the states \kq). It follows t h a t the p r o b l e m of coupling c o m m u t i n g t e n s o r o p e r a t o r s is identical m a t h e m a t i c a l l y w i t h the p r o b l e m of coupling angular m o m e n t a , reflecting t h e underlying fact t h a t in b o t h cases w e are only using the t r a n s f o r m a t i o n a l , group-theoretical, properties of t h e states or o p e r a t o r s concerned. This is t h e secret of t h e p o w e r of the irreducible tensor m e t h o d , w h i c h as we

5.6

The Construction

of Compound

Irreducible

Tensor

Operators

59

will see is n o t t o o c o n c e r n e d a b o u t any o t h e r p r o p e r t y of o p e r a t o r s and states e x c e p t their t r a n s f o r m a t i o n s u n d e r r o t a t i o n s . In general t h e n we express c o m p o u n d irreducible tensor o p e r a t o r s as ,

7 J;(l)r*j(2)

{T*i(l)®T*'(2)}£= £

(5.13)

Compare this w i t h Eq. ( 3 . 8 ) . On t h e left-hand side t h e indices k and q symbolize the rank and c o m p o n e n t of t h e c o m p o u n d o p e r a t o r . T o illustrate we n o w c o n s t r u c t all the 1 1 irreducible tensor o p e r a t o r s spanning t h e direct p r o d u c t of T 0 T , ignoring ( 3 . 4 ) and w o r k i n g from first principles. F o r k = 0 we have 1

1

I

{T ®T }8=

1^11100)^(1)7^(2)