S 2'

0 :35

nHuntQr

...... -

Vo

t--Ie e
II


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GROUP THEORY IN
QUANTUM MECHANICS
An Introduction to

its Present Usage
by

VOLKER HEINE
University of Cambridge

PERGA~fON

PRESS

LONDON · OXFORD · NEW YORK · PARIS

1960

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PERGAMON PRESS LTD.
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Headington HiU Hall, Oxford

PERGAMON PRESS INC.
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PERGAMON PRESS S.A.R.L.
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PERGAMON PRESS G.m.h.H.
Koi8e1-BtrCJ88e 76, Fran1c!'Url-am.Main

Copyright

©
1960
VOWR HEmE

Library of Congress Card Number 59-10523


Set by Santype Ltd., 45-55 Brown Street, Salisbury
Printed by J. W. Arrowsmith Ltd., Winterstoke Road, Bristol 3

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CONTENTS
FAGB

PREFACE

vii

ix

NOTATION

I. SYMMETRY TRANSFORMATIONS
1.
2.
3.
4.
5.
6.

The uses of symmetry properties
Expressing symmetry operations mathematically
Symmetry transformations of the Hamiltonian
Groups of symmetry transformations

Group representations
Applications to quantum mechanics

. II.
7.
8.
9.
10.
11.
12.
13.

14.
15.
16.
17.

THE QUANTU:M THEORY OF A FREE ATOM

Some simple groups and·representations
The irreducible representations of the full rotation group
Reduction of tho product representation D(/) X DU/)
Quantum mechanics of a free atom; orbital degeneracy
Quantum mechanics of a free atom including spin
The effect of the exclusion principle
Calculating matrix elements and selection rules

III.

1

3
6
12
24
41

48
52
67

73
78
89
99

THE REPRESENTATIONS OF FINITE GROUPS

Group oharacters
Product groups
Point-groups
The relationship between group theory a.nd tho Dirac method

113
125
128
143 .

IV. FURTHER ASPECTS OF THE THEORY OF FREE
ATOMS AND IONS
18.

19.
20.
21.

Paramagnetic ions in crystalline fields
Time-reversal and Kmmers' theorem
Wigner and Racah coefficionts
Hyperflne structure

v.

148
164

176
189

THE STRUCTURE AND VIBRATIONS OF MOLECULES

22.
23.

Valence bond orbitals and molecular orbitals
Molecular vibrations
24. Infra-red and Ram~ spectra

206

229


245

·VI. SOLID STATE PHYSICS
25.
26.
27.

Brillouin zone theory of simple structures
Further aspects of Brillouin zone theory
Tensor properties of crystals

v

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265
284304


vi

CONTE~~S

V1I.
28.
29.
30.

The isotopic spin fornudism
Nuclear forces

Reactions
VIII.

31.
32.
33.
34.

NUCLEAR PHYSICS

PAGE

313
321
334

RELA'!"IVISTIC QU"ANTUM MECHANICS

The representations of the Lorentz groLi'"
The Dirac equation
Beta decay
Positronium

351
363
384

397

APPENDICES

Matrix algebra
Hornomorphism and isomorphisln
Theorems on vector spac0s and group representations
Schur'slemrna
Irreducible representations of Abelian groups
Momenta and infinitesimal transformations
The simple harmcnic oscillator
The irreducible representations of the complete Lorentz group
Table of Wigner coefficients (jj' mtn'IJjl)
J. Notation for the thirty-two crystal point-groups
K. Charaoter ta.bles for the crysta.l point.groups
L. Character tables for the axial rotation group and derived groups

A.
B.
C.
D.
E.
F.
G.
H.
I.

LIST

O~.,

GENERAL

REF}1~RENCES,


WITH REVIEWS

404
410
412
418
420
422
424
428
432
446
448
455
457

BIBLIOGRAPHY

459

SUBJECT INDEX

464

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PREFACE
The object of this book is to introduee the t,hret:: main uses of


group theory in quantum mechanic3, which are: firstly, to label
energy levels and the corresponding e.igenEt:,atk~;:t; secondly, to discuss
qualitatively the splitting of energy levels at; one starts from Q·n
approximate Hamiltonian and adds correction terlns; and thirdly,
t.o aid in the evaluation of matrix elements of all ki.nds, and in
particular to provide general selection rule",~ for: the non-zero ones.
The theIne is to Sh01V how all this ig (1.chie'v'8d by oonsidering t.he
symmetry properties of the Hamilwninn and the wa.y in which
these 8'yrrlillptries are reflected in the "rav." funntiolls. In Chapter I
the necessary mathenlatical concepts n.r(~ introoueed in as elementary and illustrative a manner as possible. with the llroofs of some of
the fundamental theorems being relegid:ed to an appenctix. The
three U8(lf~ of group theory above are Hl~6trated. in detail in Chapter
II by a. fairly quick run through the theory of atomic energy levels
and tran8itions. This topic is particula;rly suitable for illustrative
purposes, fy:(~an8P most of the resu1fr are familiar from the usual
vector Inodel of the atoln but are o. d.ved here in a rigorous and
precise \,~ay. Also most of it, e.g. the ~f:tr.oduction of spin functions
and the exclusion principle, is fundanh~,rttal to all the later more
advanced topics. Chapter III is a rep0f..:1t,ory for the theory of group
characters, the crystallographic pOllit .. gl'OUpS and nlinor pOUlts
required in some of the later aI)plications. Thus, after selected
readjngs from chapter III according to his field of interest., the
reader is ready to jump immediately to any of the applications of
the theory covered in later chapters, nau}~ly· further tDpics in the
theory of atomic energy levels (Oh",pter I'T), t.he electronic structure
and vibrations of molecules ((Jhapt:er V \. Kolid state phYRics (Chap~
ter VI), nuclear physics (Chapter \111,l, and relativistic quantum
mechanics (Chapter VIII).
The level of t.he text is that of a course for re"earch students in

physics and chemistry, such as is now offered in many Universities.
A previous course in quantum theory, ba~ed on a text such as
Schiff Quantunt Mechanics, is assumed, but the !natrix algebra
required is included as an appendix. In selt'cting the nlaterial for
the applications in various branches of physics and chemistry in
Chapters IV to 'VIII, I have restricted mysf'lf as far as possible to
topics satisfying three criteria: (i) the topics should be simple
vii

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VBl

PREFACE

(:Lpp1ioa:t/i(~n.~

ths/t illu.strate basic principles, rather than compIica,ted
~~Jxf1:!J1ples df.~jigne,d. to overa\ve the rea,deL' with the power of group
vhe.nry; (ii) the material should be intrinsically interesting and of
title sort, that is suitable for inclnsion in a general course of advanced
,rluan~uJn mechanics; and (iii) topics nlust not involve too much
p.pecialized background kno\vledge of particular branches of physics.
!:?he .view adopted throughout is t.ha,t group theory is not just a
specialized 1;'001 for solving a few of the more difficult and intricate
problems in quantuDl tbe,)fY. In advanced quantum mechanics
practiclA,H.y all general statenlents that can be made about a com ..
plica/ted systeln depend o.n its synlmetry properties, and the use of
group representations i:~ just a systematic, unified way of thinklllg

about and exploiting Ji..;hese s~(anJnetries. For this reason I have not
heE-dtaJted to include gj e~.p::.t:· results for vvhich one could easily produce
ad hoc l}roofs froI1l fiest, pri.nciples: indeed. it must always remain
true that the use of group theory could be circumvented by detailed
aJ.gebraic considerations nn ahnost all occasions. However, the
au.tJio~~ js convinced that. tIle essential ideas of group theory are
8ldlicieutly simple to make the time spent on acquiring this way of
thio1.i.ng ,veIl worfJJ ~Yhile.
A series of ex.D,mplei~ is appended to each section. Some of these
are 8.in.l~ple drill in the concepts introduced in the section; others,
pal'tieul~~.rl,y in later ohapters, indicate extensions of the theory and
fw.,ther applic3,tions. rrhose marked with an asterisk are more
difficult or requh'~ rtdditiortal reading, and are often suitable as
topics for revie,\' eSSt~.ys (a,has t~rm papers).
\iVith th.e th.ree; criteria for selection mentioned above, it has
of conr~e been quite h~lPDB8ible to do real justice to any of the
a,pplieationf!. to variou:1 branches of physics and chemistry that are
t •.:H~~~bed on in Chap{:Pfs IV to VIII. This appears to me wlavoidable
becauBe of the amount of background knowledge required for many
applications. It nlerely highlights the fact that in each of these
specialized subjects there is a need for a monograph ",-hieh uses
group theory from the b~ginning as naturally and as freely as the
Schrodinger equation itself. In this field the chenlists have already
led the way,t and the author hopes that t~e present book may
hasten the day "then the sanle applies in physics by providing a
convenient basic reference text.
It is a pleasure to acknowledge my indebtedness to Professor
B. L. Van der Waerden \vhose elegant book first inspired nly interest
in this subject. Also I am very ,grateful to Dr. S. F. Boys,
l


t See Eyring, WaJl,er and Kimball (1944) Quanturn Ghemi8try; and Wilson,
Deoius and Cross (1955) Jf olecular Vibrations.

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ix

PREFACE

Dr. G. Chew, Dr. R. Karplus, Dr. M. A. Ruderman, Dr. M. TInkham
and Mr. D. Twose, who have either patientl),- helped me to understand
aspects of their special subject, or have read parts of ,the marIUscript and made helpful commentso I alll indebted to Mrs. M. Rogers
and Mrs. M. l\'Iiller for undertaking the typing of the Dlanuscript,
and to M".r. J. G. Collins SIld .l\fr. D. A. Goodings who have generously
helped with the correction of proofs. Dr. E. R. (~ob.en h3-5 kindly
allowed the reproduction of his tables of \Vigner coefficient.s, and
D. Van Nostrand Co. similarly a figure .
Cambridge,

v.

~Jngland,

HEINE

NOTATION
Note: e is taken as the charge on the proton: all angulf1r moment·um
operators such as 1.., == (Lx) L y , L z} hnve the dilnensions of angular

Inomentum and thus contain a factor n, (except in § 18), whereas"
the quantuIIl nunlbers L, jlfL, etc., are of course pure numbers .

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Chapter 1

SYMME'fRY TRANSFORMATIONS IN
QUANTUM MECHANICS
1. The lJ"ses of Synlmetry Pl'operties
Although this book has been titled ('Introduction to the Present
·Use of Group Theory in Quantum Mechanics" in acoordti,nce with
customary uRage, a rath.er more descrlI)tive title 'vo~~ld h.!lve been
"The Consequen(~cF\ of Symmetry in QU8utnnl JYlecIJani·'",ij1 rl--le fact
that these sytllllietry properties D)nn \vhat Ir:H~t..h~rnat..i,::jancl ha.ve
terrned "groups" lS re~nji incidental from a J?hy~i~;dst~& pO'int of view,
th.ough it is vita.l to t)>1e mathemati.cal iGl'rn of ~,l ~"theory, Tt.h~ ir.
fact the sy7nnu:;trie:.~ of quantum rne(~hani(~r-.d SYBn:;tOLt thftt \ve shaH
he lllterested in,
.
J
'11 ijstrate IJl
. a pre1·Immar~r
·
BU.np1e eXfiJmp,es
L

1. fh) to11li\V1J.1g~nrec
way ~"hat is mt~aTtt by symmetry p1'operti~& >J.nd \lvhat their ID.ain
consequences are.
(i) It can be shown that the ,\Vl1ve function~ ¥/(rv T 2 ) (without
spin) of a helitnn. atoin a.re of tV'!O type8~ ~)~~ mraetri~ and anti·
symmetric, according to whether
Y

•

"0

'"

I. "

•

or
vlhere r! and r 2 are the position vectors of the t\VO electrons (Schiff
1955, p. 234). rrhe corresponding states of the atofll are aJAo referred

to as symnletric and 8nti .. symmetrie. Thus the eige!lfuncti{)l1[;
turn out to have ,vell defined symmetry properties v-rhieh ean there"
fore bp, used in elassifying and distinguishing ;;r.H the different
eigenstates.
(U) l"here are three 2[1 \vave functioni~ for a hydrugr-n atnln~
t/J(2px) =-= xJ(r),
,,,,here f(r) is a particular fUHction of r = l rl only (Schiff 195!-"


p. 85). No\v in a free atom there 3ire no special directions and 'Vb
can choose and lahel the x·) y- and z-axes as we please, so that the
three functions (1.1) must all correspond ·to the same energy level.
If, however, W'6 apply a magnetic fi(,·id in some particular direction.
the argument no long0r holrl.s, so t.hat \ve may expect the energy
level to be ~,pHt into several different levels, up to tbree in number,
1

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2

GROUP THEORY IN QUANTUM MECILA.NICS

In this kind of way the symmetry properties of the eigenfunctions
~an determine the degeneraey of an energy level, and how such 8,
degenerate level may split a·s a result of some additional perturbation.
(ill) The probability t·hE~t, the outer electroll of a sodium atom
jumps from the state .pi to t.he state t#2 with the ernission of radiation
polarized in the x-direction is proportional to the square of
".'0

M

00

GO

J J J"1* "2 dx dy dz


=

X

(1.2)

-OCi -00 -00

(Schiff 1955, p. 253). If the two states are the 48 and 3s ones, if;l
and .p2 are functions of r only. To calculate M in this case, we
make the change of variable x' = -x in (1.2) and obtain j f = -M,
i.e. M(4s, 38) = O. This trallsition probability is therefore determined purely by symmetty. The situation is rather different when
the transition proba,bilit~l is not zero. Suppose ifJl and ¢12 are the
4p:t and 38 wave functions X/l(lt) and f2(r). 'Ithen (1.2) beconles
co

M(4pxJ 3s)

=

ro

00

J J JA*(r) x f2(r) dx dy dz.
2

.By ulaking the change of variable x'
ClrtA


be replaced by

y2

or similarly by

M(4pz, 3s)

=!

c:o

= y,
Z2.

y' = x, the x 2 in (1.3)
Thus by addition

oa

f J Jfl*(r) r'12{r) dx dy dz.
--00 -CX)

(1.3)

(1.4)

-IX)

Sirllilarly the p.robitbilities for all possible transitions from any 4p
state to the 3.9 state or vice versa, with the emission or absorption
of radia.tion, polarized circularly or linearly in any direction, can be
I·ed1.1;~ed to the integral occurring in (1.4), the simple numerical fa,ctor
in fF'ont being determined purely by the particular direetion n,nd
I.~r Htate (;hOSeD~, Symmetry properties thus establish the relative
.af\gnHAldes of 8evara.1 matrix elelncnts of the form (1.2), their
dlusolute values h~ing then deierrrti.ned by the value of one integraL
This type of arguraent cXl)lains vrhy the intensities of the various
cOHlponents of a eomlJosite spectral line are often observed to bear
8iw.ple ratios to one another.
~rhese exarupIes ~rve to· illustrate ,vhcl,t is gerlardJly true:> na m.eJy
that Rymmetry l--'r'operties alio,.,,, us to cla.fjSify and }PlIbfJ ~ke e?:qe'.a{:/;a,tes of a qnanttUH nlechanlcal systenl. 'I'hey enahie UR if) diseuf-)s
q!;.aJitat]vely \vhat 8tpliU'i;'J,gs we may expeet in a degerlerate e!1crgy·
ir.rt"eI under some pert.urbation. They help in calculating ·lransitJe)H
probahilitJies and other rnatr£x elements, and, ie particular? in setting

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3

SYMMETRY TRANSFORMATIONS

up 8election rulea stating when these quantities are zero. In {jht}
following sectiollS we shall develop these kinds of symmetry argu~
Inent in a systematic fashion,. and shall see how they can be used
for the above three purposes in situations that are less elementa..ry
than the examples giVe!l above.

The real importance of synlmetry arguments in such situations
lies in the fact that for systems of interest the Schrodinger equation
is usually too complicated to be solved analytically or even nUlneri,.
cally without. making gross ~~tI-Jproxirnations. li'or instance, for ftE
atom with n electrons the equation contains 4n -variables (includini:.
spin) which are not separable. J-Iowcyer, the ~~yanmetry "propertie:~',
of the equation may be relatively simple, so that, symmetry argu·
ments can easily be applied to the problem. ..L\.nother irnportant,
point about symmetry argulnents is that they are ba~cd on the
symnletry of the Schrodil1ger laquation itself, so that the.}'" dt) not
h"1volve approxilnations, in particular those used to obt8~in, o.;ppr~}xi.,
mate eigenfunctions of the equation. In fact the bea.uty of the
method lies in the fact that, for insronce, aI' n elect.ron p'r:oblem can
often be treated as simply and. as rigorou51y a.~ a one eleotron proolem. .J:\ t, the present time the most specta,cular in ustratio.uL~ of these
t,vo aspects of symrnetry art!uments occur in nncler;..!" f~,nd fundt1~
mental particle ph:VRics. lI'he shell-model theory of ~-.he enf:Tgy
levels of nuclei ha,s beeIl dey€!loped~ with selection f111eft f{!l' various
transitions, etc., all withDut a:n exact kn.owledge of the interaction
between two nueleons. ShnilHlrly it is possible to di;;otu:i':; \)entatively
the relRtionsbips bet.ween the various funcin·tuer.tal l)f',} dcles nnd
gi ve selection, rules for trHtl1sitions between then1) lvhj(jh are based.
'purely on symmetry ideas, ~~uch as spin, char~~·0 C(lnj).lgult:~n.;, isotopic
spin and parity, '\\ithout tho sHghtest ul1de:'sii3,.c6?ng of the fieJd
eq uations describing the interactions of all these pc',Tti~;les.

2.

l~xpressing

SYIllmetry' ()perations

lVlalh~~r:natical1y

~Iany

of the sYlumet,ry pfopect,ies that "\\'e snaJl he Cl)r~Cf~rned v.'ith
involve rotations. so thf~t we shall startl by cOl.1sir.h:.ring ho,,· 8,
physical operation such aiR rotating a system. i~ e;~presst)d mathenlatical1y.
Consider a body with a point I) on it '\vh.ieh has (;o-ordinates
(x) y, z). If 'w'e rotate the body cloek\vise by fLrl angle O! (Fig. 1),
i.e. ,ve rotate by - u. about the z-axis in thl conventional srnsc: the
point P moves to the position P'(}(, 1", Z), \vhcre
b

OA = O(} := OB COR ct '._. BP' sin a
AP -~-: OP' ==-= OB sin (t -i- .BP' cos:. C!,

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(2.1)


4:

URQUP THEORY IN QUAN'l'UM MECIIANIOS

I =X
Y sin
i y = X sin a: + },. cos ex,
L z.

~

I.e.

X

COS iX -

iX,

I

(2.2)

z =

FlO.. L

J~~ota.tiOT~

of a point P to lol'.

Here ~lH.ll~Ibev~,-h£;rl~ 'Ute ~>, 4- and z-ax?s are chosen to form a right.
hanned set. If;·)v.-~-"'\; ~~r~ iPf~t!~~aJ. I,){ rotating p~ '~'l-:~ can also consider

the hody l.ro.d [, a~ fL~ecL had refer all co~ofdinat.es to a new Dair of
,,..
J"-) -".
L
I '"

dO'
~"xe~ O.£l- ann { .1' VlnlGY.l nH.Ke an angle + ex Wlt:1 (kc an
y (pT,~.19. 2) .
~,

?

~

1

FIG.

"I

2e Rotation

•

ofax~.

V{o have ant-llogo!lsl.v to (2 1)

()E' -::--; () J) cos
E TJ ==-- 01) '-Jin

D P sin CI
+- DP cos :\

,x - IX

co~ordina,t€H (LY~ :r~ Z) of
nrt.~ relattxl to (3~, y, z) again hy (2.2).

so that the

P referfiJd to the l1t.nv axos

'Thus t.he bingl? tl'au;iforrnntiol1 (2.2) can represent eit.her the change
in the ~)o-or-..iin.a,te8 of a point Vtrhen we rotate a body hy an angle
- - (I, or the change in the co-ordinates of a fixed JXdnt ·w"hen. ,,~e
totat.e the co .. ordi'YIalf~ axes by a,n angle + t.X. The clos~ r~]tltionship
betvr--een these t\l-O opera tion~'j is directly evident f1'o111 the si.milarity
'bet,wet~n }lgs. 1 and .2. 'fhc- tV!O different points of viev;~ also ari~~e
,vherl eon.sidpring the sY1l1metry properties of a phYbical s}'Etem.

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SYMMEtrRy TRANSFORMATIONS

5

Consider for instance a perfectly round. plate without any markings
on it: we say it iR symmetrical about 3, vertical axis through it.~
centre, say the z«axis. We e.an express this more precisely by saying
that if we rotate the plate about its a.xis \ve cannot tell that we
havG rotated it because it is completely roun.d with no markings 011
it. On the other halld we could also say that for a fixed position of
the -plate, the various phYRica,l properties such as momerlts of

inertia assooiate(i with the x- and y-axes mU8t, be the same, no
matter in what directions these axes are Ch~J8enll In. this example
the first approach is perhaps more natural, but when discussing t.he
symrnetry of the Schrooinger equation fOl a ph:ysical systeln we
shaH adopt the St3COlld point of viev;r. _tlnticipating a little, we shall
be considering a given equation and the forms it take.s when expressed
in terms of different variablc~ like x, y, z and X, l"', Z which cor·
respond to using different co-ordinate axes. ~'here are two reasons
for this choice. Firstly, the Scllrodinger equ8Jtion if; a mathematical
relation and not like a plat,e so that. we cannot rotate It in quite
the same Rense, though we could, of coarse, \vrito down t.he equation
for the rotated physical system. Expressing a.n equatiol1 in terms of
different sets of co-ordinates is a more familiar concept,. Secondly,
l\-~C shall be considering some tranSfOl'Inations of cn-ordinates that
have no shnple physical analogue. l?or irlstancc, we ean carry out a
rot.ational transformation of spin co·ordinat€,s vrithont alterLllg the
position vectors rj of the electrons in an atnrrl, bilt ~hat d(1),,,,~ it.
mean physically to rotate an atom in spin spa(-;e ",TuBe holding it
fixed in ordinary space? Nevertheless the transformatic)nB of
co-ordinates lvhich we shall apply to the Srhrndinger equatioH_ ~ril1
usua,lly be suggested by and linked 'Nith the physicR.l aymmetry of
the system in an obvious way.
·When discussing linear tranSfOl"Ina tions of co~ordinates, it is
convenient to refel t.o them by a ~single ~yln 11U1 such as 11. ~"'!or
instance, ,",ye shall ca.n the tranSfOI'1l1:J,tion (2.2) the trant,~fcrrma.tio'-n( .R~
or bees.use it correRpoIlds to a rotation, lh.-e rotation R. If it is neceJ3·
sary to be apecifin hbout the angle f.Jf rotation, ,ve khalJ calt ~2.2)
the rotation R(a, z) of -.~ a. about t.he z-axis :Jeeause this sign t~Q1"­
respondtj to the change-of-axes point of vic-w ~vhich we are adopting.
We ha va already discussed .in connpction \\"i th Fig 2 t,he effeet of

applying a tra.nsformation such as .R on the cu-orrlin::ltes of a point.~
and we shall now make the following preliminary definition of what
it means to apply R(a., z) to a function of x, y. z. In § 5; rtasons·will
appear f01'> repla[;h~g this definib.on by a slig}uJ:./ enlarged concept .
. .4pplying the transfOrTI1Al,tion R(rx, z) (2.2) to a /u,ru:tior£ J (t~ y, z)
.,neans to 8uLstitute the €xIJre88io]?8 (2.:!) for ~1';> ;i! Z ~n the j"unciion
1

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6

GROUP THEORY IN QUANTUM MECHANICS

and eh'U8 expreas f in terms of X, Y, Z. This results in a function of
X, Y, Z which in general diIJplays a different functional form !rorn
f(x, y, z). For instance applying R(tX, z) to t4e function (x - y)2,
'w'e obtain
.
(x - y)t

= [(X cos a; = [X(cos tX -

Y sin tX) - (X sin C( + Y cos
sin tX) - Y(cos tX + sin tX)]2,

~)]2

(2.3)

which is a different function. of X. Y, Z. Similarly we can apply
a transformation to each side of an equation. For instance the
equation

!

(2.4)

(x - y)2 = 2(x - y)

beoomest
(cos

IX

a~ -

sin

IX

=

O~T)[X(COS
2[X(cos

tX -

IX-

sin

sin .:t) --- Y(cos

ttl ,---

JT(COS ex

IX

+ sin

1%)]2

+ sin «)],

(2.5)

which is still a correct equation as ca,n easily be verified.
PROBLEMs

2.1 Apply the transfonnation R(rx, z) (equation (2.2») to each

+

of the following functions: (a) exp x; (by (x
iy)2; (c) x 2 -t- y2 + Z2;
(d) xl(r), yf(r), zJ (-r).
2.2 Write down the linear transformation that corresponds t,O a

rotation of tX about the y-axis, and apply it to each of the functions
of problem 2.1.
2.3 The Schrodinger equation for a simple harmonie oseillator
of frequency w is

(- :~ :;: + tmw2X2)~(X) =

E",(x).

where t/J(x) is an eigenfunction belonging to the enerffjT value E.
By operating on the equation -W~HJl the tra.nsformation l' ..-= --X,
show that {I( --- x) is aJso an eigenfunction belonging to t.he sa[':~e energy
level and so are Ơ-,(x) + if;{ -x) and if;(x) - Â;( ··_-x).

3. Symlnctry Transformations of the Hamiltonian
\V(~

the

shn]l

'fJ.O\Y c~pply

tirne.,~ndijponder,t

linear transforrnations like R(a, z}' (2.2) to
Sehrodinge:r t~quation
(3.1 )
ax is

gho~"n

in

,~tny

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elementary calculus

te}~t,.


7

SYMMETRY TRA.NSFORMATIONS

where .Tt' is the Hamiltonian operator and E the energy value
belonging to the eigenfun\.t~ion tfo. It is convenient to consider
first the effect of a transformation on the Hamiltonian :Tf.
The Hamiltonian for an atom with 11, electrons, considering the
nucleus as fixed and omitting Spitl dependent terms, is (Schiff
1955, p. 284)

(3.2)
where m is the mass of an electron, e the charge on a proton, and
2

82
()2

4.-_+2

0
V,,2 _ _
Ii

-

OXl

I

Oy,2

(3.3)

&,2 1

(a.4a)

If we apply the transformation R(rx, z) (2.2) to the co-ordinates
(XI, Yt, z,) of each of the n electrons, we have
r,2

= Xt 2 + yl2 + Z~2
(X, cos C( = XI! + 1",2

:=

Y, sin «)1

+ (Xc sin. +
C(

Y, cos cx)2

+

Z,2

+ Zl~.

Similarly
r'1 2 = (Xi - XJ)2

+ (Y( --

Yj)2 -t- (Z, - Zj)2,

(3.4b)

and it can easily be shown thatt
02

62

02

02

-;-a
+ -!l..c/Yi..•..2 + v~
~.2 = ;X 2
fiX,
u (

(32

82

+ 7~y2
+ '~r;~-;;.
(, I.
v.u,'"

(3J)

Thus 8ubstitut,ing these relations into (3*2), we see that the Jlarnil~
toman has precisely the Sftme form ~dlcn ex~n'essed in t.erms of t.he
(X1' l?t, Zi) co-ordinates as in term8 of t.he (l"i, YI" Zt) ocj .. ord~tes,
i.e.
(3.6)

hy gaJyjn~ that the transformation R(a., z) leaves
,;!P(3.2) unchu,llged, cr I\((I., z) leaves ~ invariant, or.?R is invariant
'U/i'!..der R(a, zj, or li(r.x . z) is a symm.etry transformation of 7t'. A
sy'mtneJry tran·~for7natio/n (if a Jlamiltonian is defined as a linear

This is

e:x.pr(;~~~d

t The t1"an.sforrnstion of difreeentiu.1 operators is discussed in any elementary
ceJculns text.

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8

GROUP THEORY IN QUANTUM MECHANICS

traM/ormation of co-ordinates which leaves that Ilamiltonian int'ar'iunt
in the 8ense of equation (3.6).
The reason for applying linear transformations like R(a., z) (2.2)
to & Hamiltonian noW' becomes a little clearer. We have seen that
R(a., z) leaves the Hamiltonian (3.2) invariant. Howeyer, R(a., z)
applied to the eigenfunctions of the Hamiltonian docs not in general
leave them invariant.. r.JOnsider for insta.noo the 21) 'vav,:.': functions
for a hydrogen atom (example (ii) of §l). R(a., z) applied to x/(r)
gives (X cos (X - Y sin r.t) j(R) l\'hich haa a different functional
form. In particular for IX = 90 0 we obtain - .. Yf(R) so that ll!(fX, ~;)
has changed one eigenfunction into another. More generally con..
sider a Schrodinger equation
Jf7(Xt, ?/I, Zt) .pl(Xi! Yi) z,) = EtPl(Xf, !fl., Z,),

(3.7)

Applying nny symmetry trallSformation 1.1 l\1C obtain

,J'f(Xt, Y!, Zt)

tf;~(X~~

Jr{, Zd

=

E.p2(X f , J7'"

Z1)~

\vhere YJ 2 in general he.s a different functional form from

(3~8)
VJ1'

'rhus

~~(XiJ

Y t , Zt) is an eigenfwlction of £~(X(, Y,: Z,), but since
£;(Xt, }"t, Zl) and Jt'(Xj, y,~ Zt) have tJle same (orIll, we can also 8ay
from (3.8) that ¢2(Xl, '!If-t Zl) is an eigenfUftlctian.. ()f eYC'{Xil YI, z~) a.rut
belongs to the sante eigenvalue E as
An ~,}t.ernative Inet.hod of
vlording this argurnent is tlJ say that sinoe (3.8) is a differentia} eq1l8,~
tion in terms of thE: varialJles Xt~ Yt't Z" "ie can replace X(.~ X"'i, Z;
hy XI, y" Zl or any other Ret of f~ymbols throughout without upsetting
the validity of the equation. "rhus (3.8) becomes

+1·

~(Xf'

y", z,) tP2

(Xt~ !It ,Zi) :.:~> EtP2(Xt,

y" Zj),

(3.9)

which is just our previous conclusion expressed in symbols. Thus
we. see that tke aymrneiry tra~~lo-rmation,s of a Harniltoni.fllz, can be

used to relaie the different eigen;fu/lwtions of one energy le?,el to one
aTloth~r and hence to label thern. and to discuss the degree of degeneraey of the energy level. Before \fe cain pursue this further (§ 6), we
InU&t dir;cuss in grentei~ ,3etail the SYlTIJ11etry t,ransformations of
Ifauliltoruans {§§ :J and 4/~ ,'}nd their ptTect on '\~tfiVe fllnctior:.s (§ 5).
l'he Ha1uiltonia n (it2) hH,~, two ot.her type8 of sylnnletl'Y transforp.J.c.ttion besides the rota,tion R. The transforrnation

r

(Xl' Yl' Zl) =

(X"

Y g•

I (X2' Y2' Z2; ~-::: (Xl~

Z-;;-----

J'l' Zl)

l

I;--.____
(Xi, Yt, Zt) = (X i, L, Z;),
'i=~ 3, ± .. . ?t. JI
. ____ . _. _____
___________
~__.

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(3,10)


9

SYMMETRY TRANSFORMATIONS

is called the interchange or permutation of the co-ordinates 1 and 2,
a.nd is a symmetry transformation of (3.2,) as is obvious by inspection.
Similarly any permutation of the co-ordinates Xi, y.", Zt, i = 1 to n,
is a symmetry transformation. The other synlmetry transformation
is the inversion trruudormation

n

en:

Xi

=

--X..

Yt

= -Y"

Zj

= -Zt

for all

....--.

i.]

(3.11)

p;-

This can be cQrrlbined with the rotations. 'An ordinary rotation such
as (2.2) is called a proper l~otation, and the combination of a proper

rotation with the inversion n is called an i1r~proper rotation.. As a
particular example of an improper rotation, we have IIR(180°, x)
which is just tIle reflectio:n mx in the mirror plane x = 0, i.e.

~:

(X" y"

Zj)

= (-Xt,

Y.~ Z,)

for aU i.

I

(3.12)

It can easily be verified that all inlproper rotations, as well as
proper ones, leave the Hamiltonian (3.2) invariant. However,
there are many simple and important transformations that are not
symmetry transformations of (3.2), for lllstance the transformation
to cylindrical polar co-ordinates
(3.13)

This transformation is in an:r case not a linear one because it illvolv~
products of R, with trigonometric functions of
Also V,I becomes

e,.

1.

a (' t a)

B, aBc R oBt !

1

8

2

8

2

+ R,'!. 08,2 + OZ.I'

(3.14)

which is not identical in fornl with (3.3)$ so that (3.13) is Mt a
8ymmdry transformation. Of course we may wish to express the
Hamiltonian (3.2) in terms of cylindrical polar co-ordinates for some
problem, but in the future we shall refer to such a transformation
as a change to polar co-ordinates, so as to avoid confusion with
symmetry transformations which we will be considering 80 much
that it will be convenient to refer to the latter simply as transformations.

We must now indicate briefly what the symmetry transformations are for the Hamiltonians of physical systems besides free atoms
and ions which we have been considering so far. An atom has complete spherical symmetry, i.e. it is invariant to any rotation about
any axis (cf. problem 3.7), so that it has a higher degree of symmetry
than molecules and crystal lattices which are usually only invariant

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10

GROUP THEORY IN QUANTUM MECHANIOS

to certain rotations about certain axes (cf. problems 3.4 and 3.5).
Thus the latter have some of the symmetry transformations of the
atom, but not any radically new ones except for the translational
symmetry of a crystal lattice. We ha.ve therefore already mentioned
in connection with (3.2) almost all the types of symmetry transformation which we shall discuss.
To sum up, the form of a Hamiltonian remains unchanged by
certain linear transformations which are caJIed symmetry transformations of the Hamiltonian. Symmetry transformations in
general change the eigenfunctions of one energy level into one
another.
PBoBLEMS

3.1 Show that the following co-ordinate changes are not symmetry transformations of the Hamiltonian (3.2).

= (2X" 2YI, 2ZI), i = 1 to n.
(b) (Xt, Yl' %1) = (--Xl' _}7l' -. Zl)'
(Xt, Yi, z,) = (Xi, Y" Z,),
i = 2 to n.
(c) Xi = exp X(,

y, = exp Y"
z, = exp Zl,
(a) (Xl, '!iI, z,)

i = 1 to n.
(d) Xl' YI' Zt given in terms of Xl' Y I , Zl by equation (2.2),
(Xi, Yt, Zi) = (X(, }"'l, Zt),
i = 2 to n.
(e) XI = R, sin 8, cos y, = RI sin 8, sin tPt ,
%1 = RI cos 8 t ,
i = 1 to n.
3.2 Express the Hamiltonian (3.2) in terms of spherical polar
co-ordinates r, 8, cP, where

x = r sin

ecos cp,

y=

r sUi

esin 4>,

%

= r cos 8,

(Schiff 1955, p. 69).

Show that the inveI"Sion transformation II takes the form

r, = Bl,

0,

= 11' -

81,

cp~ ==

1T

+ f/J"

i

= 1 to n,

and express also the rotation (2.2) and the other symmetry transformations mentioned in § 3 in polar co--ordinates. Hence, verify
that they again leave the Hamiltonian (3.2) invariant.
3.3 Write down the Hamiltonian without spin dependent terms
for an ion of nuclear charge Z with n (not equal to Z) electrons, and
show that it has the same symmetry transformations as the Hamiltonian (3.2). Do the same for the one-electron Hartle(' equation
(Schiff 1955, p. 284) for the single valence electron of a sodium atom.
3.4 Write down the Hamiltonian without spin dependellt
terms for the two electrons in a hydrogen molecule, considering the

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11

SYMMETRY TRANSFORMATIONS

two protons as fixed at the points ± (0, 0, a).. Show that it is
(al) invariant under any rotation about the z-axis, but only to 180 0
rotations about the x .. or y-a.xes, (b) invariant, under reflections in
the plane through the origin perpendicular to the z-axis and in any
plane containing the z-axis, (c) invariant under the inversion II
and the reflection irl the z-axis
i

= 1,

2,

and (<1) invariant under the interchange of co-ordinates 1 and 2.
3.5 III problem 3.4, assume that one of the protons haS been
replaced by a deuteron, and suppose that the deuteron has a slightly
different charge from that of the proton . vVhat effect does this have
on the symmetry properties of the Hamiltonian'~ Although in
reality the deuteron and proton have the same charge, they do have
different masses and magnetic moments, and this would affect the
symmetry of the problem in a similar way to the fictitious di~rence
in charge if the interaction with the nuclear moments were included
in the Hamiltonian.
3.6 Repeat the discussion of problem 3.4 in terms of spherical
polar co-ordinates and in terms of cylindrical polar co-ordinates

(equation (3.13)). Which set of co-ordinates do you thiIJ.k is most
convenient for this problem 1
3~7 A rotation about the origin. can be defined rna.thematically
as a linear transformation of co-ordinates that leaves invariant the
distance of an arbitrary point (x, y, z) from the origin. Using this
definition, show that the Hamiltonian (3.2) i~ invariant under any
rotation about any axis. Show that the definition includes the
improper as well as the proper rotations (Margenau and Murphy
1943, p. 310).
3.8 Show that an improper rotation of 1800 about any axis is
the same as a reflection in the plane through the origin perpendicular
to that axis.
3.9 Write down the Hamiltonian for a hydrogen atom in small
uniform electric and magnetic fields parallel to the z-axis (Schiff
1955, pp. 158, 292) omitting spin dependent terms and considering
the nucleus as a fixed c~~rge. Also assume that the 2p eigenfunctions

"'1 =

x

+ iy
y2 /(r),

x - iy

"''J. =, y2

/(r),

"'8 = z/(1'),

where J{r) is given by Schiff (1955, p. 85), are still eigenfunctions
to a first approximation in the presence of the fields. Prove (a) the
2p level is three-fold degenerate in the absence of the external fields,
(b) in the presence of the electric field only, ,pI and tP2 are degenerate

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12

GltO~

TlJEOkY IN QUANTUM MECHANICS

with one another but need not be degenerate with t/1s, (c) in the
presence of the magnetic field only, symmetry arguments do not
require any of the functions .pI' ~2 and .ps to have the same energy
80 that the 2p level Dlay be split into three levels. Hint: in each of
the cases (a), (b) and (c) test whether the reflection in the plane
y = 0 and the rotation ~:.f 90 0 about the y-a~is are symmetry transformations. If they are, U8e t.hem to apply the argument of equations (3.7), (3.8), (309) to each of the functions ~l' ifi2' t#s. Try also
the inversion fl, rotations about other axes and other reflections to
ensure as far as possihle that no degeneracy required by symmetry
has been rnissed.

4. Groups of Symmetry Transformations
In this section we shall illustrate and define what is meant by
a group in the mathematical sense of the word, and shall show what
relevauce this concept has to the symnletry transformations of

Hamiltonians.
E~ample

of a group
Let us first consider the g~ymmetry properties of a particular
physical object, namely an equilateral triangle cut out of a piece
of cardboard having the same fulish on both sides and lying on the
tahle with its vertices at the point.s 1, 2 and 3 and its centre at the
origin of co~ordinates (Fig. 3). Ok, Ol, Om are three other axes

I

I

I
I
I
I
I

'" '...

,I

",'"

....... .::-----"r---~x

01

I
I
I

2~----~--------3

k
FIG.

3. Axes for an equilateral tri&Ilgle.

perpendicular to the sides, Ok being identical with the negative
y-axis. A rotation A of 1200 about the z-axis Dloves the vertex
that was at the point 1 t.o the point 2, etc., and we shall call this an
equivalent position of the triangle since it is indistinguishable from
the original position. It can easily be seen that the following

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13

SYMMETRY TRANSFORMATIONS

rotations all leave the triangle in equivalent positions and that
there are no other proper rotations that do this.

A: 1200 about z-axis
B: 240° or -120° about z-axis
K: 1800 about Ok axis

L: 1800 about Ol axis
M: 180° about O1n axis
E: 0° or 360° about any axis, I.e. no rotation.

(4.1)

If we apply two rotations successively, for instance first A and then
K, this moves the top vertex from position 1 first to position 2 and
then to 3, the vertex at the position 2 to 3 and then to 2, the vert.,:,,·
at 3 to 1 and then to 1. ThuB the combined operation A followed by
K is identical with the single rotation L. Sirnilarly K followed by A is
the same as M, and it can easily be verified that combining any pair
of the rotations (4.1) in either order gives another rotation which
is also one of the ones listed in (4.1). If the rotation F applied
First followed by S applied Second is equivalent to' the single
Combined rotation 0, we write

SF =C,

(4.2)

where it is customary to write the S before the F in analogy with
differential operators. For instance
d
x 2 dxf(x)

(

.f

~

..

":t'.v}

means first differentiating f(x) and then multJiplying the result by
Z2. This is clearly not the same as

(4.4)
and similarly when combining rotations it is inlportant to follow
the convention of (4.2). We have already seen that

XA = L,

..:4K == M,

(4.5)

and similarly it is possible to write down a whole multiplication
table (Table 1) where the rotatioll in the top row is applied first and
the rotation in the left column second. There is an important feature
of Table 1, namely that for every rotation P, t.here is also a rotation
p-l, say, which undoes the effect of P, and that P a:Iso undoes the
effect of P-l, i.e.
pp-l = p-lp ::== E.
(4.6)
In fact in every case P and p-l a,re jURt two rotations by the same

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14

GROUP THEORY IN

QUAl't~UM

MECHANICS

angle about the same axis but in' opposite directions. When the
angle is 1800 this of course makes p: and p-t identical. It can also
be verified from the multiplication table that the triple products
TABLE.l

Multiplication Table for the Group 32
E

A

B

K

L

E

A
B

B
E
A

K
M'
L

L

Prl

K
Prj

L
K

M
K
L

E

A

B
.A.

E
B

B
A

E
A
B

A
B

K

K

L

L

L

At

Jf

K

M

applied
second

E

M

applied first

E

P(QR) and (PQ)R are always the same, so that they can be written
unambiguousl~T

as PQR. Alternatively this follows directly from
the physical nature of rotations as can easily be shown. These
properties suffice to establish that the rotations E, A, B, K, L, M
(4.1) are the element8 of a, group.
Definition of a group. A group Q3 is a collection of elements
A; B, '0, D, . . . . . . . which have the propertie8 (a) to (e) below. The
elements in the simplest cases may be numbers. They may also
be any other quantities such as matrices, physical operations like
rotations, or mathematical opArations such as making a linear
transformation of cOaordinates.
(a) It must be possible to combine any pair of el.e'ments F and S
in a definite v)ay to form a comb·i1ULtion 0 which we shall write

(4.7)

a

where as before F is the first element, 8 the second element and
the combination, if the order of F and S is important. In our
example with the elements (4.1), the law of combination was "first
apply rotation F and t.hen S". With other groups the law of combination may be matrix multiplication or like addition. If for two
elements PQ = QP then P and Q are said to cmnmute, and if this
is so for every pair of elements then the law of combination is
commutative and the group is Abelian.
(b) The cornbination G = SF of any pair of elements F and S
must also be an elemen.t oj the group. Thus a multiplication table
among the group elements can always be set up like Table 1.

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SYMMETRY TRANSFORMATIONS

(c)

15

One oJ the group tlement..s, E say, must Mve the propertie8 oj a

unit element, namely

EP=PE=P

(4.8)

for every element P ~ For instance omitting all reference to E would

make it impossible to set up .. complete multiplication table for
the other rotations of (4.1) (cf. Table 1). This is related to the next
property.
(d) Every elem.ent P oj the group mwt have an in'l)erse p-l which
a'18o belongs to d3 with the property
pp-l

=

p-lp

= E.

(4.9)

(e) The triple product PQR m'tUJt be uniquely definetl, i.e .
P(QR)

=

(PQ) R = PQR.

(4.10)

This is true for all the kinds of elements and laws of combination
that we shaJI wish to deal with, but there are examples where it
does llot hold, e.g. 24 -7- (6 -7- 2) i= (24 -;- 6) -7- 21
Two simple examples of groups are all positive rational fractions
ex~luding zero with the law of combination being mult,iplication, and
all positive and negative int,egers inoluding zero with the law of

combination being addition. In the latter case it is inter~8t.ing that
zero plays the role of the unit element E. The permutations of n
objects, i.e. the operations of rearranging them and not their different
8;rrangem.ents in a row, say, form the l)errn'utation group pn also
known as the syrnmetric group Sn. The proper rotations by all
possible angles about a fixed axis form th~ axial rotation group.
This is clearly Abelian. The full rotation groulJ (Chapter II) consists
of all proper rotations about all axes through a point, and this
becomes the full rotatipn and re/lect'ion group when all improper
rotations are included. There are thir:;y.. t'~,.o groups of particular
interest forn1ed from a finite number of particular rotations about a
point and are known as point-group8 (§ 16). Thc~~e clearly do not
include all possible finite groups of rotations because, for instance,
the rotations by 360 r/n degrees about JJ fixed axi~~ "There r = 1 to n,
always form a group of n· eleluents. jill exanlple of a point.-group
is the group (4.1) which is called 32 (prnnount:,~d Htlllree t,wo", not
"thirty-two") in the international jJr)t;ation, to denote that it
includes some two·fold axes (10tation.:~; hy 18<)<,') p~rpendicular to a
three-fold axis (120°, 240°). All the pr'Yper Ei:nd improper rotations
that move a cube to an equiva,lellt pp3it,.1.0n form t·he full cubic
group m3m. In the older SchoeDf:iier:-l~; llotat.ion. th.€se two pointgroups a,re called Da and 0,.. All sqnare matrices of a given order

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