Group theory in quantum mechanics; an introduction to its present usage

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:,a

S 2' 0 :35 nHuntQr

t Ie e

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ix

2 Expressing symmetry operations mathematically 3

3 Symmetry transformations of the Hamiltonian 6

II THE QUANTU:M THEORY OF A FREE ATOM

8 The irreducible representations of the full rotation group 52

9 Reduction of tho product representation D(/) X DU/) 67

10 Quantum mechanics of a free atom; orbital degeneracy 73

11 Quantum mechanics of a free atom including spin 78

13 Calculating matrix elements and selection rules 99

III THE REPRESENTATIONS OF FINITE GROUPS

17 The relationship between group theory a.nd tho Dirac method 143

IV FURTHER ASPECTS OF THE THEORY OF FREE

ATOMS AND IONS

18 Paramagnetic ions in crystalline fields

19 Time-reversal and Kmmers' theorem

20 Wigner and Racah coefficionts

21 Hyperflne structure

22 Valence bond orbitals and molecular orbitals

23 Molecular vibrations

24 Infra-red and Ram~ spectra

·VI SOLID STATE PHYSICS

25 Brillouin zone theory of simple structures

26 Further aspects of Brillouin zone theory

27 Tensor properties of crystals

vi CONTE~~S

V1I NUCLEAR PHYSICS

28 The isotopic spin fornudism

29 Nuclear forces

30 Reactions

VIII RELA'!"IVISTIC QU"ANTUM MECHANICS

31 The representations of the Lorentz groLi'"

32 The Dirac equation

C Theorems on vector spac0s and group representations 412

E Irreducible representations of Abelian groups 420

F Momenta and infinitesimal transformations 422

H The irreducible representations of the complete Lorentz group 428

I Table of Wigner coefficients (jj' mtn'IJjl) 432

J Notation for the thirty-two crystal point-groups 446

K Charaoter ta.bles for the crysta.l point.groups 448

L Character tables for the axial rotation group and derived groups 455

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 elemen-tary 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

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.l 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 chen lists 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,

t See Eyring, WaJl,er and Kimball (1944) Quanturn Ghemi8try; and Wilson,

Deoius and Cross (1955) Jf olecular Vibrations

PREFACE ix

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 marIU-

script 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

NOTATION Note: e is taken as the charge on the proton: all angulf1r moment·um

operators such as 1 , == (Lx) Ly , L z} hnve the dilnensions of angular Inomentum and thus contain a factor n, (except in § 18), whereas" the quantuIIl nunlbers L, jlf L, etc., are of course pure numbers

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·'",ij1Y

• 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,

fh) to li\V1J.1g~nrec BU.np e eXfiJmp,es L ijstrate IJl a pre Immar~r

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

or

vlhere r! and r2 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,

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 00 GO

-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 mined purely by symmetty The situation is rather different when the transition proba,bilit~l is not zero Suppose ifJl and ¢12 are the

deter-4p:t and 38 wave functions X/l(lt) and f2(r) 'Ithen (1.2) beconles

co 00 ro

M(4pxJ 3s) = J J J A*(r) x 2 f2(r) dx dy dz (1.3)

.By ulaking the change of variable x' = y, y' = x, the x 2 in (1.3)

<rJ c:o oa

M(4pz, 3s) =! f J J fl*(r) r'12{r) dx dy dz (1.4)

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

SYMMETRY TRANSFORMATIONS 3

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 lem .J:\ t, the present time the most specta,cular in ustratio.uL~ of these

proo-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

~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 nlatical1y

mathe-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 thlb conventional srnsc: the point P moves to the position P'(}(, 1", Z), \vhcre

AP -~-: OP' ==-= OB sin (t -i- .BP' cos: C!, (2.1)

4:

I.e

URQUP THEORY IN QUAN'l'UM MECIIANIOS

I X = X COS iX - Y sin iX, I

O ,, J"-) -" ? L 1 "I • I '" dO' T,~ 2)

~"xe~ £l- ann { .1' VlnlGY.l nH.Ke an angle + ex Wlt:1 (kc an y (p 19

FIG 2e Rotation ofax~

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

()E' -:: ; () J) cos ,x - - D P sin CI

E TJ == 01) '-Jin IX +- DP cos :\

so that the co~ordina,t€H (LY~ :r~ Z) of P referfiJd to the l1t.nv axos nrt.~ relattxl to (3~, y, z) again hy (2.2)

'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

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.xis1 \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

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

= [X(cos tX - sin tX) - Y(cos tX + sin tX)]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

beoomest

(cos IX a~ - sin IX O~T)[X(COS I X - sin .:t) - Y(cos IX + sin 1%)]2

= 2[X(cos tX - sin ttl , - JT(COS ex + 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;

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 ¥-,(x) + if;{ -x) and if;(x) - ¢;( ··_-x)

3 Symlnctry Transformations of the Hamiltonian

\V(~ shn]l 'fJ.O\Y c~pply linear transforrnations like R(a, z}' (2.2) to the tirne.,~ndijponder,t Sehrodinge:r t~quation

(3.1 )

ax is gho~"n in ,~tny elementary calculus te}~t,

SYMMETRY TRA.NSFORMATIONS 7

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

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( - Y, sin «)1 + (Xc sin C( + Y, cos cx)2 + Z,2

= XI! + 1",2 + Zl~

Similarly

r'1 2 = (Xi - XJ)2 + (Y( Yj)2 -t- (Z, - Zj)2,

and it can easily be shown thatt

(3.4b)

-;-a + -!l • 2 + ~.2 = ;X 2 + 7~y2 + '~r;~-;; (3J)

fiX, c/Yi v~ u ( (, I v.u,'"

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)

This is e:x.pr(;~~~d 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

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

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)

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 = 900

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

\vhere YJ 2 in general he.s a different functional form from 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 +1· 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

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 eraey 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 trans-

degen-forp.J.c.ttion besides the rota,tion R The transforrnation

r (Xl' Yl' Zl) = (X" Yg• Z-;; - l

I (X2' Y2' Z2; ~-::: (Xl~ J'l' Zl)

I ; (Xi, Yt, Zt) = (X ~ _ _ i, L, Z;), _ J 'i=~ 3, ± ?t I

(3,10)

SYMMETRY TRANSFORMATIONS 9

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)

-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 e, Also V,I becomes

1 a (' a) 1 82 82

B, aBc Rt oBt ! + 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 trans-

formations

We must now indicate briefly what the symmetry

transforma-tions are for the Hamiltonians of physical systems besides free atoms and ions which we have been considering so far An atom has com-

plete 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

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

trans-formation which we shall discuss

To sum up, the form of a Hamiltonian remains unchanged by

certain linear transformations which are caJIed symmetry

trans-formations of the Hamiltonian Symmetry transtrans-formations in general change the eigenfunctions of one energy level into one

another

PBoBLEMS

3.1 Show that the following co-ordinate changes are not

sym-metry transformations of the Hamiltonian (3.2)

(a) (Xl, '!iI, z,) = (2X" 2YI, 2ZI), i = 1 to n

(b) (Xt, Yl' %1) = ( Xl' _}7 l' - Zl)'

(Xt, Yi, z,) = (Xi, Y" Z,), i = 2 to n

(c) Xi = exp X(, y, = exp Y" z, = exp Zl, i = 1 to n

(d) Xl' YI' Zt given in terms of Xl' YI , Zl by equation (2.2),

(Xi, Yt, Zi) = (X(, }"'l, Zt), i = 2 to n

(e) XI = R, sin 8, cos <P.", y, = RI sin 8, sin tPt ,

trans-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 tonian (3.2) Do the same for the one-electron Hartle(' equation

Hamil-(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

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

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

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 900

about the y-a~is are symmetry formations If they are, U8e t.hem to apply the argument of equa-tions (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

trans-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

FIG 3 Axes for an equilateral tri&Ilgle

perpendicular to the sides, Ok being identical with the negative y-axis A rotation A of 120 0

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

SYMMETRY TRANSFORMATIONS 13 rotations all leave the triangle in equivalent positions and that there are no other proper rotations that do this

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

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

In fact in every case P and p-l a,re jURt two rotations by the same

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

(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)

where as before F is the first element, 8 the second element and a

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 com-bination 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

SYMMETRY TRANSFORMATIONS 15

(c) One oJ the group tlement s, E say, must Mve the propertie8 oj a

unit element, namely

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

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

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 point-

groups a,re called Da and 0, All sqnare matrices of a given order

16 GltOlTP 'tUEORY IN QUAN'rUM MECHANICS

and with non-zero determinant form a group, the law of combination being matrix multiplication So do all unitary matrices (appendix ~) of given order, and likewiSe' all unit.ary matrices of given order and with deternlinant + 1, as can easily be verified Finally lineal" transformations of co-ordinates call form groups as we shall now see

The group of symmetry tro formatiolUl 01 a Hamiltonian

Consider three protons fixed at the points

r1 = [0, 2v'3a, 0], r2 = [-3a, ,/3a, 0],

ra == [3a, -y'3a, 0] (4.11)

forming an equilateral triangle about the origin (Fig 3) The

Hamiltonian for one electron Inoving in the field of the three protons is

:ff' - V2 -+- - - + +

, v - 2m Jr - rll Ir - r21 Ir - rar (4.12)

rrhis system is not one of physical importance but its symmetry is

closely related to that of an ozone moleculet or that of an ion situated between three water molecules in the hydrated crystal of a salt, to which the following discussion can easily be extended (cf problems 4.5 and 4.6) The physical system of three protons has the same rotational symmetry as the equilateral triangle already discussed, which suggests that the linear transformations E', A I J B', K', L',

M' corresponding to the rotations E, A, B, K, L, .M (4.1) may be symmetr) ;'.'··~1).sformations of the Hamiltonian (4.12) These trans-

forrnatioIl8 can easily be found from (2.2) and simple extensions of the argument of §2 FOI" instance A' is obtained from (2.2) by putting

(X = -120c1 in accordance with § 2 because A (4,,1) is a physical

rotation of +120° \Ve obtain:

r -E': 44': x (x:~, = -iX z) == (X,Y, + !V3},~ Z) - B': x == iX - !v'3Y

t Throughout t,bis book we shall assume for illustrative purpose that the

~h:ee oxygen S,tOl'IlS in ozone fornl an eq'Jila.teral triang.iP, though this is

not 80 in actual fact (see end of § 24)

whence (4.12) is invariant under A'

The result that the transformation At of (4.13) is a 8)J.lllIletry

transformation of (4.12) is actually no accident· And can be proved

as follows using the correspondulg rotation 4 wit·hout ever writing do\\~ the form of ~4.' or suhstituting in <4.12) Let the potential in

(4.12) due to the protolls he

and let P(x, y, z) be any POil1t C~ol13ider the physical operation A

of rotating the point <P a.nd the thr~e protons but not the co· ordinate axes 'rhe Rystem of protOD.8 is rnoved into an equiva.Jent

position, one proton from r 1 to r 2' one from r s to r 1 and one from

rl to rs, and P is moved to the position (X, Y, Z) During this rotation the potential at P has remained COIlStant because it depends

only on the distances of P from the three protons, and these tances have not changed because P and the protons have been rotated as a rigid whole Thus V (x, y, z) due to the initial charge

dis-didtribu.tion is equal to the potential at (X, Y, Z) due to the final

cha.rge distribution Since, however, 044 has Ino~·ed the system of

protons into an equivalent position, the initial and final charge distributions and potentiaLq are identical, so that the potential at

(X, Y, Z) due to the final charge distribut.ion is V{X, Y, Z), i.e we have

[!(x, y, z) = V(X, Y, Z)J (4.14) where according to Fig 1 of § 2, x, '1/, z and X, I"~, Z are related by equations (2.2) ,rith a == -120° It only remains t.o view (4.14) and (2.2) in terms of a change A' of co-ordinates, rather than in terms of physical rotations This oniJT involv~s u ehftPgA in the

OUO-VP THEORY IN QUANTUM MECHANICS

jnterprf~tJ,tion of (4.14) and (2.2) and does liot destroy their validity

hB Gorrect mathematical relatIons ""0 t.herefore obtain that, V(r)

i3 in variant uuder the co ,).rdinate transfnrlnatioIl J.1 j 'rhis argulnent u;pplies Himilarly t.o all the tra.nsformations (4.13), a.nd indeed to

&In}' siIniJar sit.utttion (cf problem 4.8)

\Ve ca!l also verify from (ItI3) or prove by the a/hove type of flrt!lX~~nent ~.~ that the t.ransfOlmation 4' followed bv 1./ K' is the same

;t.~ t.1u; RElgl~ tra.n~forjnatjon Lf For in det/ail th.is simply Ineans

'01 at, tirsL expl"essmg a, unctIon x, Y: Z, ill wrU1R O,.il., , ' J usmg

>~' (J.13) and thp-u in terms of g, 7]" , using K'

!la.me]y the transformation L' In 3Ylnbols K',A.' :=: L' Similal'lyt

3Jny' of the transforlnations (4.13) ean be combined, and their luultiplicatioll tahle is exactly the sanle a~ Table 1 for the point group :{2 of rotations as can be verified nl0st easily by mat.l~ix

Inultiplie;tt.ion It is also easy to sho\y that the transformatJions havo ~)Jl tIle other properties (8:) to (e) above required for them to

for!p MJ grou!>, which ,,"e shall call the point-group 32 of

transfor-rnf1·t ~c DJ";

j.)IEORE!l'.C ",re shall now generalize this result and prove that

the ~J,H~;~~UtBiry transformations of a HarniUonian always j'or1'n a groap

SUPI>Jse a Hanultonian ~ is invariant under eaell of two

syln-In.etry t~-~:nlsformations F and J '1 We shall first sho\v that the

combinetl transformation SF (first F, then S second) is also a

Elvm,n~et:iY , '\ transformation Let the co-ordinates Xl' Yl~ ' Zl' 2.'i·_~ ?!~ ;., ~ ' I ••

Zn of ihe Hamiltonian be written for convenience Ql' Q2' , q~ln., and let .F be the transformation

SYMMETRY TRANSFORMATIONS 19

in terms of the Q, using (4.15) and then to substitute for the Qt

further in terms of some new variabl~ VI where

80 that the composite transformation SF from the q, direct to the Vi

is also a symmetry tral1~formation Thus the symmetry formations satisfy the group requirements (a) and (b) above We can indeed write down the transfol·mation SF explicitly by eliminate ing the Q, from (4.15) and (4.17), i.e SF is

trans-(4.19)

Further we always have the identity tra'iUljorrnation

qf == Q" i = 1 to 3n, (4.20)

having the property (4.8) of the unit element E, which verifies (0)

As regards (d), if we substitute for the q, in the initial Hamiltonian

in terms of the Q, using F (4.15) and obtain jf'(Q,), then we can get back to ;tf'(q,) by solvillg (4.15) for the Q, and substituting into

(4.21) which undoes the effect of F, and this is therefore also a symmetry transformation In (4.21) the Q, are now the initial variables and the

q1 the new ones, and F-l is the inverse of the matrix F'1- To make the argument quite rigorous, we 110te that all the transformations

in which we are interested are unitary (cf appendix A, problem A.9), whence I FI =1= 0 and (4.15) can actually be inverted to give (4.21) This verifies (d), and (e) can easily be verified by writing out the transformation TSF

q" = F"kSldTzjQj

in full without using the summation convention and noting thaLJ d~ not matter where the brackets of (4.10) are inserted This proves the theorem It is now possible to give a precise meaning

to the expression "the s:ymmetry properties of a Hamiltonian'·' which has been used in a descriptive way up till now The symmRJry

propertie& of a HamiUonian consist of the group of all 81Jm m.etry transformations oj the HamiUonian

We shall now investigate the group of symmetry transformations

of the Hamiltonian (4.12) in greater detail Out of the six elements

20 GROUP THEORY IN QUANTUM MECHANICS

(4.13), the elements E', A' and B' form a group in themRelves a.s can easily be seen from the first three rows and columns of Table 1 These elements chosen from the Ligger group (4.13) are said to foml a 8 1

uhgroup of the larger group Another subgroup of (4.13)

is the group (E', K'), another one (E', L') etc Similarly, (4.13)

does not include all the possible symnletry transformations of the Hamiltonian, but is a subgroup of the group of all its sylnmetry transformations ~"or instance, a synlmetry traruiformation not included in (4.13) is the reflection

irrespective of their angular momentum quantum number l, unlike the more general situation in an alkali atom (Schiff 1955, p 86; Fock 1935) Thus some of the more subtle symmetry transforma-tions of certain systems have only been discovered relatively recently (Jauch and Rohrlich 1955, p 143; Baker 1956; problem 24.11) Hwe now consider n electrons in the field of the three protons

or of three identical charges of any magnitude similarly arranged, the Hamiltonian for this svstem would have the transformations ,,I 6m2

applied to each set (Xj, y" Zt), i = 1 to 1l, as symmetry

transforma-tions It also hn,s the n! pernlutatiol1 transformations of the n

variables XI etc., and all combinatioIl8 between the perlnutations and tile point-group 6m2 transformations Thus the group of all s:ymmetry transfornlations would have a large nunlber (12n!) of elements, but it would be a simple cOlllbination of the groups 6m2

and Pn

lsomorphum

It was shown above that the elements of the group (4.1), say the group ~, and t.hose of the group (4.13), 05' say, both multiply in

SY~Il\rETRY TRANS:FORl\IATIONS 21

the sarne way according to Table 1 'J'his relationship bet","een Q; and (B' is called an isornorphi&rn and can be described by saying that

the elements E, A, B, ~ of ® can be paired off ,vith the elements

a,s regards multiplication are in ever};'" way the sarne as the

relation-ships between E') 4.', B' ~ ~ Actually it requires more care to define isomorphism precisely and to distinguish it from the related concept of homomorphism (of appendix B), but the above des-

cription is sufficient for the present considerations rrha.t <5 and (5' have the saIne nlultiplication table is not accidental, but follows quite generally fronl the rela,tionship between a physical rotation and the corresponding eouordinate transformation For if we

apply t-o a system a rotation F which shifts P(x = qv Y = qal

z = 18) to P'(Ql' (J2, Q3) related by (4.15), ann then a rotation S

moving P'(Qv Q~, Q3) to P"(vl' V2) va) related by (4.17), then the cOInbined rotation SF shifts P(ql' q2) qa) to Pll( Vl' V2' va) related by (4c19) But lve had previously seen that equations (4.15), (4.17) and (4.19) represents the combination of linear transformat~ons

of co-ordinates, \vhich therefore combine in exactly the same way

as the physical rotations, this argument being quite general In

coImection with our study of the Schrodinger equation, it is always the group oj sy1nlnetry transformations of the Hamiltonian that we shall

be interested -in (§§ 2, 3) However, the group of physical operations which move a system into equivalent positions can be used ~s all aid to the imagination and to suggest what the symmetry trans-formations of the Hanliltoniarl are Thus in future we shall not distinguish the transformations (4 13) by primes from the rotations (4.1) and shall simply refer to either group as the pomt-group 32 Another exanlple of isomorphism is afforded by the permutation group Ps of o'l*d~r 3, which consists of all the transformations (ijk)

(Xl' Yl' Zl) = (Xi, Y-t Zt),

(X2' Y2' Z2) = (X" Y j , Zj),

(X3' Y3' Za) = (Xk' Y k, Z,,), (4.22) where ijk is some pennutation of the numbers 1, 2 and 3 These transformations call again be combined according to equations (4.15), (4.16), (4.17) and (4.19), where F'1, etc., are now matrices of order 9 X 9 The transformation (132) (231) means as USU:1l the permutation t.rap sformation (231) followed by the permutation

(132), and jt is equal to the perm:ut1ation (321) as can easily be verified Similarly the whole multiplication t.able can be set up (T8.blc 2) On examination, this table is seeIl to have exactly the

(123) (231) (312) (132) (321) (213) applied first (123) (231) (312) (132) (321) (213)

(231) (312) (123) (213) (132) (321) (312) (123) (231) (321) (213) (132) (132) (321) (213) (123) (231) (312) (321) (213) (132) (312) (123) (231) (213) (132) {321} (231) (312) (123)

same structure as Table 1 Indeed if we write in Table 2

isomorphic with the point~group 32 For instance, the group of numbers

1, £xp(i1T/3), exp{i21T/3), -1, exp(i4?T/3), exp(i5?T/3) (4.23)

is not, the law of combination being multiplication (cf problem

4.2) However, it can be sho'wn that any group of six distinot elements is isomorphic either with the point-group 32 or ~rith the

group (4.23)

References

All the matrix algebra required for this book is given in appendix

A Accounts of groups and their properties, more detailed than here but still iIltroductory, Inay be found in Lederman (1953),

BirkllOff and MacLane (1941) and other texts (see general references

at 'be end of the book}e

~

J Jumnwry

We have defined what is meant mathematically by a gr(Yt:lp&

We have proved that all the syrnmetry transformations of a Schr6dinger Hamiltonian form a group In the exalnples studied