and the Dirac Method
In this book we have developed the group theoretical methoo. tor sorting out and labelling a complete set of fUD.t.ttio~ usually the eigenfunctions of a Hamiltonian. rrhis oontrat-lts aãt first sight wIth the more usual procedure, developed by Dirac (1958ằ in whioh one uses as a complete set of' functions the 8imul~'\noous eigenfunotions of a setã of commuting operators (se. forex8,mple Schiff 1951$, p.143).
The purpose of the prest}nt section :is to rel::.\.te these tãwo 6:pproaohel, t
and show that they are oomplet(~iy equivalent,
In brief, Dirac proooed.~ as lollowB. Let 4s be an eigemunotJon of two operators A and B belongiIlg to eigenvalues a, b;
It follows that
A,Bt/s = Ab~ =. bat/J = rJ,B~ = lJfI4t = BAt/I,
(.4B - BA)+ :::;; o.
This suggests that simultaneous elgen£unc'bions like ~ are moat likely t.o exist if AB - BA = 0, i .. e. if th.e two operat()tS cominute.
In. fact it can be shown (Dirac 1058, p. 4£;) that the SimultaI100US
eigenfunctions of two commuting operators form a complete set of functions. By using 8e'V'eral commuting ollemtors, we can arrange it that the ieta of eigenvalues of two different function.s are always different. This then giTeS 8 definite way of achieving a sorted and labelled complete set of functions. One of the operators is usually
!Chosen to be th~ Hamiltonian, so that the eigenfunctions are sorted out according to their energies .. In thIs case the other opera..tors are t The results of this section are not mwd else'where in the book .. It haa been included for the benefit of those readers whose original introduction to qntmtuIn luechanies was through Dit'aC'a book, or wb;,) for ot,her roosoDS like to think in t i'IT:1S ':if comp'ete sets of oomm.uting op~ato~c>
144 GROUP TIIEORY IN QUANTUM MECHANICS
const.ants of the motion in the quantum me cba,ni cal sense. In detail, for any operator A not depending explicitly on tiIne, we have (Schiff 1955, p. 134)
d I
dt (A) =-= in <A~~ -2A), (17.1)
where <A) is the expectation value Jif;*AifJ dT of A. Clearly <""-4) is constant for any state ifJ(t} if A commutes with e-~, and A is called. a constant of the motion accordingly.
Continuous groups
The relatiol1shjp between the group theoretica.l and the Dirac methods of labelling eigenfunetJiol1s is very sitllple when the Hamil- tonia.Il .Yt' is invariant nnder a group of transformations fornling a contjnuous seqnence in terms of 80me co-ordinates qt- Suppose Yt' does not depend on Q1' Then if we regard ~ as a cla.ssical Hamil ..
tonian the nl0nlentU111 ~Pl conjug[~te to ql is a constant of the Illotion because
f
t~,i]antum rnechanieally, l~l commutes with all Pi and all qi except ql- Since c?lt' d~)es not in volvf\ Q1' PI also commutes with ~~te. From (l7.1) it is a CUD:;th,n1, of the motion ju~t as in the classical analysis, and. Inay be td ken ad one of the set of comlnuting operators. As a siulple exanlPJe \ve may cite t.he case of the electron of a hydrogen atom III ~t uniform magnetio field II in the z-direction~ The IIamil- tionian (neglecting spin) -gS
li2 e2 tin 0 e2
- -~- v2 .-~ - + - H _. --too - - " H2r2 8m2 8.
2m r 2mc 84> 8 rnc ~~ (17.2)
'fhis is D.ot exactly in canolllcal form,. but it does not depend exp]jcitly on cPo Thus the conjugate momentuln, naluely the angular Inornentum Lz a,bout the z.axis, is a. COllstarlt of the motion as we would expect classically. ~[,he eigenfunctions therefore ha.ve a definite value of Lz, namely 'inn, (Schiff 1955, p. 75). This illustrat,es the Dirac viewpoint.
Now group theoretically we would proceed as folloãws. Since .Ye is independent of qv it is invariant under all transformations ql = Q1 -f- L1ql' Thus £ is also invariant under the iIlfinitesimal transformation 11' Moreover all the transforlnations fOfln a con- tjnuous group. Then if we have a vector space tra.l1sfornling irreducibly under the group, we may use II to pick ont in it the
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REPRESENTATIONS OF lflNITE UROUPS 145 functions invariant under 11 and use these as base vectors. No\v these elgenfunct,ions of II are just the same ones as we ,vould have found usin.g the constant of the motion PI above, since there exjsts the rehition PI ::-::: liII (8.29) between them. There is thus a very close relationship between the classical result, the Dirac approach and tJle group theoretical procedure.
llinite gr014ps
The above discussion eomplet~ly breaks do,Vll "rhen the symnletry group of the Ifulniltonian is a finite group Q;, for in this case t.he infuritesima,] tral1.'3formations do not exist. Consider for instance the HalniJton~an (4.12) of an electron moving in the field of three protons arra.nged in an equilateral triangle. What classical constant of the nl0tion is there that corresponds to the fact that, the potential has trigonal symmetry 1 We could introduce a co-ordinate ql == 1, 2, :3. 4, 5, 6 corresponding to the six equivalent positions of the triangle. Then ~ is certainly independent of qI' but there is no eonjugate momentum because ql is not a continuous variable and differentiations such as OQl/ at, o:ff / OQl become meaningless. In fact it is not at all obvious what complete set of commuting operators to use in this problem to obtain a convenient set of simultaneous ejgenfunctions. Let us therefore adopt the group theoretical method and assume that, we have the eigellfullctions tPt(Ar> of the Hamil- tonian rYC"! sorted out to transfornl according to the hTeducible representations I)(A) of the symmetr:r group Q) of e.YF. We shall now construct a set of operators Pic of which the ~(~r) are simul- taneous eigenfunctions.
()onsider the operator
Pie = (l/hk ) L Tk
all T in class k
--=:: (ljh) 2: S'jJkS-l,
aU S in@
(17.3a) (17.3b)
\vhere S is any elernent of <B and Tk any element of the kth class . . A.I:; hI § 14, .h is the nUluber of elements of ~ and hk the 11umber in the kth class. l~ronl (17.3b), Pk comrllutes \vith every elemellt of
\l). Hellce by Schur>s lemma (appendix D) Pk is represented by
at times the unit matrix with respect to an irreducible set of base vectors 1'I,(M), i = 1, 2, . . . . Hence using (17.3a) and taking the sum of diagonal nla.trix elements
aknA = (ljhk) I x().)( T k), T7ft
== In).., fl~ ,. 4\
l~ GROUP THEORY IN QUA.1t.i'UM MEOHANIOS .... ,f) ,
where -A is the dimension of D(M. Thus tPlCAr) is an eigenfunction of
PI: witJa the eigenvalue (17.4). Since the characters Xk(A), 1c::l I, 2, .•. characterize an irreducible representation completely, the set of eigenvalues at (17.4) serve to label uniquely whioh irreduoible
representa,tio~ a given tP belongs to. Furthermore since PI: commutes with any T, trom (17.3a) it commutes with any Pl- Now any ~lement
T of ~ is a, symmetry transformation of:Yt' .. Hence operating on
J'f.p, it aBectB only the w&ye £mlction t/J and we have '1'1+ = ""'1';.
whence
T~=~T. (17.G)
Thus P is a constant gf:ã'ãthe motion, and from (17.3&) 10 is P".
Our functions rp,{Ar) tffiDsforming irreducibly according to DC.\)
are therefore simultaneous eigenfunctions of th~ set:K, Pl' p •• .••
of conunuting ~perators. The eigenvalues of the Pi distinpiah
• .rent irreducibl~ representatioIlB, and the energies (eigenvalues of 8) differentiate between different values of r, e .. g. between the 2p, 3,t>, 4.1', • • • wave functions in an atom since these all trans- form aocording to DCl) under rotations. The operators Pj; cannot distinguish between the different ""tAr) ,i ! : : 1, 2, .... of one irreducible . representation, but some more operators can easily be devised to
do thiI too (problem 17.3).
SlRnmary
The group theoretical method consists of sOI,ting out the eigen ..
functions of the IIamiltonian ~ accordin.g -to the irreducible re ..
presentations of th.e symmetry group of :Jr. We have shown that this is oompletely equivalent to the Dirac method, in which we set up 8, complete set of simultaneous eigenfunctions of ~ and other
commuting constants of the motion. .
PROBLE1\{S
17.1 In the 11otation of the text, show that 11 is a COIlBtant of the motion withou.t using the relation PI == Ii] 1.
17 .. 2 Show that the two expressions in (17.3) are equaL 1783 With the notation of § 14, cOhSttier the operators
QM, = 1 D'i(J..)*(T)T,
7.1 .
where the sutr:.matioJl is' over all T in the group, and i is not sU111med.
Show that ~iM {;('~lnlrI,1$~tes with any QfI#' any' PTe (17.3). and '~Jvith ~~i;
Hamiltonian. f\Vh~:' -;. i{~ 'the effect of QAt on an eigenfunctl(~r: ~'~.i(J.&Y~
of the Hn.rn.~utãL~.!li!;'d';') Hence discUBS the use of the QM;'-ã' tãh or
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RBPBESBNTA.TIONS OF FINITE GROUPS 147
'.
1VitbOut the Pk, as a complete set of commuting CODstantS-of the motion.
17 .. 4 An electron is moving in a potential whose maximum symmetry is the point-grou.p 32.. Set up the minimum number of oommuting operators forming a complete set 80 that a simultaneous eigenfunction is wliquely determined by its set of eigenvalues.
Give the operators and eigenvalues in detail.
Chapter Iv'