The infinitesimal rotation operatDrs
Before proceeding to derive the irreducible representations of the full rotation group, it is convenient to express all rotations in terms of three operators Ix, ly, lz known as infinitesimal rotation operators. We can then confine our attention in subsequent work simply to these three operators instead of having to deal with arbitrary rotations about arbitrary axes. Consider a rotation transformation R(a, ;) by an angle oc a,bout the axis;. From it we can define an infinitesimal rotation operator 1"( given by
L R(a,;) -- 1 °1
t = ~ ~,
ô~o ex (8.1)
or
R(a., ~) ~ 1 + iodf. .when a. ~ II (8.2)
where we llave written 1 for the identity traIlsformation E. Strictly speaking iIf; should be called the infinit.esimal rotation operator, but it is more convenient and is becoming customary to work with IF.- R(ex,;) can be expressed in terms of IF. for an arbitrary angle ex which is not necessarily small, for a rotation by ex is equal to n successive rotations by a./n. Thus from (8.2),
R(oc, ~) = tt.~oo Lt (1 + i n ~ I~)n
(S.3a)
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QUANTUM THEORY OF A FRE}4~ ATOM 53 This series can be summed in a purely formal way, and we can write R{rx, ~) = exp(iczI~). (S.3b) However, this should only be regarded as all abbreviated notation for the series (S.3a) and in any particular case the exponential has to be expanded before it ca:n operate on a function.
Consider now R(a, ;1) when rx is very sITlaH, where ;1 is an axis in the yz-plane Dlaking an angle f) with the z-axis (Fig. 5). This
ex sin () ,-1 z ~I / I
o~,---.--_J €X COS fJ
/ y
L~_cos / e
~~ asine
FIG. 5. Change of the co-ordinate axes in a small rotation R( (x, ;1) where ;1 is in the y~-plane.
rotation is approximately equal to consecutive rotations rx cos 8 about the z-axis and a. sin e abOll.t t.he y-axis, correct to terms linear in (X.. 11his can be seen froin .F"ig. 5 which ShO~TS the effect of R(rx, ~1) on three points ~~, y, z at unit dist,(-),nces along the co~ordinate
axes. R(rx cos 8, z) displaces x and y each by a cos fJ, and R(a. 8in 8, y) displaces z and x each by (f. sin fJ. '1'he total displacement of x is rx in a plane perpendicular to ~;1' antI thus the effect on all three co-ordinate axes is equivalent to t.he s~:rlgle rotation R(a., ;1) when rx is small. We have
R(rx, ;1) == R(rx sin fJ, y)R(rx cos 8, z) + O(cx.2),
and hence from (8.1), (8.3)
If;l = Iy sin 0 + Iz cos O. (8.4)
If now ; is an axis making direction cosines l = sin () cos ~,
m = sin e sin rP, n : = cos 8 with the X-, y-, z-axes, then
1'( = I' sin 8 + It. cos () where I' = Ix cos 4> + ly sin 1>,
GR01JP 'fHEOI~Y LN Qlr.Al,;'rU~M TvIEOHA.NICS
(8.5) In fact infu:dtesimal operators add like unit vectors. This result
(~orresponds to the faot t;hat sma11 rotations add veetorially to a
fu~st ord0r of approximation, and that angular velocities in merh.
anies add vectoriaU:y (Mihle 1948, p. 148; Goldstein 1950, p. 126) ..
~)f oourse l1nite rotationR do not add like vectors, since they do not even com!,li,nte (equation (4.5), problenl 8.1). In conclusioI~ we note that 1'4Tith the heJp of (8.5) and (8.3) any rot.ation can be expresserJ_ in terms of the infinitesimal rotation opera!tors Ix, 111, lz.
CC~fllt')tutation reootion,s
(;on.aid.~r now the phYfli;::aLl rotation Itot(~, ;1) when (1; is not smaH,
\~ãht)lãe;1 is the axis 8hown in J~ig. 5. ~rhis rotation can also be achieved by first rotaJt.ing hy 8 about~ Ox which brings the ;1 axis parallel to
Oz, t,llen 'rot~ting b:y tY. ab'~JutJ ()Z1 and then returning;1 to its origin.a.!
pOf:dt~~{;n by 81 rota,tion of ,,-e about ()x. Thus
Rot(rx, ~1) ::-::;- l~i\)t( - 8, x)P\!Qt(Gts z)Rot( 0, x).
JfrOYl1 § ~~ a physic<l,] rotation by an angle -l-~ is mathematicaJly equiv-aJent to 8J'rota/tional transforms,tion of co-ordinates by an allgle ãããã'"t.{c lienee in term.s of tran.~fOI'rrlations of co-ordinates~ t
(8.6) Ex.pressing this iIl t,ernlS of infiniteBimal rotation operators using (8.3) a,n.d (8,4), we ohta.in
1. _.- irx( t~l ~In. e --1-' I z ~es e) -t- ()( rx2)
= (1 -t- i8/x + 0(02)][1 - iff..lz -+- O(ot2)](1 -_. ifJIx + 0(02)].
(8.7) Now {8.6) is valid for all a and (J, 80 tha't we cnaJY express (8.7) cODlpletrely as a power series in 0 and a and eq'uate coefficients of 8lt, vvhlch gi.v~"s
-iIy = IJiJ - I"lx6 By sYlIimetry, we also obtain
--ilx ':;:'-:: I"Iy -- lylz"
-ilz = 11/1.1; - IzI1I~
(S.Sa)
(8.8b)
t This relation could ha.ve b6~n 'writt,en down diJ:eotlYt but the author finds it easier to yisualize the COlll.pc'sitii0n of physical rotations rather than that of co .. ordinate ohanges ..
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QUANTUM THEORY OF A FREE ATOM
In terms of
these commutation relations be(jorne
r~--ãã ---..-~~-'ã-l
I l a1+ -- l+J~ = J+, I
f IJ _ _ .,. I._It. =.~ --1_,
11'.1.1" 1 I .. ,,-9.' ~ 5 + -- - '- ~f' - - ~.I..Z" ,
L ____ ,--.. ...-...-..-__ ,. __ ~ The irredu.cible. rcpresenta.tiofl.1Jt
(8.9)
It was mentioned in § 4 'uJlat all proper :rotations t1bnu.t all axas through a point fOl'm t.he full rotJ}"1tion grouI), and 1;here is in faãot no difficulty' in verifying t,hat all the group :requireriH)ni£ are St~1~js ..
fled (problems 8.2 and 8.3)~ We shall no"rv take atn arhitrary veottOr space n which is invariant under the full rctatJion gTOUp, and stal't reducing it into its b,:,~,:ãetlucible 00mponen.ts. Since an.y rotation can be expressed in terms of Ix" I y, I Z'J it, is not llece"ssary to work with an arbitrary rotation but onJjr. with these three operators. More . precisely', it follows fronl (8.3) I (8.5} that, if 11 SpS,4!iJ in inv,tri~nt aIld irredl1cible under 13;, I 11~ 1 z" then it i~ ~tlsG in variant and irredu.cible Wlder all rotatious and vice vel~. Let us fn~ I~3£lUC.e n with
respect to the axial rotation group about t~he z .. axis:; and let 11,.
be any' vector tratnafOlJnl].ng aecording to ~)he' 1Jt.th representation {7uS).t Then
whence from (8.1)
(8.11)
If Una transforms t~ccording to the mth repN'.serttation of the axial rotation group, thallI +um, I._um belong respectively tc! tl18 (ãm + 1 )tb
and (m - l)th represent.eltions, for from (8.11), 8.10) Iz(I+um,} = (Ii . .1t -t- I+)um = (m + l)I+um
and similarly
(8.128)
(8.12b) When )1 is reduoed acoording to the ~x.ial rotation group, let j be the highest v'alue of m that OCCUi'a among the irreducible
t This derivation follows closely that given by Van ,dar Woorden (1932).
: It is convenient to use u instead of t/J for the vectors of 1ft 10 &rI to avoid oonfusion with the angle';.
56 GROUP TIIEORY IN QUA~"'TUl\l MECHANICS
components, and let 'llj be a vector corresponding to this value of m.
Now sinee E is invariant under rotations, I+uj also belongs to 11. and frOIn (8.12a) it has the 'i12,-vulue j + 1. However since j is already the largest tn-value found in 11., we luust have I+/u'l = o. On the
other hand by repeated use of 1_ ,ve can define from Uj a sequence of vectors ~t m. vvrith m :::::. j. j - 1" j - :2, . .. such that
(8.13)
From (S.12b) 'Un£ belongs to the eigenvalue rn of It. am is n. non- zero nunlerical constant \vhich we shall determine later such that the Um are all normalized. ]?rom (8.12a), I+um-1 belongs to the eigenvalue rn of lz) and we shall now prove from the definition (8.13) that it is actually a multiple of Um. First suppose
,vhere em is an undetermined constant. Then from (8.13) IIJtm -1 == tX1nI +l_llm == (XrnI _1 +?tm + 2rxm1z'Um
-== rt1nI -cmxm+l um+1 + 2fn0!1nU,1n
== (("m -t 2m)cx mUtm.
(8.14)
(8.15) Thus if (8.14) is true for one value of rn, then by (8.15) it is also true for the yalue n1, - 1. But (8.1-i) is true for m, == j \vith Cj = 0, so
tha t, by induction it IH true for all rn. Further, it is possible to calculate the value of ern. for from (8.15)
Cm-1 = em + 2m.
The solution of this difference equation with the boundary con- clition Cj -= 0 is t
em =j(j + 1) - m(m, + 1). (8.16)
~'urther jf E is of finite dimension, the sequence of vectors U'm must end at SOIne point. l.e. ,ve must have sonle Um = 0 with
'ltm+l =I=- O. lIenee I +UTli == 0 and from (8.16) this only happens
\vhen 'm .:::~ --j - 1 (apart from l1l = j already discussed). Thus the last of the sequence of vectors is U_J, and the number of vectors is 2j + ]. This is necessarily an integer so that j is an integer or half an odd integer.
The constants tXm can no\\" be calculated so as to make all the Um normalized. By appendix C, lemma 2, any rotation such as R(±8, x)
t The solution of difference equations is dealt with in most elementary algebra. texts, e.g. Durell and Robson (1937).
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QUANTUM THEORY OF A FREE ATOM 57 leaves the scalar product of two arbitrary vectors u and v, or
u and Rt" invariant. Thus
f u* R(O, xlv dT = f R( -0, x)[u* R(O, xlv] dT
"--= f [R( -0, x)u]*[R( --0, x)R(O, xlv) d ..
= f [R( -0, x)u]*v dT
where d-r is the volume elenlent. This is true for any value of e
so that we may substitute (8.3) for R and equate coefficients of 8.
Thus
f u*(Ixv) dT = f (Ixu)*v dr.
A similar result holds for 111, so that
f u*(I+v) dT = f (L'u)*v dT.
Now from (8.13), (8.14), (8.17) we obtain
f Um *um dr = (%m+1 f (I_um+l)*.um dT
= (Xm+1 f u;;'+11+u1/" dr
= cm ô(%m+l)2 f U~+1Um+l dr.
(S.17a)
(8.17b)
Hence, all the Um a.re simultalleously nornlalized if we choose
OCm+l = .(Cm)-1/2, al1d (8.13), (8.14) become
I +Um = y[j(j + I) - 'm('m + 1 )]1.tm+l'
= y'[(j - m){j + m + l)]u'm+l'
l_um = y[j(j + 1) - m(m - l)]uãm-1,
= V[(j -t m)(j - m + 1)]um --1,
Izum = mUmã
(8.18)
The Um are also orthogonal to one another because they transform according to diffeI'ent irreducible representations of the axial rotation group. This follows immediately from appendix C, lemma 5, or fronl the fact that in the reduction of a vector space the different irreducible subspa(les can a]waYK be nlade orthogonal t,O
58 GROUP THEORY TN QUANTUM MEOllA.NIOS
OJ1e anot,her (§ 5)~ It can a,lso be prnved V~1J''' simply directly, for by appendix C, lemma 2,
I Um*ttp dT = I R(OI., z)[um*uJ d'T
i"
= e'a(J,£-m) i ! 111. __ vv-m. *u d 7"
IJ. ,
whence
f Um *ul' d'T = 0 if m ;f= fl.-
F~m (8.18) the vector sp&oo ~(I)(ui' UJ-1' . . . 'u-l) is invariant under 1+, 1_ and Iz, .~and therefore it is also invariant under all rotations. We now show that it is also irreducible. For, suppose
!t(j) contains an invariant subspace r~ then r would also be invar ..
iant under the axial rotation group about the z-axis and would therefore be spanned by a set of the 'Um.. But from one u'm the operators 1+, 1_ generate all the other ones, so that r can only be equa.l to the whole space !{(/). Pk'U8 the vectors Um, m = j, j - 1, ...
"-~is t'fantljormi'1tg according to (8.18), form the basi3 of an irreducible
~fR.pte"entatWn 0/ the !uU rotaJion group. The different irreducible
'r~pr~ions D(1) are given, by the allowed valiUM; qfj,j = 0,1/2,1, 3/2, 21 ••• 'I and have dime1&8irYn8 2j + 1. If a space !{(j) transforms
iWcol~ing to D(J>, we s~all refer to the particular set of base vectors satiEfying (8.18) as the ata1lllard_ base vectol~ ..
In this subsectioIl we started with an arbitrarY invariant vector f3pa.oo n, and, set about reducing it. This led na/t~ally to the above
doo~')liption and definition of the irredu~"ible representations"
But it should be noted that we have here in addition a systematic
~/c;'.r of actually picking out an irreduciblf) eom.ponerit from l{~
&.t,.~trting ,rith a vector 'UJ with the highest m .. v&lue~ II.aving pioked
<,,';,~, t, ~tria ll~wucihle fJftbHpaoe~ we can then orthogonalize all remaining
';tã~;ntov(~ to it and ~tart the proceS3 again in. the remaining ve(;tor
8';ããt,,('e~ llhus R is grad.uaJIy completely reduced. This scheme is desClfibed more fully in the next section where we actually employ it.
E~mpleB
11u~ spheIica.l h.annorri(!,a can be df;;lllefl in v~,ri(H1S ways, but
llS11J-Uy arise in qua.n.tum mec,h~~}nics ae the HolutioHa of the equat30n {8.19)
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QUANTUM: THEORY 011 A. FREE .ATOM 59 spherical harmonics Ylm are particular solutions of (8.19) hr~ving
the form
(8.20)
where Nzm is (-L numerioal normalizing factor, .P,J1ml an associated Legendre polynomial, and ãmã llas t,he 2l + I integral VaJ.UB'~ li
l _. -I, . ~ . --l. The operator i.n. (8.19) is Ulvalia.nt under rotati()!u~
since it is just the angular part of the Laplf.\ciaIl Yã~:>.and henc€< t,he y l1n fnr given l span an invariant vector space R (l) (§ 6, Theor~!n J).
From (8_20) }-'-,m belongs to the 1nth repre8eut~ttion of the ulXii~J
rotation group about the z..,axis; the YZm are thel-efore o:r"b.hGgon:al to one anoth.e!~ &D.d R(U has the dimension 2l + 1 .. Now from one of them, Yu S~\l{, we can define using (8 .. 18) 21. ,- 1 vectors yll1t~'.
These span an invaI~iant subspace of ll(U whioh. musiJ be ~2.qnal to the whole space heeause it haa the same dimewlion 2l -f .. 1.. ThlU.:t tlte sphttic.fil MN1Mjt;:ic.s Y lm tram/arm accordãi~nfl to the i,'~red,",teible
repfeaentation ",DC 1) oj ~h.e .full ,oIat:ion grO'U(p. Morea 1re~c ein (.~6 Y lm, belongs to the mth represent,a,tion of the al:iaJ rotia~Li\J~ ;;~t'{}UP,
the Ylm transfonn exatJtl~r like (8.18) if we give t}l~n:~ th~~ (~Ctrrect
phase factors (Condon a"/1 Shortley 1935; ]' .. 5~~) .. rZYlw. ea.l1 also be t)xlll'e8sed as 3, F-olynoroltf~l of degrc.-,e l in !(;~ Y. !. Fnt' fI~;:rtalloo
"Y1I .. = ã~-lv(x _ .. iy\
..ILl't"-" " • I'}
These functions therefore transform according to tho?! irl'educihle representation DCl) and D(2), <d thiJ signs have actllaRv been chosen to make thenl stan.dard base vector8~ ~rhU5 ~~ 1/, z also transform aocording to l)(:u hut not as standa!'d base vootors.
An int.e:reating ieeltw-e of the representation.s DtJ) whet.) j is half
an odd i'ltte,ge'f iq tha.t they are dauble .. valued, 'by which we, mean the following~ In § 7 we d.educed the irreducible representntions (7.3) of t.he axial rotation group, and noneluded that m must he an integer becau:~e of thie eondition XC~1') :::::: X(O} and becauf.if' ",ô(OJ
60 GROUP 'J.1HEORY IN QUANTUM MECHANICS
corresponding to no rotation at all must be equal to one. However, for the standard base vectors Um (8.11), (8.18) transforming acoord- ing to D(j), we have
R(cP, z)um = exp(im</ằum
where m is half an odd integer if j is. In particular R(21T, z)um = -Um,
so that a rotation of 21r about the z-axis makes all the Um change sign. SUlce R(21T, z) is physically the same as no rotation at all, we have that the identity transformation is represented by two matrices, the unit lnatrix E alld also --E. Similarly by compound- ing any rotation R with the identity transformation, R is represented by the t,vo matrieeH l)tj)(R) and -D(l)(R). This leads to no difficulties in qualltulll 1118charucs because the wave functions t/J and -~ always repl'esent quantum mechanically the same physical state of 8) system, so that we can consider the matrices ±D(J)(R) as irlducing the same transformatioll among the base vectors. Clearly the Um cannot be ordinary single-valued functions of x, y, z because
R(27T, z)f(x, y, z) = f(x, y, z).
In fact we shall see later thOlt the l'"epresentations D(I) where j is half an odd integer only arise in connection with spin functions.
We now show ho\\" we can calculate in principle the matrix D(j)(R) which ropresents any given rotation R in the representation D<j). If R == R( rY., ; ) iM a rotation by a. about the axis ;, the matrix
~r)<j)(R) is cumpletely determined by (8.18), (8.9), (8.5) and (8.3).
\Vhen discussing such a general rotation it is usually convenient not to ,vork in terms of th(~ aI~.gle a and the direction c03ines of the
; axis, but to express the rotation parametrically in terms of three Eulerian angles 4>, fJ, X defined in Fig. 6. Any rotation can be considered u.s shifting the z .. axis to OP making polar angles </>, 0, plus a rotatjon by X about this axis. Thus if the physical rotation
Rot(a~ ;) corresponds to the Eulerian angles (~, 0, X), wãe have . from Fig. 6
Rot(<x, ;) = Rot(~, z)Rot(O, y)Rot(X, z).
Note that OP is not the axis ;; it is the direction into which Oz is rotated by R(a., ;). In § 2 it was seen that a physical rotation by an angle a is mathematically equivalent to a rotational transforma- tion of co-ordinates by ~--~. Hence
R( -a, ;) = R(-t/>, z)R( -(), y)R( -x' z), R(a, ;) == [R( -IX, ;)]-1
~-= R(X, z)R(O, y)R(,p, z). (8.22)
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QUANTUM THEORY OF A FREE ATOM 61
z p
~---y
X
FIG. 6. The Eulerian angles X' 8, cp.
We can now calculate the matrix representing R(~, ~) in terms of the three parameters ~, 0, x. As a definite example let us consider the representation D(1/2). From (8.18)
I.e. Ix = 1 [. 1] 2 1 . ' Iy = 2i --1 1 [ . ~].
and Iz2 = 1112 = Iz'l. = ! 1,
where we have written D(1/2)(I+) etc. as 1+ for short, and 1 for the Imit matrix E. From (8.3) the matrix representing R(c/>, z) is
Similarly
D(1/21(c/>. z) = 1 + ~ (i~~)"
1&
= 1 + 1 ~ (t~)n + 2Iz 2J(t~)n
neven nodd~
= 1 cos it/> + 2Iz;i sin tcfo
D(1/2)(9, y) = 1 cos iO + 2111i sin 19
(
COS iO sin to]
= - sin to cos to .
62 GROUP tB:E~BY IN QUA.NTUM MEOHANIOS
Hence from (8.22),
D(l/l)(oc, ;) = ± [et•lx e_e,x.u] [ oos i8 sin 19] [t'.W
III - sin 16 cos to
[
eIKt#) cos til ell<K-~) 8in ,8 ]
= ± -e-mrlằ sin 19 e--fl<x+<fằ cos 19 ã (8.24)
The ± has been included beca.use j = t. It can in faat be seen
explicitl~ that adding 21T t,() X or q, changes the sign of the matrix.
RelGtWlUbJp to an.gult.ir momentum
The infinitesimal rotation oper~t()m 08,11 be expre.ssed directly in terms of co-ordinates. . If f is a fWl(~ti()n of several sets of co- ordinates X'h '1/11, Z'h then frQm (2,2) and using the notation of (5.3),
R(<<, z)f(zn, 1/'h zn)
== f(x,-. 60S r.x - Y. sin (x, Xn sin (X + 'Un COS at zn)
,. Of 8/)
= !(zn, 1In'l znJ + {t L..t ~ (Xft ... _-- -Oy 1; 'Un ~ OX,. -t- O(~I),
. "
whence from (8.1.)
,~( 0 a )
1~ 1l:rb = - i L.., . x. Oyn - VI; Ox~ •
fa
Here 'we have 'written Iv, orb be08Juse as we shall see in § 11. Iz can a,Iso operate on the spin co.ordinate of an electron. Until we introduce spin oo-ordinat,es, Iz and I,. <lrb can be considered as identical. In spherical polar co-ordinates
"'" 8
l/JOfb = - i L.., ar/J" ã
n
(8.26)
Now the quantum ulechanical operator for the z-component of the orbital angula;r momentum L is (Schiff 1955, p. 74)
b f ãc 2 ... ãã.. '8 or-., W.lence tern (0. n t, ~, vuU f
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QUANTUM TRF)JBY OF A nBE ATOll 63 This result is a, special case of tbe general relation due to Dirac
(8.29)
for the momentum operator p, canonir.ally oonjugate to the co- ordinate q. The relation is prov~Al ill appendix F. The most oommon example of this relation is tho us-aal linear momentum
Ii 8
p~ = -;"'-,
" ox
the i coming in through the flefinition (8.1) of an infinitetJinla,l operator.
Equation (8.29) is very important, izJ disOlL.Wrig the NlgUlar momenã
tum due to the electron spin. ,In the 0.r.\8e of the ttiomentum due to orbital motion, we have ãthe classical expression L, = xPu - ~/Px
from Which (8.21) is derived, 80 tlhat we have verijkd (8.28). Row~
ever there is no classical ~nalogue to spin angul&r momentum, a.nd the only satisfactory way of defining it is via (8.29) (ot. Schiff 1955, p.. 142; Dirao 1968, p. 142)" Thus anticipating a little~ we can define the total angular momentunl vector J by
where I~, 111, I~ apply to an orhita.I (01' spa.tial) co .. ord.inates Xn.,
Y'h ;. and all spin co .. ordinates (§ 1}). I
If we are transfonning .(1, functiol) of tuariY OO .. ãoff.lllates Xn, fin') Zn~ we can consider rotational tranclfol"lJlatiou.s _Rn.{/)'f,.J ;) of the 00 ..
ordinates Xfh Yf., Zfh keeping .all tho other Xm, tlnh ?:nl fi~ed. 'Ifb~:.se
are not symmetry transformations of <!l.2). TheIl
fJorrespondingljr ,va can define il1finh:4;siniaJ trp'yDBformations If,n
a;l1(l l~n o!';:~ de~n(ling 911 whet-her we inolude or exclude the spin co"ordhla,t,*, in the rot~tionJ 1~hen from (8J31), (8.2),
r~'~""ããã~---ãã--ããã-'---ãã
t 1ã _ .. _"" 1 J -~~ ~ I
I "!l -_. ;: ~'l;n' l:; or!) - -; lin 01'11"
" - -... _~ _ _ _ ..--.-,._,~ ... _ _ _ ~ ~ _ _ If .M~"""~,..,.,_.\f-'_ ... ,,.,. ... _ , . . , , _
(8.32) The i&tter de~~olllpos.itlon is clearly sec:n in (8~25), (8.26). Analo ..
gonsl~? t.,c) (8 .. 28), (S.30) we. define the angular momentum vectors
itt.{lz~~ lz'lt) and j'nl,(i~1i' fjyn~ );::u) f nth particle bJI
64 GROUP THEORY IN QUANTUM MECHANICS
These operators should not be confused with the quantum numbers l a,nd j. From (8.32) we also have
(8.34)
Let ,pm(J) be a wave functioll transforming according to D<i>
under rotations. Then from (8.18),
(1~2 + 1'1/2 + Iz2)l/Jm<J) == (11+1_ + ll_I+ + 1z,2).pm{J)
= j(j + 1 )t/;mU). (8.35)
Thus ~m (1) describes an eigenstate of the total angular momentum with the eigenvalue y![j(j + 1)]1£ and with z-conlponent mli.
Similarly if a function transforms according to D<z> ul1der rotational transforlnations of the orbit.al co-ordinates alone (lea ving spin co-ordinates fixed), thell it correRponds to a stat.e with orbit.al angular momentum v[l(l + l)]/i.
~urther reJferences
A more detailed discussion of infinitesimal rotations, Eulerian angles, the representation .D( 1/2) and its rela,t.ion to the Cayley.
Klein parameters is given by Goldstein (1950, .chapter 4) from a somewhat different point of ,,-iew. .its regards the irreducible representations of the rotation group, there are also three main different approaches as exemplified by Van der Waerden (1932), Weyl (1931), Wigner (1931), Murnaghall (1938), Boerner (1955).
Sumrrwry
The irreducible representations of the full rotatien group are called D(1) ,vhere j is an lllteger or half an odd lllteger. DU) is of dimension 2j + 1, and the base vectors are conventionally chosen to t.ransforlll according to (8.18) and labelled by 'In == j, j - 1, ... , -j. A wave function transforming according to D<i) describes a state with total angular momentum v/[j(j + 1)]1i.
PROBLEMS
S.l Start with a book in some definite position and apply successively the rotations Rot(90°, x) and Rot(90°, y) about two perpendicular axes Ox, Oy. Show that t.he final position depends on the order of applying the rotat.ions, but that in either nase it does not correspond to a single rotation by 90yl2 degrees about
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