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Generalized Schrödinger representations III
ANNALS
OF PHYSICS:
57, 145-173 (1970)
Generalized
Schrtidinger
Representations
III
JOHN S. NODVIK Department of Physics, University of Southern California, Los Angeles, California 90007 Received July 17, 1969
The three-dimensional three-coordinate angular momentum problem is analyzed in terms of the formalism of generalized Schriidinger representations developed in two previous papers. It is shown that the three-coordinate problem may be factored in terms of two spinors. The classical and quantum formulations of the problem of the coupling of two three-coordinate angular momenta are given and simple solutions are obtained in terms of relations betweenangularmomentumcreationand annihilation
operators. I. INTRODUCTION The purpose of this paper is to present a discussionof various aspectsof the 3-dimensional 3-coordinate angular momentum problem in terms of the formalism of generalized Schrodinger representations which has been developed in two previous papers [l], [2], hereafter referred to as I and II. In I and II we have discussedthe following representations of angular momentum vectors in three dimensions: (1) A one-coordinate representation wherein one coordinate and the corresponding conjugate momentum are used to describe an angular momentum vector J = (J1, J, , J3) which is constrained in such a way that its magnitude is a constant: / J / = J,, . A simple description is obtained by choosing the momentum to be (say) p, = J, and the coordinate to be y’J = tan-l(J,/.&). (2) A 2-coordinate representation wherein two coordinates and the two corresponding conjugate momenta are used to describe an angular momentum vector J (whose magnitude is not restricted) together with two associated vectors r and I” which are such that r, I”, and J are mutually orthogonal with / r I = 1r’ ] = / J 1. A simple description is obtained by choosing the momenta to be pJ = J3 and PJ = 1J /; the corresponding coordinates are ?J = tan-l(J,/J,) and an angle QJ which orients the vectors r and r’ in a plane perpendicular to J. In this paper we shall be concerned with: (3) A 3-coordinate representation wherein three coordinates and the three 145 595/57/r-ro
146
NODVIK
corresponding conjugate momenta are used to describe a set of orthonormal vectors n(l) , nb) , qs) fixed in a rotating rigid body, together with the angular momentum of the body J and its associated vectors I’, I”.
II. THE EULER-ANGLE
REPRESENTATION
A three-coordinate angular momentum problem is encountered when one considers the three-dimensional rotational motion of a rigid body. The orientation of the body may be specified in terms of an orthogonal transformation matrix Aij , depending on three Euler angles1 01,/?, y, which connects a set of orthonormal vectors n(,) fixed in the body to a set of orthonormal vectors eb) fixed in space, this connection being given by the relation2 (2.1)
We shall write the transformation
matrix Aij in the form (2.2)
where R,(a) =
cos cy sin (Y 0 -sin OL cos 01 0 i 0 0 11
(2.3)
etc. Specifically we have A,
=
%)i
=
Wl
w2
w3
n(2),
n(2),
n(2)3
( n(3)l
n(3)2
n(3)3
(2.4)
3 i
where (Wl
3 w2 9 w3) = (cos a cos
p cos y - sin 01sin y, sin 01cos /3 cos y + cos a: sin y, -sin fi cos y),
h2)l
3 n(2)2
3 n(2)3)
(2.5)
= (-cos a cos /3 sin y - sin 01cos y, -sin OLcos /3 sin y + cos a cos y, sin /3 sin y), @(,)I
3 n(3)2
3 %3,3)
= (cos a! sin /3, sin OLsin /3, cos 8). 1 The notation in this section is consistent with that of Edmonds [3]. ( The summation convention is used throughout; Latin indices run 1 to 3.
I
GENERALIZED
147
REPRESENTATIONS
In what comes later we shall make use of the fact that the components of n(,) , w , and n(,) (i.e., the elements of the transformation matrix Aij = FZ(,)~)may be written in the factored form PI(,)+ = A*A* - BB, n(,).-. = AA - B*B* , n(,), = -AB - A*B*,
q,), = i(A*A* + BB), n(,)- = -i(AA + B*B*), n(,),, = i(AB - A*B*),
n(,)+ = 2A*B, n(,)- = 2AB*, q,), = AA* - BB* I
(2.6)
where B e-ii(a+vl / A z cos -zB =: sin $
(2.7)
eji(ol-v)
and qiji = qin + iqijz . For simplicity we consider the force-free rotational motion of a rigid body about its center of mass (assumed to be the origin of the body axes). The space components wi = CO* eti) of the angular velocity w are given by
and the body components ~5~= w * n(,) by Gi = &Eijkri(i)m/rtk)m , where the dot denotes differentiation with respect to time. The Lagrangian is given by L = +ZijBiGj , where Zij = Zji are the components of the inertia tensor with respect to the body axes, and the momenta prr , pB , pr conjugate to a, /I, y are given by pa = aL/&i = ZijGj(abi/adr), etc. Upon eliminating the time derivatives &, @, f, we obtain the following expressions for the space components Ji = J * eu) and body components Ji = J * qi) = ZijB, of the angular momentum J of the body. J1= J,=
J-e(,) J-e(,)
= -p,cosacot/?-p,sinor+p,cosolcsc/$ = -p,sinacot/?+p,cosol+p,sinacsc/?,
J3 = J * ec3)= pa,
W-9
I
jl = J * n(,) = -pa csc /3 cos y + pB sin y + p,, cot /3 cos y, j3 = J . n(,) = pa csc fl sin y + ps cos y - pv cot /? sin y, j3 = J * q3) = py . I
(2.9)
These components are connected by the relations (2.10)
148
NODVIK
with Aij given by (2.4)-(2.5). verified.
The following
Poisson bracket relations may be
[Ji , Jjl = ~ijrcJ~e , Vi
, Ai
=
-kJk
IJ$ , Jjl = 0,
(2.11)
, i
P,i , Jlcl = &rn~ik > [Aii , jJ = --~m&~ , I
(2.12)
[A, A*] = [B, B*] = [A*, B] = [A, B] = 0.
(2.13)
The minus sign appearing in the Poisson bracket relations between j, , jZ , and ja should be noted: Even if the space and body axes are both right-handed the body components behave like components with respect to a left-handed system insofar as Poisson bracket relations are concerned. Quantization
In a (CL,/3, y)-Schriidinger representation we take the volume element in the normalization integral to be sin p dol d/3 dy, the ranges of the variables being n, 0 < y < 27r. The Hermitian operators corresponding O
pa=f$
3 (2.14)
ps =;(++;COtp)’ ,=ia
i 3y ’
and the operators corresponding to Ji and ji are obtained by substituting into (2.8) and (2.9) and symmetrizing products of operators: J1=
-cosacotjg(T-&)-sina(qG)+cosacscfi($$),
J2 = -sin a cot /I (+ &) J
3
jl
=!ia
(2.14)
i
+ cosa(f+)
+sinucsc/3(+-+-),
+ sin y (+ -&)
+ cot p cos y (f
acY'
= -csc p cos y (f +-)
J2=csc/3siny(+-&)+cosy(++)-cotasinr($$),
$), (2.16)
GENERALIZED
The operator corresponding
149
REPRESENTATIONS
to J2 is given by
J2 = cT12+ Jt2 + J3p = jlz + j,z + js2
=-
a2 29 acuay + ay2 i j-
-2cos/3-
If it is assumed that (01, /3, y) are observables, we may take the functions a basis with respect to which J2, J, , and j; are diagonal to be1 CV~‘“(a, f!l, y) = (1/2n) dj = (1/2~) dj
+ 3 &&(f~,
comprising
/I, y)
+ 4 d,$!,@) eimarfi**
(m, Gi = j,j
-
(2.17)
l,..., -j
(2.18)
and j = 0, 1, 2,... ),
where d~l(P)
=
(j + %I! (j - ~)!]1’2 (:cos ~)“‘”
[(j
+
m) ! (j
The Jacobi polynomial p(aqx) n
= 2”nr C-1)
_
m) !
(sin ~)“-“”
(2.19)
on the right of (2.19) is defined by (1 - x>P (1 + x)-~-g
((1 - x)“+n (1 + X)B+n}.
The expression given by (2.18) may also be written q@(a,
P,!~~~,-n”‘“+m)(cosPI.
in the form
p, y) = (1/2?T)[(j + &)(j + m)! (.j - m)! (j + fi)! (j -
s s!(s-tm+%)!(j-m-s)!(j--C-s)!
(2.20)
?ji)!]1/2
'
where A and B are given by (2.7) and the summation on the right of (2.21) is over positive integers s such that the arguments of the factorials in the denominator are non-negative. The functions given by (2.18) or (2.21) satisfy
(2.22)
(2.23)
150
NODVIK
where & = J1 & iJ, and j, = I1 f ij, . It should be noted that j+ lowers r?i by unity [which is a consequence of the minus sign appearing on the right of (2.1 l)]. It may also be noted that “Yy’“($fi,
9yi(a,
y) = (2~)-l/2 Y,“@, a),
(2.24)
/I, y) = (- I)” (27$1/Z Yj6@, r), 1
“yj”*“(a, j?, y>* = (-I)+*
9’j”*-“(CX, /3, y),
where the Yim in (2.24) are the spherical harmonics. The operators corresponding to Aij = n(,jj may be obtained together with the relations
(2.25)
by using (2.6)
1 3+t= jI ) (2.26) fi)III2gy+i,ah-A - M.i++1) 3-i 2, / +[ (jWC% 3++ [(j+m+I>(j+6+I>I1/2g/m+i,a+i\
(j - m + l)(j - 6. + 1) lj2 gF-b,a-l A${lbfi 3 = (2i + 2Dj + 1)
(2
+ +I) 2p.j
> _[(j+Mj+-1) 3 Il/2gF--:,a+l 3--t=3 /
(2.27)
cw?j
where A+, B+ denote the operators corresponding to A*, B*. These relations are obtained by muhiplying *~~‘“(oI, fi, r) by the expressions given by (2.7) and then using the recursion relations for the Jacobi polynomials. It may be noted that the operators A, B, A+, B+ all commute with each other [cf. (2.13)]: (A, A+) = (B, B’) = (A+, B) = (A, B) = 0.
(2.28)
The introduction of gy*” with half-integral j, m, fi on the right of (2.26) and (2.27) is for mathematical convenience onIy:3 Since Aij = n(,jj involves bi-linear combina3 The relations given by (2.18)-(2.23) and (2.25H2.27) also apply to ?Jy,“‘(a, 8, 7) with halfintegral j, m, ti but the normalization integral in this case is infinite.
GENERALIZED
151
REPRESENTATIONS
tions of A, B, A+, and B+, operating with Aij will give expressions involving ?Vyint*fi with integral j, m, fi only. We shall not list these expressions here.
III.
THE FACTORIZABLE
REPRESENTATION
A simple representation similar to that used in the a-coordinate problem discussed in I is obtained by making use of the variables (QJ, q, , qr ; P, , p, , jJ) where the momenta PJ, pJ , and jJ are defined by
PJ = I J I = [ ps2 + PJ = Js = pa 3
zn+
(pa” - 2p,p, cos /3 + py$‘2,
(3.1)
Al = J-2= Pv ’
Expressions for the coordinates
@I, vJ, $jJ in terms of the variables (01,,!?, y;
pa , p6 , p,) may be obtained by using a generating function F((Y, /3, y; PJ , p, , FJ)
which satisfies pCt=g,
PF,B’
aF
pv = E %J ’
(3.2)
and aF
aF +J
Combining
=
vJ.
(3.3)
(3.1) and (3.2) we find that a particular solution for F is given by Fb, 8, Y; pJ9 pry $9 = PJ tan-%WW
~0sB - P,IUI
PII> + #AY + tan-‘bl(p, - FJcosBN>,
+ P&
+ tan-%/(&
- ~9s
(3.4)
where p = [P,” sin2 j3 -- pJ2 + 2~,@~ccos /3 - fiJ2]1,‘z.
Making
(3.5)
use of (3.3) then gives QJ = tan-l[P,p, VJ = 01+ tan-lb FJ = Y + tan-lb
sin /?/(PJ2 cos p - p,p,)], sin P/(p, - pa cos /$I, sin kVBl(p,- py cos /$I,
(3-e)
152
NODVIK
where PJ on the right of (3.6) is given by (3.1). Combining and (2.9) we obtain
(3.1) and (3.6) with (2.8)
(3.7)
jl = d/p,”- fiJ/”cos+J, j2 = -dPJ” - @J2 sin $J) j2
=
fiJ.
I
The minus sign in the expression for j2 should be noted. Similarly, combining and (3.6) with (2.7) yields - pJ)(PJ - fin)#*J + ~(PJ+ PJ)(PJ + $J)C@@J} X &i(pJ++JJ’, (1/2PJ)(--2/(PJ $-pJ)(PJ - $J)ePioJ + l/(PJ - pJ)(PJ + FJ) e+@J}
(3.8) (3.1)
A = (1/2PJ){,@J B =
The corresponding into (2.6).
expressions
for Aij = nujj are obtained by substituting
(3.9)
(3.9)
Factorization As we have seen in I, the components may be written in the factored form
Ji given by (3.7) and the magnitude
J+ = a*b, J- = ab*, J, = $(aa* - bb*), I J I = PJ = +(aa* + bb*),
/J 1
(3.10)
where (2 = z/PJ f PJ &?+‘@J+“J’,
(3.11)
b = z/p, - pJ e-ii’@J-mJj. i Under rotations of the space axes the quantities a and b transform like the components, with respect to the space axes, of a spinor of first order (a brief discussion of transformation properties is given in Appendix A), and satisfy the Poisson bracket relations [a, a*] = [b, b*] = -i, [a, b] = [a, b*] = [a*, b] = [a*, b*] = 0.1
(3.12)
GENERALIZED
REPRESENTATIONS
153
The body components of J given by (3.8) may also be written in factored form in terms of the quantities (z, b”which are the components, with respect to the body axes, of the spinor whose components with respect to the space axes are a, b. The relation between the space components a, b and body components d, d is given by
(ij = u(i), where, in the Euler angle representation, the 2 u
=
(3.13) x
2 unitary matrix U is given by
e+ivoae&9u,e~im7s~
(3.14)
In (3.14) CJ~, u2 , uQare the Pauli matrices. Using the relation e”“f = cos h + ioj sin h we obtain
u= t “;I
“kj,
(3.15)
and, hence, d = aA* -j- bB*, 6 = -aB + bA. 1
Finally, substituting
(3.16)
(3.9) and (3.11) into (3.16) gives (3.17)
It is to be noted that a” and b” do not depend on @I and do not satisfy simple Poisson bracket relations of the form of (3.12). The body components I< given by (3.8) may now be written as k I”;& cz *, j, = @a* - &*). 1 The Associated
(3.18)
Vectors
As we have seen in I, with any a-coordinate angular momentum J there is associated a pair of vectors I?, I” such that I?‘, r, and J are mutually orthogonal forming a right-handed triad, with I r I = I I” 1 = 1J /, and such that
154
NODWK
In the case at hand, one may also introduce a pair of vectors I’, I” associated with J; the space components of these vectors satisfy the Poisson bracket relations (3.19) and are defined in the same way as are the components of the associated vectors introduced in the 2-coordinate angular momentum problem discussed in I: r+ = - &i(u*u* + bb) = eiOJ[-iP, cos QJ + p, sin GJ], r- = &i(aa + b*b*) = eeiwJ[iP,cos QJ + pJ sin CD,],
(3.20)
r, = $i(u*b* - ub) = - l/PJ2 - pJ2 sin cD~, l-,’ = $(bb - ~*a*) = e+[--iP, r-’
sin QJ - pJ cos aJ],
= &(b*b* - au) = e-iQJ[iPJ sin @, - pr cos CD,],
r,’ = $(u*b* + ab) = l/P+? - pJ2cos @J.
(3.21) I
The body components of r and I” are obtained from transformation having the form of (2.10),
equations (3.22)
and are given by c+ = - &i(g*d* + 66) = - iP,e-i3,
r- = $i(aii + b*b"*) = iP,f+J, F2 = +i@i*b"*- ai) = 0,
(3.23)
I
p+' = i(66 - c*z*) = -$Je-i+J, pet = +(l*J* - 55) = -fiJJeG,,
(3.24)
i-y = ga"*B* + nb) = y/P,2-@,2. I It may be noted that in virtue of (2.10) and (3.22) we may write Aij = qiji = (l/P,2)(p&
+ Fi’rj’
+ iiJj).
(3.25)
In the 2-coordinate angular momentum problem for the case in which J is the orbital angular momentum r x p of a particle whose position and linear momentum are r and p, respectively, the associated vectors enter into the expressions for r and p [cf., Eqs. (5.11)-(5.12) of I]. In the case of the rotational motion of a rigid body one is usually concerned only with the angular momentum J of the body and the vectors no) , n(,) , ng directed along the body axes. In the latter case, the associated vectors r and r’ do not play a physically significant role; nevertheless, the introduction of these vectors serves a useful purpose from the standpoint of
GENERALIZED
155
REPRESENTATIONS
understanding the overall mathematical momentum problem.
structure of the 3-coordinate
angular
Quantization
In a generalized Schrodinger (QjJ, vJ, qJ)-representation the functions comprising a basis with respect to which J2, J3, and Js are diagonal are given by xyy@J,
yJ,
+j,)
=
(m, A = j,j
(2nj-3/3
ei(i~~J+~~.r+fi@~)
- l,..., -j andj
= 0, l,..., orj = + , Q ,...)
(3.26)
which are to be compared with the basis functions given by (2.18). The operators corresponding to PJ, pr , and fi$ are (3.27) and the operators corresponding to Ji and ji are obtained by writing the trigonometric functions in (3.7) and (3.8) in exponential form and then splitting the exponentials, as discussed in I: (3.28) (3.29) When applied to XT*” the operators given by (3.28) and (3.29) lead to expressions identical to those given by Eqs. (2.22) and (2.23) with gy*” replaced by ~7,“. In the (Gr, p?r, fJ) representation, however, we can accomodate either integral or half-integral j. As discussed in I, we may take the operators corresponding to the classical quantities a, b given by (3.11) to be a =
e-ti(@J+wr)
dpJ
+
pJ e-P(@J+@J),
b
e-$i(@J-aaJ)
dpJ
_
pJ
=
These satisfy the commutator relations (3.12),
(3.30)
e-ii(@J-wJ)J,.
relations corresponding
to the Poisson bracket
(a, a+) = (b, b+) = k, (a, b) = (a, b+) = (a+, b) = (a+, b+) = 0, I
(3.31)
156
NODVIK
and are such that the operators given by (3.28) result when a, b, a*, b* in (3.10) are replaced by the operators a, b, a’, b+, respectively; the corresponding replacement in (3.20)-(3.21) yields operators representing the components of I’ and I” but these are not of interest in the problem at hand. The Poisson bracket relations satisfied by the body components 5, 6, d*, 6*, do not have a simple form and it is not clear how the corresponding operators ought to be ordered. However, if we are interested only in the quantities Ji , IL , and /I(~)~we may define (somewhat arbitrarily)
in which case the expressions for Ji given by (3.29) may be written as
J+= a+&, j- = d+n, J3 = &?+c? - b’h),
(3.33)
I
and the expressions for the operators A, B, A+, B+ to be inserted in (2.6) are [cf. (3.9)] A = AJ
[d+u + b+b] d&J,
B = -&
[d+b - a’61 d&J,
(3.34)
A+ = -&
[~+a” + b+b] -&
,
B+ = &
[b+i? -
-&+a] &;
.
(3.35)
The operators given by (3.34)-(3.35) all commute with one another [cf. (2.28)] and, when applied to xy*“, lead to expressions identical to those given by (2.26) and (2.27) with C?Yy**replaced by x7*“. It may be noted that the basis functions ~7’” given by (3.26) may be written [in a set of units in which fi = I] as (j - iii)! (j+Cz)!(j+m)!(j-mm)!
(ut)i+m (bt)i-m 1"'(d+c)fi x;,O
(3.36)
in the case of integral j, m, 5, and as XJP*
=
[(j
+
m)
! (j
-
m)
!]-I/2
&%,(u+)i+m
(b+)jd
x",,O
(3.37)
GENERALIZED
in the case of integral or half-integral p”
= [(I2+ ‘7
j, m, A. Another form is given by
l)(j+m)!(j-m)!(j+ (-l)j-e-s
157
REPRESENTATIONS
fi)!(j-
(At)s+m+fi
(A)s
fE)!]llz
(Bt)j-m-s
(B)j-fi-s
s!(s+m+iii)!(,j-m-s)!(j--G-S)!
(3.38)
x”o*
Inasmuch as there is a one-to-one correspondence between the (QJ, v,, $jJ) and (a, /3, y)-representations, it follows that Eq. (3.38) must hold in the (GJ, vI, $,)representation if Eq. (2.21) holds in the (cy,p, y)-representation.
IV. STRUCTURE OF THE ~-COORDINATE
ANGULAR
MOMENTUM
PROBLEM
The structure of the 3-coordinate angular momentum problem being considered here may be described simply as follows: All the quantities of interest depend on two spinors whose components are expressible in terms of three coordinates and three corresponding conjugate momenta. One spinor has components a, b which are given by (3.11) and satisfy the Poisson bracket relations (3.12), the second has components A, B which are given by (3.9) and satisfy the Poisson bracket relations (2.13). The components a, b determine, via Eqs. (3.10), (3.20), (3.21), a set of orthogonal vectors J, I’, I” having the same magnitude, and the components A, B determine, via Eq. (2.6), a set of orthogonal vectors nu) , nc2), n(,) having unit magnitude. In the next two sections we shall be dealing with a number of different 3-coordinate angular momenta. These will be labelled with a superscript and all the related quantities will be labelled with the same superscript or subscript. Thus, the associated vectors corresponding to the 3-coordinate angular momentum J@) will be denoted by IQ), JW’, and the related body vectors by n{;i , n$‘, , n$ . Usually, but not always, Aij’ will denote the transformation matrix connecting the rn’(i! and a set of vectors eo) fixed in space, A!?) z, = nil,’ * e (3)’
(4.1)
and usually, but not always, jtv) will be defined by4 jiu)
=
J(v)
- n[;,’ .
In any case, in a (CD”, y”, $5”; P, , py , &)-representation, the momenta defined by $” = p, py = J(v) P, = I J(“) I, 3 ’ 4 The summation convention does not apply to Y in Eqs. (4.2)<4.9).
(4.2) will be (4.3)
158
NODVIK
the classical expressions for the space components a,, 6, and body components 12,) 6” of the spinor relating to the triad J(“), W), I’(~)’ will be given by
and the space components A,, B, of the spinor relating to the triad n$ , n$‘, , n{$ will be given by A, = (W’Nv*av
+ b,*&),
B, = (1/2P,)(a”,*b,
- a,*&). I
In the corresponding quantum theory, the momentum
(4.6)
operators will be taken to be (4.7)
and the operators a,, b, will be ordered in the same way as those given by (3.30), the operators d, , 6” in the same way as those given by (3.32), and the operators A, , B, in the same way as those given by (3.34). These operators will be combined to form the space and body components of the various vectors in the manner described in Section III. In Section VI we shall also make use of the fact that Jicy) and J”:” can also be written in factored form in terms of certain conjugate quantities avC,b,” and Zvc,Lvc; if, in the classical case, a,, b, and d, , &, are defined by (4.4)-(4.5) then the corresponding conjugate quantities are defined by avC
=
a e'iav Y 2
=
d/p, + py e-i%,
b,” = b,.&% = 1/P, - py etiw,,
(4.8) i
In the quantum case, the operators {a” c, b,“) are ordered in the same way as are the operators {Z, 6+} given by (3.32), and the operators {c?,~,gvc} are ordered in the same way as are the adjoints {a+, b+} of the operators given by (3.30). As discussed in Appendix B, the conjugate quantities enter in when the roles of eti) and II(~)
GENERALIZED
159
REPRESENTATIONS
are interchanged, i.e., when one chooses to consider the e(i) to be fixed in the body and the n(,) to be fixed in space.
V. ANGULAR
MOMENTUM
ADDITION
In the classical case, the problem of the addition of two angular momenta J(l) and J”’ refers to the problem of setting up a representation in which two of the momentum variables are the magnitude and (say) the 3-component of the sum J = J(l) + Jc2). The corresponding problem in the quantum case consists in finding a basis with respect to which J2 and J3 are diagonal. In this section and the next we shall consider two such sum representations which apply to 3-coordinate angular momenta J(l) and Jf2’. The (I J(l) 1, / Jc2’ I, I J 1,J3, ji’), j;“)
Variables
The sum representation considered in this section is characterized by the equations J. = ,!l) + $2) t 3 p:
=
+,
(5.1)
3i(z’,= 3p, and employs the set of canonical variables (c&‘, Q&‘, QJ, yJ, qI’, q2’; PI’, P2’, PJ , pr, A’, A’) which are connected to the variables (Q , v1 , @I ; PI , pl, A) and (Q2, y2, F2 ; P, , p2 , A) referring to J(l) and JC2’ by the relations \
PJ = 1J / = 1J(l) + J(‘) j ~= [PI” + P2” + 2p,p, + 2 dPl2 - p121/p,z - p22cos(y1 - q4]1/2,
> (5.2)
pJ = J, = 5:) + .I?) = p1 + p2 , PI’ = 1J(l) / = PI, P,’ = j J(‘) I = P, ,
/ j&1’=
A’ =
3;) = p1 ) 33(2’ = F2 )
41 = @I2 q2’ = $j2.1
In this case J. Ji”, and J,!“’ are, respectively, the components with respect t’o vectors eti) fixed in space, 32” the components to the body vectorsni”, and Jj2’ the components of Jt2) with vectors I$“’ [cf. (4.2)]. Since & , e2 , A , fi2 are unchanged, the
(5.3) of J, J(l), and J@’ of J(l) with respect respect to the body new representation
160
NODVIK
is very similar to that employed in II in connection with the addition of two 2-coordinate angular momenta; most of the results we require may be carried over from II. In a generalized Schr6dinger (c&‘, &‘, DI, p?r, c&‘, $,‘)-representation the functions comprising a basis with respect to which J(l)‘, J(2)2, J3, J, , Jill, J.$‘) are diagonal are given by 1-h j2 j * fil f%> = - (2$2 (I
fii,
I
<.A3
and we take the momentum
expb(j&’
I fi2
+
I
j202’
$-
jQJ
I * I
+
*qJ
+
*p&
+
&&‘)I
jl - j2 I Gj <.A + j2),
(5.4)
operators to be pJ=;&+;,
(5.5) pJ=;&,
The operators corresponding to a, , b, , a,, 6, are given by [cf. (1.47)-(1.50) of II] a, = -${a,[(P,
+ PI’)” - Pk2]l12 + bJt[Pi2 - (P, - Plf)2]1/2} -&
e--t%‘, J
bl = --&{b,[(P,
+ PI’)” - Pi2]l12 - aJt[P12 - (P, - Pl’)2]1/2) d2k
U, = TkJ{aJ[(PJ
+ P2’)2 - P;2]l12 - b,+[P;Z - (PJ - P2’)2]1/23 $
b2 = -&
3
{b,[(P, + P,‘)” - Pi2]l/2 + a,+[P;z - (pJ - P2’)2]l12> -& .I
where aJ and bJ are given by (3.30). The form of the operators unchanged,
(5.6) e-ti@tl’, e-t*%‘, (5.7) e-S%‘,
J
i
& , & , 6,
J2 is
(5.8)
and the operators Al , Bl , As, B, are obtained by substituting (5.6)-(5.8) into the expressions
GENERALIZED
161
REPRESENTATlONS
In a generalized Schrbdinger (@‘, , v1 , & , QZ, v2 , &)-representation, the functions comprising a basiswith respect to which J(l)‘, @, j;cl), J@, Ji2), j;2J are diagonal are given by
(ml y iz, = j, ,.jl -
1)...)-j,
and m2 , Fi2 = j, , j, - l,..., -j,
and j, , j, = 0, l,..., orjl , j, = 4, $ ,... ),
(5.10)
which may also be written [in a set of units in which X = I] as
I jl ml 1% .i2 m2 fi2> = [(j, + ml)! (jl - m,)! (j, + m,)! (j, - me)!]-1/2 xe ~~,~l+~~z~,(alt)~l+ml (&yl-T
(a2t)j,+% (~2t)~2-%, 0).
(5.11)
The corresponding basisfunctions in the (@,‘, Q2’, @$, vJ, &‘, +,‘)-representation are obtained by substituting (5.6)--(5.7) and replacing &, q2 by qI’, e2’ on the right of (5.1 I). In either representation ) O> = 1427~)~is the state in which all the quantum numbers are zero. The coupling coefficients relating the two setsof basis functions are given by
( jl’ j,’ j m El’ @ii,’ 1j, ml Fzl j, m2 $> = 6..3131’ 6.3232’ 6~tl~&+2ilzwz41 j2 j nl I jl ml j2 m2h
(5.12),
where the Clebsch-Gordon coefficient (j, j, jnz I j, ml j, m,) is the same as the one which enters into the coupling of two 2-coordinate angular momenta; as shown in Appendix A of II, an explicit expression for the latter may be obtained in a straightforward manner by expanding the right side of (5.11).
VI. ANGULAR
MOMENTUM
ADDITION
(CON'T.)
In this section we consider another sum representation for two 3-coordinate angular momenta J(l) and J@). In this representation the momenta consist of the magnitudes and certain components of one of these vectors, say Jc2), and of the sum vector J, which we shall also denote by Jf3), $3’
595/57/I-11
_
J
=
J’l’
+
J’3’.
(6.1)
162
NODVIK
Specifically, we shall be concerned with the components Jj3), ji3), Ji4), Jj4) defined by Jp’ = J * eci) = (J(l) + 5”‘)
* e(,) ,
j!3) I
=
J
.
5!4) 2
=
J(3)
. ,,# .
=
2 d,’
(5’1’
+
5’3’)
nk’
,
(6.2)
3
Jy' = Jt2) . n(t;,) ,
i
where, as before, the eti) are orthonormal vectors fixed in space, the n’($orthonormal vectors fixed in the body whose angular momentum is J(l), and the I$‘, orthonormal vectors fixed in the body whose angular momentum is Jf2). However, the notation here differs from that of previous sections in that the Jj4) are components with respect to the first set of body axes and not with respect to the space axes. The following relations are readily verified. (Eq. (2.12) proves useful in this regard.) Jp)Ji(3’
=
]i(3yp’
=
, J(1)
+
J(3)
13,
(6.3) I
[Ji’3’, Ji’“‘]= EiikJp, [p’,J/3’] = -EijrJf’, [Ji’3’,pq
(6.4)
= 0,
[Ji’4’,Ji’4’] = cijkJf’, [J,!“‘, J/4’] = -Q.J~‘,
[Ji’3’,
Ji’4’]
=
[Ji’3’,
JF’]
=
[Jp’,
Ji’4’]
(6.5)
=
[Ji’a),
ij’“‘]
=
0.
(6.6)
Thus, Jj3), Jj3J, Jj4), J”k4) satisfy the same formal relations as Jil), jjl), Jj2), /i2), respectively, and we now have two 3-coordinate angular momenta one of which is the sum given by (6.1). The sum representation characterized by (6.2) may be generated in the following manner. In an Euler-angle representation the body vectors n$‘, and $1 are connected to the space vectors e(i) by the relations
GENERALIZED
163
REPRESENTATIONS
where A\:’ (IY~, & , rJ in place of orientation tion from
depends on Euler angles (a1 , /31, yI) and A$’ depends on Euler angles in the manner specified by Eqs. (2.4)-(2.5). Now suppose we introduce, the set (LYE, j!& , y2), a set of Euler angles (Q , ,f& , y,J which specifies the of the n’$ relative to then&‘; specifically suppose we make a transformathe representation in which the coordinates are the Euler angles az , /I$ , yJ to a representation in which the coordinates are the Euler (a1 7 A 3 71, angles (a3 , P3 , y3 , a4 , P4 , r4) such that (6.8)
where AiT) depends on the Euler angles (01~)p3, r3) and Ai:) on the Euler angles (a4, 134, y4) in the manner specified by (2.4)-(2.5). Comparing (6.7) and (6.8) we have (6.9)
so that the set (a3 , /I$ , y3) is the same as the set (01~, fil , -&while the set (cx~, p4, y4) depends on both (01~, /II , rl) and (01~, p2 , y2). Using (2.6) we find that the transformation given by (6.9) is equivalent to
A, = A,,
A, = A,A, - B,*B, ,
& = B, ,
B2
=
&A4
+
A,*4,
(6.10) i
where A, and B, depend on (01,) pp, yy) and have the form given by Eq. (2.7). The transformation connecting (o(1, /3, , y1 , a2, ,&, y2) and (aa, ,&, y3, a4, p4, y4) is a simple point transformation and the transformation equations for the corresponding momenta are readily obtained. If the components Jj3) and Jy) depend on ((Y~, &, y3) and the corresponding momenta in the manner specified by Eqs. (2.8) and (2.9), respectively, and similarly for .Ij4) and J”i4), then a straightforward calculation shows that these components may be expressed in the form
(6.11)
The expressions given by (6.11) are equivalent to those given by (6.2).
164
NODVIK
From this point on we shall use the set of variables [cf. Section IV] (@+ , p)r , $& , @2 9 P2 2 Q2 ; PI, pl, A, Pz , p2, A) for @, jz!l), Ji2), jjz) and the set (@, , p3, “(3), Jj4), jj4). In relating these $53 2 @a 3 p)4 9 $54 ; Pa, p3, h , P4 , p4 , 15~)for Ji3’, Ji two sets of variables the simplest procedure is to work through the intermediary of the set PI’, G2’, QJ, vJ, %‘, q2’; PI’, P2’, PJ, p, , A’, A’) considered in Section V: The angular momentum coupling problem characterized by Eq. (6.11) may be broken down into two separate problems of the type considered in Section V, the first being characterized by the equations given by (5.1) and the second by the equations J))’ = p
- Ji(4),
Jj2)’ = p,
(6.12)
J.z = J!3) z The solution to the first part is given by Eqs. (5.6)-(5.8) and the solution to the second part may be obtained by following the same algebraic procedure which leads to (5.6)-(5.8). In the second part, however, the components ]!I)‘, ji3), Jj3), and Ji must be factored in terms of conjugate quantities having the form given by (4.8)-(4.9).
The equivalence of the coupling problem characterized by (6.12) and that characterized by (5.1) becomes evident if, on the left side of (5.1), we make the replacement (@l’,
@J 2 h’, -
(@J
9.l , @l’,
(@2’,
@2’;
; pl’, =
PJ -
P2’,
, $1’~
p?r , n $2’)
-+
PJ) +I’; P2’,
PJ
, PI’,
$52’;
P$‘,
-PJ A’)
(6.13)
, -A’) I
and on the right side of (5.1), the replacement (~j1,9)1,~1;pr.,Pl,~~l)j(~3,~---3,rr--3;p3,-~3,-p3) P2
3 F2
2 $2
; p2
3 P2 , $2)
(614) +
(@4,
9)4 , $54 ; p4
, p4
, B4>.
i
Under the replacement given by (6.13), the components (Ji , ]I”‘, J”j2)‘} on the left of (5.1) go over into {-Ji -(l)‘, - Ji , fi”)‘> and under the replacement (6.14) the components (Jil), Ji2), Jil), ji2)} on the right of (5.1) go over into {-Ii3), Ji4’, - Jj3), J”:“)}; the set of Eqs. (5.1) then goes over into the set (6.12). The solution to the coupling problem characterized by (6.12) may now be obtained by making the replacement (6.14) on the left sides, and the replacement (6.13) on the right sides, of Eqs. (5.6)-(5.8).
GENERALIZED
165
REPRESENTATIONS
In this way we find that a solution to (6.12), analogous to that given by (5.6)-(5.8), is5 c= aJ”, (6.15) ;Ic = bJc,
- C= [email protected]/:P,, {[(P,’ + P,)” - Py %
ci;”
\
- [Pi” - (Pl’ - PJ)2]l’2 d2+, &C= eii@J& __ {[(P,’ + PJ)2- P;y
b;c
+ [PZ - (P,’ - PJ)2]l/2(qq+> -&
(6.16)
, 1
a4 = - &
>
)
{(ii?)+ [(PI’ + P,‘)” - PJz]ll’ 1
-
Z;"[P,2
-
(P
1
' -
-dipl,
P2')2]1/2}
e-ii%',
(6.17) b4
=
d2;l,
{a~)+
[Pl'
+
P,')2
-
PJz]l/2
-&, + QP,’ -211’2> ii, = (z2’,j b4= tT2’.)
e-$i@z’ 9 (6.18)
In the (Q3, y3, &, Q4, y,, @,,)-representation the quantities {sac, bat} and {ZSc, sac} appearing on the left side of Eqs. (6.15)-(6.18) are obtained by setting v = 3 in Eqs. (4.8)-(4.9), and {a, , b4} and {Cz,, i4} by setting v = 4 in Eqs. (4.4)(4.5); in the (G1’, Q2’, vJ, &‘, +,‘)-representation the quantities {Z2’, g2’} appearing on the right side of (6.15)-(6.18) are given by (5.8) and the expressions for (arc, bJc} are obtained by replacing (@” , cpy; P, , pv) in (4.8) by (GJ, yJ ; PJ , p,), those for {Zp, 62 } by replacing (CD”, & ; P, , A) in (4.9) by (c&‘, $jl’; PI’, A’). The corresponding operators must be ordered in the manner described in Sections III and IV. @J,
5 The solution to (6.12) given by (6.15)-(6.18) is unique to within arbitrary constant phase factors for the pairs {sac, b,” }, {ESc,gac}, {ad, Lr4), and {Cd, &}. Actually, with the replacement (6.13)-(6.14) one obtains {--ia*, -i&} on the left of (6.17) instead of {a4 ,6,); we have changed the phase of {a,, b4)1in order to make the result obtained by combining (5.7)-(5.8) and (6.15)(6.18) consistent with the result (6.10) obtained in the Euler-angle representation.
166
NODVIK
the functions comprising a basis In the (Q3, 9~~, $j3, @, , P)~, &)-representation with respect to which J(3)2, Jp), jp), Jt4)*, .I?), IL*) are diagonal are given by
m3F3+ fi3$j3+j4@, + n?,~, + %+54)1
lj3 m3 fi3 j4 m4 fi4) = -evP(j3@3+ &).
(6.19) which may also be written
[in a set of units in which fi = l] as
lj3 m3 fi3 .i4 m4 fi4)
= [(j, + fi3)! (j, - fi3)! (j, + m,)! (j, - m4)!]-1/2 x eim,m,+irii,e,(d3cy,+fi, (~3c)j,-fi, (a4t)j,+nl, (b4t)j,-m, / 0). The corresponding basis functions are obtained by setting y3 = yj, and {u4, b4} on the right of (6.20). be obtained by making use of the have
(6.20)
in the (@r’, Q2’, QJ, vX, &‘, $,‘)-representation e4 = F2 and using (6.16)-(6.17) for {d3c, b3C} The explicit expressions for these functions may results of Section V. From (5.11) and (5.12) we
Cl + ml) ! (A - ml) ! (j2 + m2)!(j2 - m2>!l-1/2 x p,@,‘+irii,~,’(alt)‘l+n”1(blt)k? (a,y+“z (b2t)e% 10) = ig, exp[Wh’
+ j2@,’ + j’@, + m’yJ + file,
+ fi2q2’)]
(6.21)
where {a, ,6,} and {a2 , b,j are given by the right sides of (5.6) and (5.7). If, in (6.21), we make the replacement (6.13) together with the replacement 13 3
w
.
? “13
J2 9 m2
9 fi2)
-
(j,
,-
fi3
, -%
, Ji
, 1114
,
54)
we obtain (j [(j3
-
fi3)!
(j, +
fi3)
! (j4
+ m,) ! (j, - m4)!]-1/2 exp[--im,(n
- p?l) + ijjt4~2’]
(-+’zn/2a4t)i4+m* (e-in/2b4t)j4-m, j 0) = ,g, (j3.i4j’ m’I j3- fi3 .i4ma> x
(&n/21;3cy~-l,
(e-in/2d3Cy3+%‘3
exp{i[j3@J + j4G2’ + .i’@4’ + m’(r
- &‘) - m3(n - vJ) + iii,@‘]} (6.22)
GENERALIZED
REPRESENTATIONS
167
where {Zsc,Z&c>and {a, , b4} on the left of (6.22) are given by (6.16)-(6.17). Comparing with (6.20) we find
I j3 nz3fi3 j4 m4fi4) = (- l)j4+m4 ez(7rr’+fi3-m4)~(j3 j, j’m’ L,
1j, -#ii, j, m4> jj’ j, j, m3-ni
Fz4), (6.23)
so that (j,’ j,’ j m ei,’ iii,’ 1j, m3 E3 j, m4 Fi4) = (- lP+m4 sij,8mn,,sj,‘j,slii,,~4(j3 j, j,’ - fi,’ j j, - fi3 j, m4), (6.24) where we have made use of the fact that the Clebsch-Gordon coefficient on the right of (6.23) vanishes unless m’ = - fi3 + m4. Finally, combining (6.24) and (5.12) we obtain (A ml fh .i2 m2 % I j3 m3 s3 j4 m4 fi4) = (-l)j4+m4 Sj2j4Sf12fi4CA j3 j3 nf3 I jl ml j2 @ x (.j3 j4 jl - % I j3 - E3 j4 m4>.
APPENDIX
A.
TRANSFORMATION
(6.25)
PROPERTIES
In this appendix we present a brief discussion of the way in which various quantities appearing in Sections II and III transform under a change from one (observer’s) orthonormal set of spacevectors et,) to another (observer’s) orthonorma1set of spacevectors e& . Assuming that thesetwo setshave the samehandedness we may write e& = A!?)e 2, (J)
(A.11
where A$’ is a real, orthogonal, constant matrix with unit determinant, A!O’A!O’ = A’O’A’O’ Zk 3k ki ki
det(d$)) = 1,
= 6
ii
’
Ai:) = real.
64.2)
Any matrix having the properties given by (A.2) may be written in factored form in terms of complex quantities A, , B. which satisfy AoAo* + BOB,* = 1.
(A.3)
168
NODVIK
The specific expressions for A$) may be taken to be those given by (2.6) with 11~~)~ on the left replaced by A$’ and {A, B} on the right replaced by {A,, B,}, e.g. Ai;’ = A,A,* - B ,,B ,,*. If the orientation of the eii, relative to the eci) is specified in terms of Euler angles (LYE,/IO, ~~0)then the expressions for A, and B,, are obtained by replacing by P, r> in (2.7) by (ho, is, , R,). In order that a quantity V be a vector, it is necessary that its components (V, , Yz , P’J with respect to the set et<) and its components (Vi’, V,‘, V,‘) with respect to the set eii, be related by Vi’ = A$‘V, .
G4.4)
Similarly, if a quantity u is to be a spinor its components (ul , UJ with respect to the set e(,) and its components (ul’, u,‘) with respect to the set eii, must be related by ’ = A,*u, + B,*u,, z&” = -B,u, + A,u, . I
Ul
(A.5)
It may be noted that if (ur , UJ transforms according to (A.5) then so does (u?*, -u1*>. If ntijj = Aij are the components of the body vectors nci) with respect to the set eci) , and PZ;~,~= Aij are the components with respect to the set eii, , we must have
64.6)
Combining
(A.6) with (2.6) we obtain (to within an overall minus sign) A’ = A,*A + B,,*B, B’ = -B,,A + A,B, I
which imply components Similarly we (3.21) are to
64.7)
that the quantities A, B appearing in the factorization of the space of the triad n(,) , n(,) , nc3) transform like the components of a spinor. find that if the quantities Ji, ri, and ri’ given by (3.10), (3.20), and transform like vector components, Ji’ = A$‘J,
64.8)
a’ = A,*a + Bo*b, b’ = -Boa + A,b, I
(A.9
etc., then we must have
GENERALIZED
REPRESENTATIONS
169
which is to say that the quantities a, b appearing in the factorization of the space components of the triad J, I?, I” must transform like spinor components. Finally, if we combine (A.7), (A.9), and (3.16) we obtain 2’ = a’jf’* 6’ = -a’#
+ b/B’* = aA* + bB* = d, + b’A’ = -aB + bA = 6, I
(A. 10)
which imply that the quantities 5, d appearing in the factorization of the body components of the triad J, I’, r’ transform like scalars (as do the body components of these vectors themselves). In particular, PJ , ~5~) and qJ transform like scalars. Suppose now that the vector components (V, , V’, , Vs) and/or the spinor components (ul , u,), with respect to the set e(,) , can be expressed in terms of a set of canonical variables (qi, pi). Then in general there will be a transformed set of canonical variables (qj’, pi’) related to the set (qi , pi) in such a way that the components (VI’, V,‘, V3’) and/or (UI’, u,‘) with respect to the set e;,, depend on (qi’, p,‘) in exactly the same way as (VI , vz , VJ and/or (v ,4 depend on hi , PA. (If this were not the case there would be certain observers singled out as preferred over other observers and some of the physical results obtained would depend on the particular observer who formulates the problem.) The observer using the basis vectors eii, may then use a Hamiltonian formulation in terms of the variables (qi , p,), or a formulation in terms of the variables (qi’, p,‘). In the case of the representation discussed in Section III, the connection between the variables PJ, p?r , GJ ; PJ, pJ, $$J) and (@i, yJ’, +J’; Pi, pi, &‘I corresponding to the transformation (A.l) is given by (cf. (A.9) and (A.lO)) &‘J’
+ p; e-+i(@J’+mJ’) = A,* 2/pJ + pJ e-li(@J+mJ, (A.ll) ___+ A, dPJ - pJ e-ii(3-~r), (A.12)
The observer using the basis vectors eii, may set up a formulation in terms of the variables (QJ, Pi, $ J ; PJ , pJ , fiJ), in which case he would factor the components Jl = J . e& in terms of quantities a’, b’ given by the right sides of (A. 1 l), or a formulation in terms of the variables (CD;, vi, @J’; Pi, pi, ji), in which case he would factor Ji’ = J * eii, in terms of quantities a’, b’ given by the left sides of (A.1 l), etc. It may be verified that the transformation specified by (A.1 l)-(A.12) is a canonical transformation for arbitrary complex constants A, , B, which satisfy (A.3).
170
NODVIK
In the quantum theory there is a correspondence, but no direct connection between the (@$, yJ, qJ)- and (@pJ’, p)J), $;)-representations. In the (@i, 9?;, +,‘)representation the functions comprising a basis with respect to which J2, J - e;a, , and J * n(,) are diagonal, are given by x;.“(@,‘,
?,‘,
,&‘)
and the corresponding [(j
+
Ill)! =
(j [(j
x
+
i
! (j
eifi6J(ut)j+m'
=
[(j
eiWJ’+m’aJ’+%J’) +
m)!
&“@J(x&d -
m)
(j
-m)!]-l/2
&Z+J’(a’t)i+m
(b’t)j-ht
1 0)
in the (cD~, v, , @j)-representation +
&bt)‘+”
(--B,,*d
+
Lf,,*bt)‘-”
are 1 0)
(Qs+m’+m
(Ao*)8 (&)j-m’-s
(&*)Gm-s
1
s! (s + m’ + m)! (j - m’ - s)! (j - m - s)! (Qh'
(*‘13)
!]--1i2
c (-l)j-m-s
rn’=-j L s x
&-3/S
basis functions
WZ)!]-1/2 m)
z
(A. 14)
/ 0).
The right side of (A. 14) is obtained by multiplying together the expansions
(A.15) i and then making a change of summation variable from k to m’, where k = s + IH + m’. Writing A, and B, in terms of Euler angles(aO, so, y,,) as in (2.7) and making use of Eq. (2.21), we obtain the correspondence
In the Euler-angle representation, Eq. (A.6) becomes
and the relation given by (A.16) is consistent with the result
GENERALIZED
APPENDIX
B.
171
REPRESENTATIONS
THE CONJUGATE
FORMULATION
Up to this point we have considered the orthonormal set eci)to be fixed in space and the orthonormal setn(,) to be fixed in the body whose angular momentum is J. From the physical standpoint these two setshave rather different characteristics: The choice of the particular set eci)to be used is left to the observer who is formulating the problem, and in general it is assumedthat another observer using a different set eii, [related to the e(,) by (A.])] will obtain the samephysical results as the first observer. The set n(,) , on the other hand, is one and the same for all observers-only the components of the nci) differ for different observers: n(,) = we(j) = &e;,, . That the setse(,) and n(,) have the characteristics just mentioned is simply a matter of definition. We could just as well adopt a point of view in which the roles of eu) and n(,) are interchanged, with the n(,) being considered as some observer’s basis vectors fixed in spaceand the eu) asvectors fixed in the body whoseangular momentum is J. In such a case we would be dealing with the inverse of (2.1) and the resulting formulation, which we shall refer to as the conjugate formulation, is somewhat different from the one considered in Sections II and III in that the components of the associatedvectors and various other quantities must be redefined. Quantities which refer to the conjugate formulation will be labelled with a superscript c. Supposethat L?,are (vector or spinor) components with respect to the set et,) , and ai the corresponding components, with respect to the set nci, , when the e(,) are regarded as some observer’s basisvectors and the nci) as body vectors; when the roles of et,) and n(,) are interchanged, we shall define the analogous components with respect to the set e(,) to be Qic = C{aii,} and the analogous components, with respect to the setn(,) , to be &i)ic= C{sZ,>,where C is the operator whose effect in the classicaltheory is to replace
(01,A Yi Pa5Pa7PJ
by C-Y, -B, --ol; pv 3PO, ~a)
in the Euler-angle representation and
in the (QJ, v,, eJ)-representation. This definition is such that Jic = Ji and Jgc = ji , i.e., the components of J are the sameno matter which point of view one takes; however, in the conjugate formulation these components are factored in a different way. In the conjugate formulation we factor the components e (Z)j =
eti,
- ntj,
=
(A-l)ij
=
A&
(B.1)
172
NODVIK
in terms of the quantities -J;‘, B” given by z& = C(A) = A*, ) B’ = C(B) = -B. j
VW
Equation (B.2) is consistent with the fact that if Aij is factored in terms of {A, B} then (A-l)ij [which, in the Euler-angle representation, is obtained by replacing (cy,/3, r) by (-y, -/?, -m) in Aij] is factored in terms of (A*, -B}. The components of J and its associated vectors rc, I”~ are factored in terms of the quantities ac, bc and 3, bc given by ac = C(C) = l/pJ
+ pJ e-tim,
)
= eti@J a,
(8.3)
b’ = C(d) = d/pJ - pJ eiiQJ = eti3 b, (B-4)
Thus the components, with respect to the set eu) , are J+” = (ac)* (6”) = dPJ2 - pJ2 e2m.r = J+ , r+’ = - &i[(a”)* (ac)* + (bc)(bc)] = [email protected],
(B-5)
I
etc., and the components, with respect to the set nt,) , are l+’ = ($)* (&) = z/pJ2 - jJ2 r+c = - &i[(Z)* (S)* + (@)(@] = &+J
=
I+
e-@J[-ip,
,
~0s
QJ
-
fiJ
sin
QJ],
I
(B*6)
etc. Now the Poisson bracket relations involving 9, b”“,etc., have a simple form, while those involving ac, bc, etc. do not. The form is the same, except for a minus sign in front of each Poisson bracket, as that of the relations involving a, b, etc., listed in Sections II and III, e.g. where one has [a, a*] = -i and [r, , rj] = --eijrJk one obtains [3, Lie*] = +iand [pje, p/] = +ciisJti. It may be noted that the associatedvectors whose components are Tic, ric are physically different from those whose components are ri , ri’: When the e(,) are regarded as an observer’s basisvectors and then(,) as body vectors, the definition of the vectors r, I” is such that these are perpendicular to J with r’, l’, J forming a right-handed triad, and such that @Jis the angle measuredcounter-clockwise from J x e(,) to r. In the conjugate formulation where the roles of eti) and nci) are interchanged the definition of F, Fe differs in that GJ becomesthe angle measured clockwise from rc to J x nc3) .
GENERALIZED
REPRESENTATIONS
173
REFERENCES 1. J. S. NODVIK, Ann. Phys. 51 (1969), 251. 2. J. S. NODVIK, Ann. Whys. 54 (1969), 505. 3. A. R. EDMONDS, “Angular Momentum in Quantum Mechanics,” 2nd ed. Princeton University Press, Princeton, New Jersey, 1960.
OF PHYSICS:
57, 145-173 (1970)
Generalized
Schrtidinger
Representations
III
JOHN S. NODVIK Department of Physics, University of Southern California, Los Angeles, California 90007 Received July 17, 1969
The three-dimensional three-coordinate angular momentum problem is analyzed in terms of the formalism of generalized Schriidinger representations developed in two previous papers. It is shown that the three-coordinate problem may be factored in terms of two spinors. The classical and quantum formulations of the problem of the coupling of two three-coordinate angular momenta are given and simple solutions are obtained in terms of relations betweenangularmomentumcreationand annihilation
operators. I. INTRODUCTION The purpose of this paper is to present a discussionof various aspectsof the 3-dimensional 3-coordinate angular momentum problem in terms of the formalism of generalized Schrodinger representations which has been developed in two previous papers [l], [2], hereafter referred to as I and II. In I and II we have discussedthe following representations of angular momentum vectors in three dimensions: (1) A one-coordinate representation wherein one coordinate and the corresponding conjugate momentum are used to describe an angular momentum vector J = (J1, J, , J3) which is constrained in such a way that its magnitude is a constant: / J / = J,, . A simple description is obtained by choosing the momentum to be (say) p, = J, and the coordinate to be y’J = tan-l(J,/.&). (2) A 2-coordinate representation wherein two coordinates and the two corresponding conjugate momenta are used to describe an angular momentum vector J (whose magnitude is not restricted) together with two associated vectors r and I” which are such that r, I”, and J are mutually orthogonal with / r I = 1r’ ] = / J 1. A simple description is obtained by choosing the momenta to be pJ = J3 and PJ = 1J /; the corresponding coordinates are ?J = tan-l(J,/J,) and an angle QJ which orients the vectors r and r’ in a plane perpendicular to J. In this paper we shall be concerned with: (3) A 3-coordinate representation wherein three coordinates and the three 145 595/57/r-ro
146
NODVIK
corresponding conjugate momenta are used to describe a set of orthonormal vectors n(l) , nb) , qs) fixed in a rotating rigid body, together with the angular momentum of the body J and its associated vectors I’, I”.
II. THE EULER-ANGLE
REPRESENTATION
A three-coordinate angular momentum problem is encountered when one considers the three-dimensional rotational motion of a rigid body. The orientation of the body may be specified in terms of an orthogonal transformation matrix Aij , depending on three Euler angles1 01,/?, y, which connects a set of orthonormal vectors n(,) fixed in the body to a set of orthonormal vectors eb) fixed in space, this connection being given by the relation2 (2.1)
We shall write the transformation
matrix Aij in the form (2.2)
where R,(a) =
cos cy sin (Y 0 -sin OL cos 01 0 i 0 0 11
(2.3)
etc. Specifically we have A,
=
%)i
=
Wl
w2
w3
n(2),
n(2),
n(2)3
( n(3)l
n(3)2
n(3)3
(2.4)
3 i
where (Wl
3 w2 9 w3) = (cos a cos
p cos y - sin 01sin y, sin 01cos /3 cos y + cos a: sin y, -sin fi cos y),
h2)l
3 n(2)2
3 n(2)3)
(2.5)
= (-cos a cos /3 sin y - sin 01cos y, -sin OLcos /3 sin y + cos a cos y, sin /3 sin y), @(,)I
3 n(3)2
3 %3,3)
= (cos a! sin /3, sin OLsin /3, cos 8). 1 The notation in this section is consistent with that of Edmonds [3]. ( The summation convention is used throughout; Latin indices run 1 to 3.
I
GENERALIZED
147
REPRESENTATIONS
In what comes later we shall make use of the fact that the components of n(,) , w , and n(,) (i.e., the elements of the transformation matrix Aij = FZ(,)~)may be written in the factored form PI(,)+ = A*A* - BB, n(,).-. = AA - B*B* , n(,), = -AB - A*B*,
q,), = i(A*A* + BB), n(,)- = -i(AA + B*B*), n(,),, = i(AB - A*B*),
n(,)+ = 2A*B, n(,)- = 2AB*, q,), = AA* - BB* I
(2.6)
where B e-ii(a+vl / A z cos -zB =: sin $
(2.7)
eji(ol-v)
and qiji = qin + iqijz . For simplicity we consider the force-free rotational motion of a rigid body about its center of mass (assumed to be the origin of the body axes). The space components wi = CO* eti) of the angular velocity w are given by
and the body components ~5~= w * n(,) by Gi = &Eijkri(i)m/rtk)m , where the dot denotes differentiation with respect to time. The Lagrangian is given by L = +ZijBiGj , where Zij = Zji are the components of the inertia tensor with respect to the body axes, and the momenta prr , pB , pr conjugate to a, /I, y are given by pa = aL/&i = ZijGj(abi/adr), etc. Upon eliminating the time derivatives &, @, f, we obtain the following expressions for the space components Ji = J * eu) and body components Ji = J * qi) = ZijB, of the angular momentum J of the body. J1= J,=
J-e(,) J-e(,)
= -p,cosacot/?-p,sinor+p,cosolcsc/$ = -p,sinacot/?+p,cosol+p,sinacsc/?,
J3 = J * ec3)= pa,
W-9
I
jl = J * n(,) = -pa csc /3 cos y + pB sin y + p,, cot /3 cos y, j3 = J . n(,) = pa csc fl sin y + ps cos y - pv cot /? sin y, j3 = J * q3) = py . I
(2.9)
These components are connected by the relations (2.10)
148
NODVIK
with Aij given by (2.4)-(2.5). verified.
The following
Poisson bracket relations may be
[Ji , Jjl = ~ijrcJ~e , Vi
, Ai
=
-kJk
IJ$ , Jjl = 0,
(2.11)
, i
P,i , Jlcl = &rn~ik > [Aii , jJ = --~m&~ , I
(2.12)
[A, A*] = [B, B*] = [A*, B] = [A, B] = 0.
(2.13)
The minus sign appearing in the Poisson bracket relations between j, , jZ , and ja should be noted: Even if the space and body axes are both right-handed the body components behave like components with respect to a left-handed system insofar as Poisson bracket relations are concerned. Quantization
In a (CL,/3, y)-Schriidinger representation we take the volume element in the normalization integral to be sin p dol d/3 dy, the ranges of the variables being n, 0 < y < 27r. The Hermitian operators corresponding O
pa=f$
3 (2.14)
ps =;(++;COtp)’ ,=ia
i 3y ’
and the operators corresponding to Ji and ji are obtained by substituting into (2.8) and (2.9) and symmetrizing products of operators: J1=
-cosacotjg(T-&)-sina(qG)+cosacscfi($$),
J2 = -sin a cot /I (+ &) J
3
jl
=!ia
(2.14)
i
+ cosa(f+)
+sinucsc/3(+-+-),
+ sin y (+ -&)
+ cot p cos y (f
acY'
= -csc p cos y (f +-)
J2=csc/3siny(+-&)+cosy(++)-cotasinr($$),
$), (2.16)
GENERALIZED
The operator corresponding
149
REPRESENTATIONS
to J2 is given by
J2 = cT12+ Jt2 + J3p = jlz + j,z + js2
=-
a2 29 acuay + ay2 i j-
-2cos/3-
If it is assumed that (01, /3, y) are observables, we may take the functions a basis with respect to which J2, J, , and j; are diagonal to be1 CV~‘“(a, f!l, y) = (1/2n) dj = (1/2~) dj
+ 3 &&(f~,
comprising
/I, y)
+ 4 d,$!,@) eimarfi**
(m, Gi = j,j
-
(2.17)
l,..., -j
(2.18)
and j = 0, 1, 2,... ),
where d~l(P)
=
(j + %I! (j - ~)!]1’2 (:cos ~)“‘”
[(j
+
m) ! (j
The Jacobi polynomial p(aqx) n
= 2”nr C-1)
_
m) !
(sin ~)“-“”
(2.19)
on the right of (2.19) is defined by (1 - x>P (1 + x)-~-g
((1 - x)“+n (1 + X)B+n}.
The expression given by (2.18) may also be written q@(a,
P,!~~~,-n”‘“+m)(cosPI.
in the form
p, y) = (1/2?T)[(j + &)(j + m)! (.j - m)! (j + fi)! (j -
s s!(s-tm+%)!(j-m-s)!(j--C-s)!
(2.20)
?ji)!]1/2
'
where A and B are given by (2.7) and the summation on the right of (2.21) is over positive integers s such that the arguments of the factorials in the denominator are non-negative. The functions given by (2.18) or (2.21) satisfy
(2.22)
(2.23)
150
NODVIK
where & = J1 & iJ, and j, = I1 f ij, . It should be noted that j+ lowers r?i by unity [which is a consequence of the minus sign appearing on the right of (2.1 l)]. It may also be noted that “Yy’“($fi,
9yi(a,
y) = (2~)-l/2 Y,“@, a),
(2.24)
/I, y) = (- I)” (27$1/Z Yj6@, r), 1
“yj”*“(a, j?, y>* = (-I)+*
9’j”*-“(CX, /3, y),
where the Yim in (2.24) are the spherical harmonics. The operators corresponding to Aij = n(,jj may be obtained together with the relations
(2.25)
by using (2.6)
1 3+t= jI ) (2.26) fi)III2gy+i,ah-A - M.i++1) 3-i 2, / +[ (jWC% 3++ [(j+m+I>(j+6+I>I1/2g/m+i,a+i\
(j - m + l)(j - 6. + 1) lj2 gF-b,a-l A${lbfi 3 = (2i + 2Dj + 1)
(2
+ +I) 2p.j
> _[(j+Mj+-1) 3 Il/2gF--:,a+l 3--t=3 /
(2.27)
cw?j
where A+, B+ denote the operators corresponding to A*, B*. These relations are obtained by muhiplying *~~‘“(oI, fi, r) by the expressions given by (2.7) and then using the recursion relations for the Jacobi polynomials. It may be noted that the operators A, B, A+, B+ all commute with each other [cf. (2.13)]: (A, A+) = (B, B’) = (A+, B) = (A, B) = 0.
(2.28)
The introduction of gy*” with half-integral j, m, fi on the right of (2.26) and (2.27) is for mathematical convenience onIy:3 Since Aij = n(,jj involves bi-linear combina3 The relations given by (2.18)-(2.23) and (2.25H2.27) also apply to ?Jy,“‘(a, 8, 7) with halfintegral j, m, ti but the normalization integral in this case is infinite.
GENERALIZED
151
REPRESENTATIONS
tions of A, B, A+, and B+, operating with Aij will give expressions involving ?Vyint*fi with integral j, m, fi only. We shall not list these expressions here.
III.
THE FACTORIZABLE
REPRESENTATION
A simple representation similar to that used in the a-coordinate problem discussed in I is obtained by making use of the variables (QJ, q, , qr ; P, , p, , jJ) where the momenta PJ, pJ , and jJ are defined by
PJ = I J I = [ ps2 + PJ = Js = pa 3
zn+
(pa” - 2p,p, cos /3 + py$‘2,
(3.1)
Al = J-2= Pv ’
Expressions for the coordinates
@I, vJ, $jJ in terms of the variables (01,,!?, y;
pa , p6 , p,) may be obtained by using a generating function F((Y, /3, y; PJ , p, , FJ)
which satisfies pCt=g,
PF,B’
aF
pv = E %J ’
(3.2)
and aF
aF +J
Combining
=
vJ.
(3.3)
(3.1) and (3.2) we find that a particular solution for F is given by Fb, 8, Y; pJ9 pry $9 = PJ tan-%WW
~0sB - P,IUI
PII> + #AY + tan-‘bl(p, - FJcosBN>,
+ P&
+ tan-%/(&
- ~9s
(3.4)
where p = [P,” sin2 j3 -- pJ2 + 2~,@~ccos /3 - fiJ2]1,‘z.
Making
(3.5)
use of (3.3) then gives QJ = tan-l[P,p, VJ = 01+ tan-lb FJ = Y + tan-lb
sin /?/(PJ2 cos p - p,p,)], sin P/(p, - pa cos /$I, sin kVBl(p,- py cos /$I,
(3-e)
152
NODVIK
where PJ on the right of (3.6) is given by (3.1). Combining and (2.9) we obtain
(3.1) and (3.6) with (2.8)
(3.7)
jl = d/p,”- fiJ/”cos+J, j2 = -dPJ” - @J2 sin $J) j2
=
fiJ.
I
The minus sign in the expression for j2 should be noted. Similarly, combining and (3.6) with (2.7) yields - pJ)(PJ - fin)#*J + ~(PJ+ PJ)(PJ + $J)C@@J} X &i(pJ++JJ’, (1/2PJ)(--2/(PJ $-pJ)(PJ - $J)ePioJ + l/(PJ - pJ)(PJ + FJ) e+@J}
(3.8) (3.1)
A = (1/2PJ){,@J B =
The corresponding into (2.6).
expressions
for Aij = nujj are obtained by substituting
(3.9)
(3.9)
Factorization As we have seen in I, the components may be written in the factored form
Ji given by (3.7) and the magnitude
J+ = a*b, J- = ab*, J, = $(aa* - bb*), I J I = PJ = +(aa* + bb*),
/J 1
(3.10)
where (2 = z/PJ f PJ &?+‘@J+“J’,
(3.11)
b = z/p, - pJ e-ii’@J-mJj. i Under rotations of the space axes the quantities a and b transform like the components, with respect to the space axes, of a spinor of first order (a brief discussion of transformation properties is given in Appendix A), and satisfy the Poisson bracket relations [a, a*] = [b, b*] = -i, [a, b] = [a, b*] = [a*, b] = [a*, b*] = 0.1
(3.12)
GENERALIZED
REPRESENTATIONS
153
The body components of J given by (3.8) may also be written in factored form in terms of the quantities (z, b”which are the components, with respect to the body axes, of the spinor whose components with respect to the space axes are a, b. The relation between the space components a, b and body components d, d is given by
(ij = u(i), where, in the Euler angle representation, the 2 u
=
(3.13) x
2 unitary matrix U is given by
e+ivoae&9u,e~im7s~
(3.14)
In (3.14) CJ~, u2 , uQare the Pauli matrices. Using the relation e”“f = cos h + ioj sin h we obtain
u= t “;I
“kj,
(3.15)
and, hence, d = aA* -j- bB*, 6 = -aB + bA. 1
Finally, substituting
(3.16)
(3.9) and (3.11) into (3.16) gives (3.17)
It is to be noted that a” and b” do not depend on @I and do not satisfy simple Poisson bracket relations of the form of (3.12). The body components I< given by (3.8) may now be written as k I”;& cz *, j, = @a* - &*). 1 The Associated
(3.18)
Vectors
As we have seen in I, with any a-coordinate angular momentum J there is associated a pair of vectors I?, I” such that I?‘, r, and J are mutually orthogonal forming a right-handed triad, with I r I = I I” 1 = 1J /, and such that
154
NODWK
In the case at hand, one may also introduce a pair of vectors I’, I” associated with J; the space components of these vectors satisfy the Poisson bracket relations (3.19) and are defined in the same way as are the components of the associated vectors introduced in the 2-coordinate angular momentum problem discussed in I: r+ = - &i(u*u* + bb) = eiOJ[-iP, cos QJ + p, sin GJ], r- = &i(aa + b*b*) = eeiwJ[iP,cos QJ + pJ sin CD,],
(3.20)
r, = $i(u*b* - ub) = - l/PJ2 - pJ2 sin cD~, l-,’ = $(bb - ~*a*) = e+[--iP, r-’
sin QJ - pJ cos aJ],
= &(b*b* - au) = e-iQJ[iPJ sin @, - pr cos CD,],
r,’ = $(u*b* + ab) = l/P+? - pJ2cos @J.
(3.21) I
The body components of r and I” are obtained from transformation having the form of (2.10),
equations (3.22)
and are given by c+ = - &i(g*d* + 66) = - iP,e-i3,
r- = $i(aii + b*b"*) = iP,f+J, F2 = +i@i*b"*- ai) = 0,
(3.23)
I
p+' = i(66 - c*z*) = -$Je-i+J, pet = +(l*J* - 55) = -fiJJeG,,
(3.24)
i-y = ga"*B* + nb) = y/P,2-@,2. I It may be noted that in virtue of (2.10) and (3.22) we may write Aij = qiji = (l/P,2)(p&
+ Fi’rj’
+ iiJj).
(3.25)
In the 2-coordinate angular momentum problem for the case in which J is the orbital angular momentum r x p of a particle whose position and linear momentum are r and p, respectively, the associated vectors enter into the expressions for r and p [cf., Eqs. (5.11)-(5.12) of I]. In the case of the rotational motion of a rigid body one is usually concerned only with the angular momentum J of the body and the vectors no) , n(,) , ng directed along the body axes. In the latter case, the associated vectors r and r’ do not play a physically significant role; nevertheless, the introduction of these vectors serves a useful purpose from the standpoint of
GENERALIZED
155
REPRESENTATIONS
understanding the overall mathematical momentum problem.
structure of the 3-coordinate
angular
Quantization
In a generalized Schrodinger (QjJ, vJ, qJ)-representation the functions comprising a basis with respect to which J2, J3, and Js are diagonal are given by xyy@J,
yJ,
+j,)
=
(m, A = j,j
(2nj-3/3
ei(i~~J+~~.r+fi@~)
- l,..., -j andj
= 0, l,..., orj = + , Q ,...)
(3.26)
which are to be compared with the basis functions given by (2.18). The operators corresponding to PJ, pr , and fi$ are (3.27) and the operators corresponding to Ji and ji are obtained by writing the trigonometric functions in (3.7) and (3.8) in exponential form and then splitting the exponentials, as discussed in I: (3.28) (3.29) When applied to XT*” the operators given by (3.28) and (3.29) lead to expressions identical to those given by Eqs. (2.22) and (2.23) with gy*” replaced by ~7,“. In the (Gr, p?r, fJ) representation, however, we can accomodate either integral or half-integral j. As discussed in I, we may take the operators corresponding to the classical quantities a, b given by (3.11) to be a =
e-ti(@J+wr)
dpJ
+
pJ e-P(@J+@J),
b
e-$i(@J-aaJ)
dpJ
_
pJ
=
These satisfy the commutator relations (3.12),
(3.30)
e-ii(@J-wJ)J,.
relations corresponding
to the Poisson bracket
(a, a+) = (b, b+) = k, (a, b) = (a, b+) = (a+, b) = (a+, b+) = 0, I
(3.31)
156
NODVIK
and are such that the operators given by (3.28) result when a, b, a*, b* in (3.10) are replaced by the operators a, b, a’, b+, respectively; the corresponding replacement in (3.20)-(3.21) yields operators representing the components of I’ and I” but these are not of interest in the problem at hand. The Poisson bracket relations satisfied by the body components 5, 6, d*, 6*, do not have a simple form and it is not clear how the corresponding operators ought to be ordered. However, if we are interested only in the quantities Ji , IL , and /I(~)~we may define (somewhat arbitrarily)
in which case the expressions for Ji given by (3.29) may be written as
J+= a+&, j- = d+n, J3 = &?+c? - b’h),
(3.33)
I
and the expressions for the operators A, B, A+, B+ to be inserted in (2.6) are [cf. (3.9)] A = AJ
[d+u + b+b] d&J,
B = -&
[d+b - a’61 d&J,
(3.34)
A+ = -&
[~+a” + b+b] -&
,
B+ = &
[b+i? -
-&+a] &;
.
(3.35)
The operators given by (3.34)-(3.35) all commute with one another [cf. (2.28)] and, when applied to xy*“, lead to expressions identical to those given by (2.26) and (2.27) with C?Yy**replaced by x7*“. It may be noted that the basis functions ~7’” given by (3.26) may be written [in a set of units in which fi = I] as (j - iii)! (j+Cz)!(j+m)!(j-mm)!
(ut)i+m (bt)i-m 1"'(d+c)fi x;,O
(3.36)
in the case of integral j, m, 5, and as XJP*
=
[(j
+
m)
! (j
-
m)
!]-I/2
&%,(u+)i+m
(b+)jd
x",,O
(3.37)
GENERALIZED
in the case of integral or half-integral p”
= [(I2+ ‘7
j, m, A. Another form is given by
l)(j+m)!(j-m)!(j+ (-l)j-e-s
157
REPRESENTATIONS
fi)!(j-
(At)s+m+fi
(A)s
fE)!]llz
(Bt)j-m-s
(B)j-fi-s
s!(s+m+iii)!(,j-m-s)!(j--G-S)!
(3.38)
x”o*
Inasmuch as there is a one-to-one correspondence between the (QJ, v,, $jJ) and (a, /3, y)-representations, it follows that Eq. (3.38) must hold in the (GJ, vI, $,)representation if Eq. (2.21) holds in the (cy,p, y)-representation.
IV. STRUCTURE OF THE ~-COORDINATE
ANGULAR
MOMENTUM
PROBLEM
The structure of the 3-coordinate angular momentum problem being considered here may be described simply as follows: All the quantities of interest depend on two spinors whose components are expressible in terms of three coordinates and three corresponding conjugate momenta. One spinor has components a, b which are given by (3.11) and satisfy the Poisson bracket relations (3.12), the second has components A, B which are given by (3.9) and satisfy the Poisson bracket relations (2.13). The components a, b determine, via Eqs. (3.10), (3.20), (3.21), a set of orthogonal vectors J, I’, I” having the same magnitude, and the components A, B determine, via Eq. (2.6), a set of orthogonal vectors nu) , nc2), n(,) having unit magnitude. In the next two sections we shall be dealing with a number of different 3-coordinate angular momenta. These will be labelled with a superscript and all the related quantities will be labelled with the same superscript or subscript. Thus, the associated vectors corresponding to the 3-coordinate angular momentum J@) will be denoted by IQ), JW’, and the related body vectors by n{;i , n$‘, , n$ . Usually, but not always, Aij’ will denote the transformation matrix connecting the rn’(i! and a set of vectors eo) fixed in space, A!?) z, = nil,’ * e (3)’
(4.1)
and usually, but not always, jtv) will be defined by4 jiu)
=
J(v)
- n[;,’ .
In any case, in a (CD”, y”, $5”; P, , py , &)-representation, the momenta defined by $” = p, py = J(v) P, = I J(“) I, 3 ’ 4 The summation convention does not apply to Y in Eqs. (4.2)<4.9).
(4.2) will be (4.3)
158
NODVIK
the classical expressions for the space components a,, 6, and body components 12,) 6” of the spinor relating to the triad J(“), W), I’(~)’ will be given by
and the space components A,, B, of the spinor relating to the triad n$ , n$‘, , n{$ will be given by A, = (W’Nv*av
+ b,*&),
B, = (1/2P,)(a”,*b,
- a,*&). I
In the corresponding quantum theory, the momentum
(4.6)
operators will be taken to be (4.7)
and the operators a,, b, will be ordered in the same way as those given by (3.30), the operators d, , 6” in the same way as those given by (3.32), and the operators A, , B, in the same way as those given by (3.34). These operators will be combined to form the space and body components of the various vectors in the manner described in Section III. In Section VI we shall also make use of the fact that Jicy) and J”:” can also be written in factored form in terms of certain conjugate quantities avC,b,” and Zvc,Lvc; if, in the classical case, a,, b, and d, , &, are defined by (4.4)-(4.5) then the corresponding conjugate quantities are defined by avC
=
a e'iav Y 2
=
d/p, + py e-i%,
b,” = b,.&% = 1/P, - py etiw,,
(4.8) i
In the quantum case, the operators {a” c, b,“) are ordered in the same way as are the operators {Z, 6+} given by (3.32), and the operators {c?,~,gvc} are ordered in the same way as are the adjoints {a+, b+} of the operators given by (3.30). As discussed in Appendix B, the conjugate quantities enter in when the roles of eti) and II(~)
GENERALIZED
159
REPRESENTATIONS
are interchanged, i.e., when one chooses to consider the e(i) to be fixed in the body and the n(,) to be fixed in space.
V. ANGULAR
MOMENTUM
ADDITION
In the classical case, the problem of the addition of two angular momenta J(l) and J”’ refers to the problem of setting up a representation in which two of the momentum variables are the magnitude and (say) the 3-component of the sum J = J(l) + Jc2). The corresponding problem in the quantum case consists in finding a basis with respect to which J2 and J3 are diagonal. In this section and the next we shall consider two such sum representations which apply to 3-coordinate angular momenta J(l) and Jf2’. The (I J(l) 1, / Jc2’ I, I J 1,J3, ji’), j;“)
Variables
The sum representation considered in this section is characterized by the equations J. = ,!l) + $2) t 3 p:
=
+,
(5.1)
3i(z’,= 3p, and employs the set of canonical variables (c&‘, Q&‘, QJ, yJ, qI’, q2’; PI’, P2’, PJ , pr, A’, A’) which are connected to the variables (Q , v1 , @I ; PI , pl, A) and (Q2, y2, F2 ; P, , p2 , A) referring to J(l) and JC2’ by the relations \
PJ = 1J / = 1J(l) + J(‘) j ~= [PI” + P2” + 2p,p, + 2 dPl2 - p121/p,z - p22cos(y1 - q4]1/2,
> (5.2)
pJ = J, = 5:) + .I?) = p1 + p2 , PI’ = 1J(l) / = PI, P,’ = j J(‘) I = P, ,
/ j&1’=
A’ =
3;) = p1 ) 33(2’ = F2 )
41 = @I2 q2’ = $j2.1
In this case J. Ji”, and J,!“’ are, respectively, the components with respect t’o vectors eti) fixed in space, 32” the components to the body vectorsni”, and Jj2’ the components of Jt2) with vectors I$“’ [cf. (4.2)]. Since & , e2 , A , fi2 are unchanged, the
(5.3) of J, J(l), and J@’ of J(l) with respect respect to the body new representation
160
NODVIK
is very similar to that employed in II in connection with the addition of two 2-coordinate angular momenta; most of the results we require may be carried over from II. In a generalized Schr6dinger (c&‘, &‘, DI, p?r, c&‘, $,‘)-representation the functions comprising a basis with respect to which J(l)‘, J(2)2, J3, J, , Jill, J.$‘) are diagonal are given by 1-h j2 j * fil f%> = - (2$2 (I
fii,
I
<.A3
and we take the momentum
expb(j&’
I fi2
+
I
j202’
$-
jQJ
I * I
+
*qJ
+
*p&
+
&&‘)I
jl - j2 I Gj <.A + j2),
(5.4)
operators to be pJ=;&+;,
(5.5) pJ=;&,
The operators corresponding to a, , b, , a,, 6, are given by [cf. (1.47)-(1.50) of II] a, = -${a,[(P,
+ PI’)” - Pk2]l12 + bJt[Pi2 - (P, - Plf)2]1/2} -&
e--t%‘, J
bl = --&{b,[(P,
+ PI’)” - Pi2]l12 - aJt[P12 - (P, - Pl’)2]1/2) d2k
U, = TkJ{aJ[(PJ
+ P2’)2 - P;2]l12 - b,+[P;Z - (PJ - P2’)2]1/23 $
b2 = -&
3
{b,[(P, + P,‘)” - Pi2]l/2 + a,+[P;z - (pJ - P2’)2]l12> -& .I
where aJ and bJ are given by (3.30). The form of the operators unchanged,
(5.6) e-ti@tl’, e-t*%‘, (5.7) e-S%‘,
J
i
& , & , 6,
J2 is
(5.8)
and the operators Al , Bl , As, B, are obtained by substituting (5.6)-(5.8) into the expressions
GENERALIZED
161
REPRESENTATlONS
In a generalized Schrbdinger (@‘, , v1 , & , QZ, v2 , &)-representation, the functions comprising a basiswith respect to which J(l)‘, @, j;cl), J@, Ji2), j;2J are diagonal are given by
(ml y iz, = j, ,.jl -
1)...)-j,
and m2 , Fi2 = j, , j, - l,..., -j,
and j, , j, = 0, l,..., orjl , j, = 4, $ ,... ),
(5.10)
which may also be written [in a set of units in which X = I] as
I jl ml 1% .i2 m2 fi2> = [(j, + ml)! (jl - m,)! (j, + m,)! (j, - me)!]-1/2 xe ~~,~l+~~z~,(alt)~l+ml (&yl-T
(a2t)j,+% (~2t)~2-%, 0).
(5.11)
The corresponding basisfunctions in the (@,‘, Q2’, @$, vJ, &‘, +,‘)-representation are obtained by substituting (5.6)--(5.7) and replacing &, q2 by qI’, e2’ on the right of (5.1 I). In either representation ) O> = 1427~)~is the state in which all the quantum numbers are zero. The coupling coefficients relating the two setsof basis functions are given by
( jl’ j,’ j m El’ @ii,’ 1j, ml Fzl j, m2 $> = 6..3131’ 6.3232’ 6~tl~&+2ilzwz41 j2 j nl I jl ml j2 m2h
(5.12),
where the Clebsch-Gordon coefficient (j, j, jnz I j, ml j, m,) is the same as the one which enters into the coupling of two 2-coordinate angular momenta; as shown in Appendix A of II, an explicit expression for the latter may be obtained in a straightforward manner by expanding the right side of (5.11).
VI. ANGULAR
MOMENTUM
ADDITION
(CON'T.)
In this section we consider another sum representation for two 3-coordinate angular momenta J(l) and J@). In this representation the momenta consist of the magnitudes and certain components of one of these vectors, say Jc2), and of the sum vector J, which we shall also denote by Jf3), $3’
595/57/I-11
_
J
=
J’l’
+
J’3’.
(6.1)
162
NODVIK
Specifically, we shall be concerned with the components Jj3), ji3), Ji4), Jj4) defined by Jp’ = J * eci) = (J(l) + 5”‘)
* e(,) ,
j!3) I
=
J
.
5!4) 2
=
J(3)
. ,,# .
=
2 d,’
(5’1’
+
5’3’)
nk’
,
(6.2)
3
Jy' = Jt2) . n(t;,) ,
i
where, as before, the eti) are orthonormal vectors fixed in space, the n’($orthonormal vectors fixed in the body whose angular momentum is J(l), and the I$‘, orthonormal vectors fixed in the body whose angular momentum is Jf2). However, the notation here differs from that of previous sections in that the Jj4) are components with respect to the first set of body axes and not with respect to the space axes. The following relations are readily verified. (Eq. (2.12) proves useful in this regard.) Jp)Ji(3’
=
]i(3yp’
=
, J(1)
+
J(3)
13,
(6.3) I
[Ji’3’, Ji’“‘]= EiikJp, [p’,J/3’] = -EijrJf’, [Ji’3’,pq
(6.4)
= 0,
[Ji’4’,Ji’4’] = cijkJf’, [J,!“‘, J/4’] = -Q.J~‘,
[Ji’3’,
Ji’4’]
=
[Ji’3’,
JF’]
=
[Jp’,
Ji’4’]
(6.5)
=
[Ji’a),
ij’“‘]
=
0.
(6.6)
Thus, Jj3), Jj3J, Jj4), J”k4) satisfy the same formal relations as Jil), jjl), Jj2), /i2), respectively, and we now have two 3-coordinate angular momenta one of which is the sum given by (6.1). The sum representation characterized by (6.2) may be generated in the following manner. In an Euler-angle representation the body vectors n$‘, and $1 are connected to the space vectors e(i) by the relations
GENERALIZED
163
REPRESENTATIONS
where A\:’ (IY~, & , rJ in place of orientation tion from
depends on Euler angles (a1 , /31, yI) and A$’ depends on Euler angles in the manner specified by Eqs. (2.4)-(2.5). Now suppose we introduce, the set (LYE, j!& , y2), a set of Euler angles (Q , ,f& , y,J which specifies the of the n’$ relative to then&‘; specifically suppose we make a transformathe representation in which the coordinates are the Euler angles az , /I$ , yJ to a representation in which the coordinates are the Euler (a1 7 A 3 71, angles (a3 , P3 , y3 , a4 , P4 , r4) such that (6.8)
where AiT) depends on the Euler angles (01~)p3, r3) and Ai:) on the Euler angles (a4, 134, y4) in the manner specified by (2.4)-(2.5). Comparing (6.7) and (6.8) we have (6.9)
so that the set (a3 , /I$ , y3) is the same as the set (01~, fil , -&while the set (cx~, p4, y4) depends on both (01~, /II , rl) and (01~, p2 , y2). Using (2.6) we find that the transformation given by (6.9) is equivalent to
A, = A,,
A, = A,A, - B,*B, ,
& = B, ,
B2
=
&A4
+
A,*4,
(6.10) i
where A, and B, depend on (01,) pp, yy) and have the form given by Eq. (2.7). The transformation connecting (o(1, /3, , y1 , a2, ,&, y2) and (aa, ,&, y3, a4, p4, y4) is a simple point transformation and the transformation equations for the corresponding momenta are readily obtained. If the components Jj3) and Jy) depend on ((Y~, &, y3) and the corresponding momenta in the manner specified by Eqs. (2.8) and (2.9), respectively, and similarly for .Ij4) and J”i4), then a straightforward calculation shows that these components may be expressed in the form
(6.11)
The expressions given by (6.11) are equivalent to those given by (6.2).
164
NODVIK
From this point on we shall use the set of variables [cf. Section IV] (@+ , p)r , $& , @2 9 P2 2 Q2 ; PI, pl, A, Pz , p2, A) for @, jz!l), Ji2), jjz) and the set (@, , p3, “(3), Jj4), jj4). In relating these $53 2 @a 3 p)4 9 $54 ; Pa, p3, h , P4 , p4 , 15~)for Ji3’, Ji two sets of variables the simplest procedure is to work through the intermediary of the set PI’, G2’, QJ, vJ, %‘, q2’; PI’, P2’, PJ, p, , A’, A’) considered in Section V: The angular momentum coupling problem characterized by Eq. (6.11) may be broken down into two separate problems of the type considered in Section V, the first being characterized by the equations given by (5.1) and the second by the equations J))’ = p
- Ji(4),
Jj2)’ = p,
(6.12)
J.z = J!3) z The solution to the first part is given by Eqs. (5.6)-(5.8) and the solution to the second part may be obtained by following the same algebraic procedure which leads to (5.6)-(5.8). In the second part, however, the components ]!I)‘, ji3), Jj3), and Ji must be factored in terms of conjugate quantities having the form given by (4.8)-(4.9).
The equivalence of the coupling problem characterized by (6.12) and that characterized by (5.1) becomes evident if, on the left side of (5.1), we make the replacement (@l’,
@J 2 h’, -
(@J
9.l , @l’,
(@2’,
@2’;
; pl’, =
PJ -
P2’,
, $1’~
p?r , n $2’)
-+
PJ) +I’; P2’,
PJ
, PI’,
$52’;
P$‘,
-PJ A’)
(6.13)
, -A’) I
and on the right side of (5.1), the replacement (~j1,9)1,~1;pr.,Pl,~~l)j(~3,~---3,rr--3;p3,-~3,-p3) P2
3 F2
2 $2
; p2
3 P2 , $2)
(614) +
(@4,
9)4 , $54 ; p4
, p4
, B4>.
i
Under the replacement given by (6.13), the components (Ji , ]I”‘, J”j2)‘} on the left of (5.1) go over into {-Ji -(l)‘, - Ji , fi”)‘> and under the replacement (6.14) the components (Jil), Ji2), Jil), ji2)} on the right of (5.1) go over into {-Ii3), Ji4’, - Jj3), J”:“)}; the set of Eqs. (5.1) then goes over into the set (6.12). The solution to the coupling problem characterized by (6.12) may now be obtained by making the replacement (6.14) on the left sides, and the replacement (6.13) on the right sides, of Eqs. (5.6)-(5.8).
GENERALIZED
165
REPRESENTATIONS
In this way we find that a solution to (6.12), analogous to that given by (5.6)-(5.8), is5 c= aJ”, (6.15) ;Ic = bJc,
- C= [email protected]/:P,, {[(P,’ + P,)” - Py %
ci;”
\
- [Pi” - (Pl’ - PJ)2]l’2
b;c
+ [PZ - (P,’ - PJ)2]l/2(qq+> -&
(6.16)
, 1
a4 = - &
>
)
{(ii?)+ [(PI’ + P,‘)” - PJz]ll’ 1
-
Z;"[P,2
-
(P
1
' -
-dipl,
P2')2]1/2}
e-ii%',
(6.17) b4
=
d2;l,
{a~)+
[Pl'
+
P,')2
-
PJz]l/2
-&, + QP,’ -
e-$i@z’ 9 (6.18)
In the (Q3, y3, &, Q4, y,, @,,)-representation the quantities {sac, bat} and {ZSc, sac} appearing on the left side of Eqs. (6.15)-(6.18) are obtained by setting v = 3 in Eqs. (4.8)-(4.9), and {a, , b4} and {Cz,, i4} by setting v = 4 in Eqs. (4.4)(4.5); in the (G1’, Q2’, vJ, &‘, +,‘)-representation the quantities {Z2’, g2’} appearing on the right side of (6.15)-(6.18) are given by (5.8) and the expressions for (arc, bJc} are obtained by replacing (@” , cpy; P, , pv) in (4.8) by (GJ, yJ ; PJ , p,), those for {Zp, 62 } by replacing (CD”, & ; P, , A) in (4.9) by (c&‘, $jl’; PI’, A’). The corresponding operators must be ordered in the manner described in Sections III and IV. @J,
5 The solution to (6.12) given by (6.15)-(6.18) is unique to within arbitrary constant phase factors for the pairs {sac, b,” }, {ESc,gac}, {ad, Lr4), and {Cd, &}. Actually, with the replacement (6.13)-(6.14) one obtains {--ia*, -i&} on the left of (6.17) instead of {a4 ,6,); we have changed the phase of {a,, b4)1in order to make the result obtained by combining (5.7)-(5.8) and (6.15)(6.18) consistent with the result (6.10) obtained in the Euler-angle representation.
166
NODVIK
the functions comprising a basis In the (Q3, 9~~, $j3, @, , P)~, &)-representation with respect to which J(3)2, Jp), jp), Jt4)*, .I?), IL*) are diagonal are given by
m3F3+ fi3$j3+j4@, + n?,~, + %+54)1
lj3 m3 fi3 j4 m4 fi4) = -evP(j3@3+ &).
(6.19) which may also be written
[in a set of units in which fi = l] as
lj3 m3 fi3 .i4 m4 fi4)
= [(j, + fi3)! (j, - fi3)! (j, + m,)! (j, - m4)!]-1/2 x eim,m,+irii,e,(d3cy,+fi, (~3c)j,-fi, (a4t)j,+nl, (b4t)j,-m, / 0). The corresponding basis functions are obtained by setting y3 = yj, and {u4, b4} on the right of (6.20). be obtained by making use of the have
(6.20)
in the (@r’, Q2’, QJ, vX, &‘, $,‘)-representation e4 = F2 and using (6.16)-(6.17) for {d3c, b3C} The explicit expressions for these functions may results of Section V. From (5.11) and (5.12) we
Cl + ml) ! (A - ml) ! (j2 + m2)!(j2 - m2>!l-1/2 x p,@,‘+irii,~,’(alt)‘l+n”1(blt)k? (a,y+“z (b2t)e% 10) = ig, exp[Wh’
+ j2@,’ + j’@, + m’yJ + file,
+ fi2q2’)]
(6.21)
where {a, ,6,} and {a2 , b,j are given by the right sides of (5.6) and (5.7). If, in (6.21), we make the replacement (6.13) together with the replacement 13 3
w
.
? “13
J2 9 m2
9 fi2)
-
(j,
,-
fi3
, -%
, Ji
, 1114
,
54)
we obtain (j [(j3
-
fi3)!
(j, +
fi3)
! (j4
+ m,) ! (j, - m4)!]-1/2 exp[--im,(n
- p?l) + ijjt4~2’]
(-+’zn/2a4t)i4+m* (e-in/2b4t)j4-m, j 0) = ,g, (j3.i4j’ m’I j3- fi3 .i4ma> x
(&n/21;3cy~-l,
(e-in/2d3Cy3+%‘3
exp{i[j3@J + j4G2’ + .i’@4’ + m’(r
- &‘) - m3(n - vJ) + iii,@‘]} (6.22)
GENERALIZED
REPRESENTATIONS
167
where {Zsc,Z&c>and {a, , b4} on the left of (6.22) are given by (6.16)-(6.17). Comparing with (6.20) we find
I j3 nz3fi3 j4 m4fi4) = (- l)j4+m4 ez(7rr’+fi3-m4)~(j3 j, j’m’ L,
1j, -#ii, j, m4> jj’ j, j, m3-ni
Fz4), (6.23)
so that (j,’ j,’ j m ei,’ iii,’ 1j, m3 E3 j, m4 Fi4) = (- lP+m4 sij,8mn,,sj,‘j,slii,,~4(j3 j, j,’ - fi,’ j j, - fi3 j, m4), (6.24) where we have made use of the fact that the Clebsch-Gordon coefficient on the right of (6.23) vanishes unless m’ = - fi3 + m4. Finally, combining (6.24) and (5.12) we obtain (A ml fh .i2 m2 % I j3 m3 s3 j4 m4 fi4) = (-l)j4+m4 Sj2j4Sf12fi4CA j3 j3 nf3 I jl ml j2 @ x (.j3 j4 jl - % I j3 - E3 j4 m4>.
APPENDIX
A.
TRANSFORMATION
(6.25)
PROPERTIES
In this appendix we present a brief discussion of the way in which various quantities appearing in Sections II and III transform under a change from one (observer’s) orthonormal set of spacevectors et,) to another (observer’s) orthonorma1set of spacevectors e& . Assuming that thesetwo setshave the samehandedness we may write e& = A!?)e 2, (J)
(A.11
where A$’ is a real, orthogonal, constant matrix with unit determinant, A!O’A!O’ = A’O’A’O’ Zk 3k ki ki
det(d$)) = 1,
= 6
ii
’
Ai:) = real.
64.2)
Any matrix having the properties given by (A.2) may be written in factored form in terms of complex quantities A, , B. which satisfy AoAo* + BOB,* = 1.
(A.3)
168
NODVIK
The specific expressions for A$) may be taken to be those given by (2.6) with 11~~)~ on the left replaced by A$’ and {A, B} on the right replaced by {A,, B,}, e.g. Ai;’ = A,A,* - B ,,B ,,*. If the orientation of the eii, relative to the eci) is specified in terms of Euler angles (LYE,/IO, ~~0)then the expressions for A, and B,, are obtained by replacing by P, r> in (2.7) by (ho, is, , R,). In order that a quantity V be a vector, it is necessary that its components (V, , Yz , P’J with respect to the set et<) and its components (Vi’, V,‘, V,‘) with respect to the set eii, be related by Vi’ = A$‘V, .
G4.4)
Similarly, if a quantity u is to be a spinor its components (ul , UJ with respect to the set e(,) and its components (ul’, u,‘) with respect to the set eii, must be related by ’ = A,*u, + B,*u,, z&” = -B,u, + A,u, . I
Ul
(A.5)
It may be noted that if (ur , UJ transforms according to (A.5) then so does (u?*, -u1*>. If ntijj = Aij are the components of the body vectors nci) with respect to the set eci) , and PZ;~,~= Aij are the components with respect to the set eii, , we must have
64.6)
Combining
(A.6) with (2.6) we obtain (to within an overall minus sign) A’ = A,*A + B,,*B, B’ = -B,,A + A,B, I
which imply components Similarly we (3.21) are to
64.7)
that the quantities A, B appearing in the factorization of the space of the triad n(,) , n(,) , nc3) transform like the components of a spinor. find that if the quantities Ji, ri, and ri’ given by (3.10), (3.20), and transform like vector components, Ji’ = A$‘J,
64.8)
a’ = A,*a + Bo*b, b’ = -Boa + A,b, I
(A.9
etc., then we must have
GENERALIZED
REPRESENTATIONS
169
which is to say that the quantities a, b appearing in the factorization of the space components of the triad J, I?, I” must transform like spinor components. Finally, if we combine (A.7), (A.9), and (3.16) we obtain 2’ = a’jf’* 6’ = -a’#
+ b/B’* = aA* + bB* = d, + b’A’ = -aB + bA = 6, I
(A. 10)
which imply that the quantities 5, d appearing in the factorization of the body components of the triad J, I’, r’ transform like scalars (as do the body components of these vectors themselves). In particular, PJ , ~5~) and qJ transform like scalars. Suppose now that the vector components (V, , V’, , Vs) and/or the spinor components (ul , u,), with respect to the set e(,) , can be expressed in terms of a set of canonical variables (qi, pi). Then in general there will be a transformed set of canonical variables (qj’, pi’) related to the set (qi , pi) in such a way that the components (VI’, V,‘, V3’) and/or (UI’, u,‘) with respect to the set e;,, depend on (qi’, p,‘) in exactly the same way as (VI , vz , VJ and/or (v ,4 depend on hi , PA. (If this were not the case there would be certain observers singled out as preferred over other observers and some of the physical results obtained would depend on the particular observer who formulates the problem.) The observer using the basis vectors eii, may then use a Hamiltonian formulation in terms of the variables (qi , p,), or a formulation in terms of the variables (qi’, p,‘). In the case of the representation discussed in Section III, the connection between the variables PJ, p?r , GJ ; PJ, pJ, $$J) and (@i, yJ’, +J’; Pi, pi, &‘I corresponding to the transformation (A.l) is given by (cf. (A.9) and (A.lO)) &‘J’
+ p; e-+i(@J’+mJ’) = A,* 2/pJ + pJ e-li(@J+mJ, (A.ll) ___+ A, dPJ - pJ e-ii(3-~r), (A.12)
The observer using the basis vectors eii, may set up a formulation in terms of the variables (QJ, Pi, $ J ; PJ , pJ , fiJ), in which case he would factor the components Jl = J . e& in terms of quantities a’, b’ given by the right sides of (A. 1 l), or a formulation in terms of the variables (CD;, vi, @J’; Pi, pi, ji), in which case he would factor Ji’ = J * eii, in terms of quantities a’, b’ given by the left sides of (A.1 l), etc. It may be verified that the transformation specified by (A.1 l)-(A.12) is a canonical transformation for arbitrary complex constants A, , B, which satisfy (A.3).
170
NODVIK
In the quantum theory there is a correspondence, but no direct connection between the (@$, yJ, qJ)- and (@pJ’, p)J), $;)-representations. In the (@i, 9?;, +,‘)representation the functions comprising a basis with respect to which J2, J - e;a, , and J * n(,) are diagonal, are given by x;.“(@,‘,
?,‘,
,&‘)
and the corresponding [(j
+
Ill)! =
(j [(j
x
+
i
! (j
eifi6J(ut)j+m'
=
[(j
eiWJ’+m’aJ’+%J’) +
m)!
&“@J(x&d -
m)
(j
-m)!]-l/2
&Z+J’(a’t)i+m
(b’t)j-ht
1 0)
in the (cD~, v, , @j)-representation +
&bt)‘+”
(--B,,*d
+
Lf,,*bt)‘-”
are 1 0)
(Qs+m’+m
(Ao*)8 (&)j-m’-s
(&*)Gm-s
1
s! (s + m’ + m)! (j - m’ - s)! (j - m - s)! (Qh'
(*‘13)
!]--1i2
c (-l)j-m-s
rn’=-j L s x
&-3/S
basis functions
WZ)!]-1/2 m)
z
(A. 14)
/ 0).
The right side of (A. 14) is obtained by multiplying together the expansions
(A.15) i and then making a change of summation variable from k to m’, where k = s + IH + m’. Writing A, and B, in terms of Euler angles(aO, so, y,,) as in (2.7) and making use of Eq. (2.21), we obtain the correspondence
In the Euler-angle representation, Eq. (A.6) becomes
and the relation given by (A.16) is consistent with the result
GENERALIZED
APPENDIX
B.
171
REPRESENTATIONS
THE CONJUGATE
FORMULATION
Up to this point we have considered the orthonormal set eci)to be fixed in space and the orthonormal setn(,) to be fixed in the body whose angular momentum is J. From the physical standpoint these two setshave rather different characteristics: The choice of the particular set eci)to be used is left to the observer who is formulating the problem, and in general it is assumedthat another observer using a different set eii, [related to the e(,) by (A.])] will obtain the samephysical results as the first observer. The set n(,) , on the other hand, is one and the same for all observers-only the components of the nci) differ for different observers: n(,) = we(j) = &e;,, . That the setse(,) and n(,) have the characteristics just mentioned is simply a matter of definition. We could just as well adopt a point of view in which the roles of eu) and n(,) are interchanged, with the n(,) being considered as some observer’s basis vectors fixed in spaceand the eu) asvectors fixed in the body whoseangular momentum is J. In such a case we would be dealing with the inverse of (2.1) and the resulting formulation, which we shall refer to as the conjugate formulation, is somewhat different from the one considered in Sections II and III in that the components of the associatedvectors and various other quantities must be redefined. Quantities which refer to the conjugate formulation will be labelled with a superscript c. Supposethat L?,are (vector or spinor) components with respect to the set et,) , and ai the corresponding components, with respect to the set nci, , when the e(,) are regarded as some observer’s basisvectors and the nci) as body vectors; when the roles of et,) and n(,) are interchanged, we shall define the analogous components with respect to the set e(,) to be Qic = C{aii,} and the analogous components, with respect to the setn(,) , to be &i)ic= C{sZ,>,where C is the operator whose effect in the classicaltheory is to replace
(01,A Yi Pa5Pa7PJ
by C-Y, -B, --ol; pv 3PO, ~a)
in the Euler-angle representation and
in the (QJ, v,, eJ)-representation. This definition is such that Jic = Ji and Jgc = ji , i.e., the components of J are the sameno matter which point of view one takes; however, in the conjugate formulation these components are factored in a different way. In the conjugate formulation we factor the components e (Z)j =
eti,
- ntj,
=
(A-l)ij
=
A&
(B.1)
172
NODVIK
in terms of the quantities -J;‘, B” given by z& = C(A) = A*, ) B’ = C(B) = -B. j
VW
Equation (B.2) is consistent with the fact that if Aij is factored in terms of {A, B} then (A-l)ij [which, in the Euler-angle representation, is obtained by replacing (cy,/3, r) by (-y, -/?, -m) in Aij] is factored in terms of (A*, -B}. The components of J and its associated vectors rc, I”~ are factored in terms of the quantities ac, bc and 3, bc given by ac = C(C) = l/pJ
+ pJ e-tim,
)
= eti@J a,
(8.3)
b’ = C(d) = d/pJ - pJ eiiQJ = eti3 b, (B-4)
Thus the components, with respect to the set eu) , are J+” = (ac)* (6”) = dPJ2 - pJ2 e2m.r = J+ , r+’ = - &i[(a”)* (ac)* + (bc)(bc)] = [email protected],
(B-5)
I
etc., and the components, with respect to the set nt,) , are l+’ = ($)* (&) = z/pJ2 - jJ2 r+c = - &i[(Z)* (S)* + (@)(@] = &+J
=
I+
e-@J[-ip,
,
~0s
QJ
-
fiJ
sin
QJ],
I
(B*6)
etc. Now the Poisson bracket relations involving 9, b”“,etc., have a simple form, while those involving ac, bc, etc. do not. The form is the same, except for a minus sign in front of each Poisson bracket, as that of the relations involving a, b, etc., listed in Sections II and III, e.g. where one has [a, a*] = -i and [r, , rj] = --eijrJk one obtains [3, Lie*] = +iand [pje, p/] = +ciisJti. It may be noted that the associatedvectors whose components are Tic, ric are physically different from those whose components are ri , ri’: When the e(,) are regarded as an observer’s basisvectors and then(,) as body vectors, the definition of the vectors r, I” is such that these are perpendicular to J with r’, l’, J forming a right-handed triad, and such that @Jis the angle measuredcounter-clockwise from J x e(,) to r. In the conjugate formulation where the roles of eti) and nci) are interchanged the definition of F, Fe differs in that GJ becomesthe angle measured clockwise from rc to J x nc3) .
GENERALIZED
REPRESENTATIONS
173
REFERENCES 1. J. S. NODVIK, Ann. Phys. 51 (1969), 251. 2. J. S. NODVIK, Ann. Whys. 54 (1969), 505. 3. A. R. EDMONDS, “Angular Momentum in Quantum Mechanics,” 2nd ed. Princeton University Press, Princeton, New Jersey, 1960.