ph 870465

A u s t . J . P h y s . , 1 9 8 7 , 4 0 , 4 6 5 - 9 7

A G u i d e t o R o t a t i o n s i n Q u a n t u m M e c h a n i c s

M i c h a e l A . M o r r i s o n a n d G r e g o r y A . P a r k e r

D e p a r t m e n t o f P h y s i c s a n d A s t r o n o m y , U n i v e r s i t y o f O k l a h o m a ,

N o r m a n , O K 7 3 0 1 9 , U . S . A .

A b s t r a c t

T o l a y a f o u n d a t i o n f o r t h e s t u d y a n d u s e o f r o t a t i o n o p e r a t o r s i n g r a d u a t e q u a n t u m m e c h a n i c s

a n d i n r e s e a r c h , a t h o r o u g h d i s c u s s i o n i s p r e s e n t e d o f r o t a t i o n s i n E u c l i d e a n t h r e e s p a c e ( R

3

)

a n d o f t h e i r e f f e c t o n k e t s i n t h e H i l b e r t s p a c e o f a s i n g l e p a r t i c l e . T h e W i g n e r D - m a t r i c e s a r e

o b t a i n e d a n d u s e d t o r o t a t e s p h e r i c a l h a r m o n i c s . A n e x t e n s i v e r e a d y - r e f e r e n c e a p p e n d i x o f t h e

p r o p e r t i e s o f t h e s e m a t r i c e s , e x p r e s s e d i n a c o n s i s t e n t n o t a t i o n , i s p r o v i d e d . C a r e f u l a t t e n t i o n i s

p a i d t h r o u g h o u t t o v a r i o u s c o n v e n t i o n s ( e . g . a c t i v e v e r s u s p a s s i v e v i e w p o i n t s ) t h a t a r e u s e d i n

t h e l i t e r a t u r e .

T a b l e o f C o n t e n t s

P a r t I . I n t r o d u c t i o n

P a r t I I . R o t a t i o n s i n ~3

1 . S i m p l e R o t a t i o n s a b o u t a n A r b i t r a r y A x i s

( a ) A c t i v e a n d P a s s i v e R o t a t i o n s

( b ) R o t a t i o n M a t r i c e s

2 . E u l e r A n g l e R o t a t i o n s i n ~3

( a ) D e f i n i t i o n s o f t h e E u l e r A n g l e s

( b ) A c t i v e E u l e r A n g l e R o t a t i o n s

( c ) P a s s i v e E u l e r A n g l e R o t a t i o n s

( d ) T h e E u l e r A n g l e R o t a t i o n M a t r i c e s

P a r t I I I . R o t a t i o n O p e r a t o r s i n H i l b e r t S p a c e

1 . S i n g l e A n g l e R o t a t i o n s o f a n A r b i t r a r y K e t

2 . E u l e r A n g l e R o t a t i o n s o f a n A r b i t r a r y K e t

3 . E u l e r A n g l e R o t a t i o n s o f a P o s i t i o n E i g e n k e t

4 . E u l e r A n g l e R o t a t i o n s o f a n A n g u l a r M o m e n t u m E i g e n k e t

5 . P h y s i c a l S i g n i f i c a n c e o f a R o t a t e d A n g u l a r M o m e n t u m E i g e n k e t

6 . T r a n s f o r m a t i o n o f t h e S p h e r i c a l H a r m o n i c s

P a r t I V . C o n c l u d i n g R e m a r k s

A c k n o w l e d g m e n t s

R e f e r e n c e s

A p p e n d i x A . T h e C l a s s A n g l e P r o o f

A p p e n d i x B . T h e W i g n e r R o t a t i o n M a t r i c e s

A p p e n d i x C . T h e P a s s i v e C o n v e n t i o n

4 6 6

4 6 6

4 6 6

4 6 9

4 7 0

4 7 3

4 7 3

4 7 3

4 7 4

4 7 4

4 7 5

4 7 5

4 7 5

4 7 6

4 7 6

4 7 8

4 7 8

4 8 0

4 8 0

4 8 0

4 8 1

4 8 2

4 9 6

0 0 0 4 - 9 5 0 6 / 8 7 / 0 4 0 4 6 5 $ 0 2 . 0 0

466 M. A. Morrison aud G. A. Parker

Part 1. INTRODUCTION

Rotation operators, their matrix representations, and their effect on quantum states are an essential part of the quantum mechanics of microscopic systems. In graduate-level treatments of this topic (cf. Gottfried 1966; Shankar 1980; Messiah 1966; Schiff 1968; Merzbacher 1970; Sakurai 1985; Bohm 1979; Cohen-Tannoudji et at. 1977) the introduction of the rotation operator in Hilbert space and the Euler angle parametrization of rotations leads in short order to the Wigner rotation matrices.* These matrices, in turn, are widely used in a variety of applications (cf. Weissbluth 1979).

In our experience, newcomers to the theory of angular momentum, be they stu-dents or practicing physicists, are frequently confused by differences in the conven-tions and viewpoints used by various authors and by the lack of clarity in many introductory texts. Moreover, many have difficulty relating their intuitive (classi-cal) notion of a rotation to the quantum mechanics of rotations in Hilbert space -primarily, we think, because their training in rotations in ~3 is poor. Although a few recent papers have dealt with rotations in Euclidean three space (~3) (Leubner 1980) and with rotation matrices (Bayha 1984), few have addressed the rotation operators per se (see, however, Wolf 1969).

Our objec1:ive is to lay a foundation for the study of rotations in quantum me-chanics and to provide a compendium of essential results of the theory of angular momentum. We begin in II by discussing geometrical rotations in ~3 and the corresponding rotation matrices, both for a general rotation and in the Euler angle parametrization. (Goldstein 1980). \Ve also discuss the difference between active and passive rotation conventions - a notorious source of confusion.

There follows in III an introduction to rotations in Hilbert space. This leads to the \Vigner rotation matrices, which are illustrated by rotating the familiar spherical harmonics. We have not included proofs of the properties of these matrices, since such proofs can be found in many texts and treatises on angular momentum (Brink and Satchler 1962; Edmonds 1960; Rose 1957; Biedenharn and Louck 1981; Judd 1975; Normand 1980; Silver 1976; Cushing 1975; Wybourne 1970).t To compensate for this lack, we have listed the important properties of the Wigner matrices in a "ready-reference appendix." The material in this appendix was gathered from a wide range of sources, but is here presented in a consistent notation and viewpoint.

Part II. ROTATIONS IN ~3 1. Simple Rotations about an Arbitrary Axis

A rotation in ~3 is a geometrical transformation. A particular rotation can be fully specified by two parameters: a rotation angle 6 and a rotation axis fu, the axis about which the rotation occurs. We shall denote a rotation in ~3 by Ru (6). Unless otherwise indicated, all rotations are in the positive sense, as illustrated in Fig. 1.

* Rotations are treated in several advanced undergraduate quantum mechanics texts, including Dicke and Wittke (1960), Saxon (1968), Liboff (1980), and Powell and Crasemann (1961).

t Group theory is applied to rotations in Tinkham (1964) and in Hammermesh (1962); see also Harter et al. (1978).

R o t a t i o n s i n Q u a n t u m M e c h a n i c s

4 6 7

e

z

e

u

~

,

a

8

\~

~

e

y

e x

F i g . 1 . A r o t a t i o n i n ~3 i s d e f i n e d b y a r o t a t i o n a n g l e 8 a n d a r o t a t i o n a x i s e u . F o r 8 > 0 , t h e

r o t a t i o n R , , ( 8 ) a p p e a r s c o u n t e r c l o c k w i s e t o a n o b s e r v e r l o o k i n g t o w a r d s t h e o r i g i n .

e

z

R~a)(I5)

1 5 > 0

e

u

e

y

--~ P 2

e x

F i g . 2 . A n a c t i v e r o t a t i o n t h r o u g h a p o s i t i v e a n g l e 8 a b o u t t h e z a x i s . T h e o p e r a t o r t h a t e f f e c t s

t h i s r o t a t i o n i s R~ a ) ( 8 ) . N o t e t h a t 8 = ' P 2 - ' P l .

468 M. A. Morrison and G. A. Parker

ez = ez ,

RiP) (b)

ey,

ey

ex' Fig. 3. A passive rotation through a positive angle Ii about the z axis. The operator that effects this rotation is RV')(Ii). This rotation is not equivalent to the active rotation in Fig. 2.

ez ez,

Rip) ( -b)

ey

PI ey,

ex'

ex Fig. 4. The passive rotation that is equivalent to R ~ a) (Ii) of Fig 1. The operator for this rotation is R~p)(-li). Note that Ii = 'P~ - 'Pl.

R o t a t i o n s i n Q u a n t m n M e c h a n i c s

4 6 9

( a J A c t i v e a n d P a s s i v e R o t a t i o n s

T o d e s c r i b e t h e e f f e c t o f a r o t a t i o n i n ? J ? 3 , w e f i r s t d e f i n e a r i g h t - h a n d e d c o o r -

d i n a t e s y s t e m , s u c h a s t h e o n e i n F i g . 1 . B e c a u s e t h i s s y s t e m d o e s n o t c h a n g e i t s

o r i e n t a t i o n , w e c a l l i t t h e s p a c e - f i x e d s y s t e m . T h e o r t h o g o n a l u n i t v e c t o r s t h a t

d e f i n e t h e s p a c e - f i x e d s y s t e m a r e ( e x , e

y

, e

z

) .

A r o t a t i o n i n ? J ? 3 c a n b e d e s c r i b e d u s i n g t w o v i e w p o i n t s : a c t i v e o r p a s s i v e . ( D a v y -

d o v 1 9 7 6 ; B i e d e n h a r n a n d L o u c k 1 9 8 1 ) . A n a c t i v e r o t a t i o n i s a r o t a t i o n o f a

v e c t o r i n a f i x e d c o o r d i n a t e f r a m e . W e s h a l l u s e a s u p e r s c r i p t ( a ) t o e m p h a s i z e t h a t

a r o t a t i o n i s a c t i v e e . g . n~a)(8). I n t h e a c t i v e r o t a t i o n a b o u t t h e z a x i s ( e

u

= e

z

)

s h o w n i n F i g . 2 , t h e v e c t o r r l f r o m t h e o r i g i n t o p o i n t P

1

i s r o t a t e d b y n~a)(8) i n t o

t h e v e c t o r r 2 t o p o i n t P

2

:

r 2 = n~a)( 8 ) r l .

( 1 )

T h e i n v e r s e o f a n a c t i v e r o t a t i o n t h r o u g h a n g l e 8 a b o u t e

u

i s a n a c t i v e r o t a t i o n

t h r o u g h - 8 a b o u t e

u

o r , e q u i v a l e n t l y , t h r o u g h 8 a b o u t - e

u

:

n~a)-1(8) = n~a)( - 8 ) = n~2(8). ( 2 )

T h u s , t h e i n v e r s e o f t h e r o t a t i o n n~a) ( 8 ) s h o w n i n F i g . 2 t a k e s r 2 i n t o r l , i . e .

r l = n~a)-1(8)r2' ( 3 )

I n c o n t r a s t , a p a s s i v e r o t a t i o n i s a r o t a t i o n o f t h e c o o r d i n a t e s y s t e m . I n a

p a r t i c u l a r r e f e r e n c e f r a m e , a p o i n t ( t h e t i p o f a v e c t o r f r o m t h e o r i g i n ) i s l a b e l l e d

b y a t r i p l e o f n u m b e r s - e . g . ( x , y , z ) o r ( r , B , ) . U n d e r a p a s s i v e r o t a t i o n ,

t h e v e c t o r r e m a i n s f i x e d , b u t t h e p o i n t i t d e f i n e s r e c e i v e s n e w l a b e l s . T h a t i s , t h e

n u m b e r s t h a t d e s c r i b e t h e l o c a t i o n o f t h e p o i n t c h a n g e f r o m t h o s e a p p r o p r i a t e t o

t h e ( i n i t i a l ) u n r o t a t e d f r a m e t o t h e ( f i n a l ) r o t a t e d f r a m e . W e s h a l l u s e a s u p e r s c r i p t

( p ) t o d e n o t e a p a s s i v e r o t a t i o n , e . g . n } ! ) ( 8 ) .

A p a s s i v e r o t a t i o n t h r o u g h a p o s i t i v e a n g l e 8 a b o u t t h e z a x i s i s s h o w n i n F i g . 3 .

D e n o t i n g t h e l o c a t i o n o f t h e p o i n t P

1

i n t h e f i x e d f r a m e b y u n p r i m e d c o o r d i n a t e s

a n d t h a t o f t h e s a m e p o i n t i n t h e r o t a t e d f r a m e b y p r i m e d c o o r d i n a t e s , w e c a n

w r i t e t h e e f f e c t o f a p a s s i v e r o t a t i o n a s

r ; = n~p)(8)rl' ( 4 )

T h e r o t a t i o n s s h o w n i n F i g s 2 a n d 3 a r e n o t e q u i v a l e n t . T h e y d i f f e r b e c a u s e

b o t h a r e d e f i n e d a s p o s i t i v e r o t a t i o n s t h r o u g h a p o s i t i v e a n g l e ( 8 > 0 ) . T h e

p a s s i v e r o t a t i o n o f c o o r d i n a t e s t h a t i s e q u i v a l e n t . t o a n a c t i v e r o t a t i o n t h r o u g h 8

a b o u t e

u

m u s t d o t o t h e c o o r d i n a t e a x e s t h e o p p o s i t e o f w h a t n~a\8) d o e s t o a

v e c t o r . S o t h e p a s s i v e r o t a t i o n t h a t i s e q u i v a l e n t t o n~a)(8) o f F i g . 2 i s n~p)( - 8 ) ,

a s i l l u s t r a t e d i n F i g . 4 .

T h e r e i s t h e r e f o r e a n e q u i v a l e n c e ( a n i s o m o r p h i s m ) b e t w e e n a c t i v e a n d p a s s i v e

r o t a t i o n s , i . e .

n~a)(8) ~ n } ! ) ( - 8 ) ,

( 5 a )

w h e r e ~ s t a n d s f o r " i s e q u i v a l e n t t o " . A s i m i l a r a r g u m e n t e s t a b l i s h e s t h e

c o r r e s p o n d i n g e q u i v a l e n c e

n } ! ) ( 8 ) ~ n~a)( - 8 ) .

( 5 b )


These equivalences will play an important role in the subsequent discussions of Euler angle rotations ( II.2) and ofrotation operators in Hilbert space ( III).

To summarize: we can describe a rotation mathematically using either the active or passive convention, because according to Eqs (5), these rotations are equivalent provided we remember to change the sign of the rotation angle. The essential point is that n~a)(8) and n~)(-8) are merely two ways to describe the same geometrical rotation. In an active rotation the "final" coordinate system is the same as the initial one; in a passive rotation the final system is (ex" ey " ez'). But the proximity of the physical system to the final coordinate system is the same regardless of whether we carry out the rotation in the active or the (equivalent) passive viewpoint.

(b) Rotation Matrices Operationally, we describe the effect of a rotation in ~3 in terms of the compo-

nents of the vector being rotated. For example, we can write T1 in terms of the unit vectors that define the fixed reference frame as

T1 X1ex + Y1ey + Zlez. (6) The Cartesian components Xl, Y1, and Zl are just the direction cosines of T1 with respect to the fixed axes, e.g. Xl = ex T1. We can conveniently represent T1 by the column vector

,,= GJ (7) To write Eq. (6) in matrix notation, we must first define the basis

~ = (i), ~ = (~), ~ = (;) (8) Then the matrix counterpart of (6) is Eq. (7),

r1 X1ex + Y1ey + Zlez (9) The representation (7) depends critically on the coordinate system - this column vector is the correct representation of T1 only in the basis (8) which, in turn, pertains only to the fixed frame (ex, ey , e z ).

A vector T2 that is obtained via an active rotation of T1 [Eq. (1)] can similarly be represented in the basis (8) by

" = (~) (10) How can the components of T2 in the fixed basis (8) be obtained directly from those of T1 in this basis? The components of a vector before and after rotation are related by a rotation matrix; for each rotation nSa) (8) in ~3 there exists a matrix R u ( 8) that appears in the matrix counterpart of Eq. (1),

r2 = R u (8) r1 (11)


4 7 1

T o f i n d t h e e l e m e n t s o f R u ( 0 ) w e f i r s t w r i t e b o t h s i d e s o f ( 1 ) i n t h e f i x e d f r a m e

a n d t h e n t r a n s l a t e t h e r e s u l t i n g e q u a t i o n i n t o m a t r i x n o t a t i o n u s i n g t h e b a s i s ( 8 ) .

W e b e g i n b y l e t t i n g n~a)(o) a c t o n b o t h s i d e s o f E q . ( 6 ) :

r 2 = Xln~a)(o)ex + Yln~a)(o)ey + Zln~a)(o)ez. ( 1 2 )

L e t u s d e n o t e t h e ( o r t h o g o n a l ) u n i t v e c t o r s o b t a i n e d w h e n e x , e

y

, a n d e

z

i n ( 1 2 )

( a ) " , .

a r e r o t a t e d V i a n u ( 0 ) b y f x , f

y

, a n d f z :

I x = n~a)(o)ex,

I y = n~a)(o)ey,

I z = n~a)(o)ez.

I n t e r m s o f t h e s e r o t a t e d u n i t v e c t o r s , E q . ( 1 2 ) i s s i m p l y

r 2 = x d x + y dy + z d z '

( 1 3 a )

( 1 3 b )

( 1 3 c )

( 1 4 )

B u t o u r g o a l i s t o o b t a i n t h e c o m p o n e n t s o f r : i i n t h e f i x e d b a s i s . T o d o s o , w e m u s t

w r i t e e a c h u n i t v e c t o r i n ( 1 3 ) i n t e r m s o f e x , e

y

, e

z

j e . g .

I x = ( e x ' I x ) e x + ( e

y

I x ) e

y

+ ( e

z

. I x ) e

z

.

( 1 5 )

W e n o w d e f i n e t h e d i r e c t i o n c o s i n e s i n ( 1 5 ) t o b e e l e m e n t s o f t h e f i r s t c o l u m n o f

a 3 x 3 m a t r i x R :

R l 1 = ( e x I x ) ,

R 2 l = ( e

y

I x ) ,

R 3 l = ( e

z

. I x ) .

( 1 6 a )

( 1 6 b )

( 1 6 c )

T h e e l e m e n t s o f t h e s e c o n d a n d t h i r d c o l u m n s o f R a r e j u s t t h e d i r e c t i o n c o s i n e s

o f I y a n d I z , r e s p e c t i v e l y . U s i n g t h e e l e m e n t s o f t h e m a t r i x R , w e c a n w r i t e t h e

r i g h t - h a n d s i d e o f ( 1 4 ) i n t h e f i x e d f r a m e , v i z .

r 2 = X l R l 1 e x + X l R 2 l e

y

+ X l R 3 l e

z

+ Y l R 1 2 e x + Y l R 2 2 e y + Y l R 3 2 e

z

( 1 7 )

+ Z l R 1 3 e x + Z l R

2 3

e

y

+ Z l R 3 3 e

z

W h e n w e t r a n s l a t e ( 1 7 ) i n t o m a t r i x n o t a t i o n u s i n g t h e b a s i s ( 8 ) , w e r e c o g n i z e R

a s t h e r o t a t i o n m a t r i x o f E q . ( 1 1 ) :

(

e x . I x ) ( e x ' I y ) ( e x ' I z ) )

R u ( 0 ) = ( e y ) x ) ( e y ) y ) ( e y ) z ) .

( e

z

. f x ) ( e

z

: f y ) ( e

z

f z )

( 1 8 )

T h i s m a t r i x i s o r t h o g o n a l , b e c a u s e n~a)(o) i s a n o r t h o g o n a l t r a n s f o r m a t i o n . I n

1 1 . 2 w e s h a l l w r i t e d o w n t h e r o t a t i o n m a t r i x f o r a n a r b i t r a r y r o t a t i o n i n t h e

( p a r a m e t r i z e d ) E u l e r a n g l e f o r m .

T o i l l u s t r a t e h o w o n e d e t e r m i n e s a r o t a t i o n m a t r i x , l e t u s c o n s i d e r a n a c t i v e

r o t a t i o n t h r o u g h a n g l e 0 a b o u t t h e Z a x i s ( F i g . 2 ) . F o r e

u

= e

z

, t h e r o t a t i o n m a t r i x

( 1 8 ) i s

(

C O S O

R z ( o ) = si~ 0

- s i n o

c o s O

o

n

( 1 9 )


For example, if 0 = sin-l(t) ~ 53.1301, then

(3/5 -4/5

Rz(o) = 4/5 3/5 o 0 n (20)

Suppose we rotate the vector rl with components

r, = en (21) through 53.1301 about ez . To determine the resulting rotated vector, we multiply the matrix (20) by the column vector (21), viz.

r, = R.(6)r, = (~1). (22) We could, of course, carry out the rotation n~a)(o) using the passive viewpoint.

To do so in ~3, we would have to use the equivalent passive rotation operator nr,r)(-o). But we can determine the components of rl in the rotated frame using the rotation matrix Ru( 0) of Eq. (18). To see why, let us act on rl of (6) with

nr,r\~o): r{ = nr,r) ( -0) rl (23a)

= xlnr,r)(-o) ex + Ylnr,r) (-0) ey + Zlnr,r) ( -0) ez . (23b) Now, to work out the effect of nr,r)(-o) on the unit vectors in (23b), we replace this operator with its active equivalent, viz.

I -n(a)(c) A -n(a)(c) , -n(a)(c) , rl = Xl"-u u ex + Y1/~u u ey + zl/~u u ez . (24) The right-hand side 0/(24} is now identical to that 0/{12}. So Ru(o) of (18) gives us the components of rl in the rotated frame, e.g.

X{ = Rll(O) Xl + R12(0) Yl + R13(0) Zl. (25) The fact that the same rotation matrix is used in the active rotation (17) and in

the passive rotation (25) should not come as a surprise, since by construction o/the equivalent passive rotation, the components of rl in the rotated frame are related to those of the rotated vector r2 in the fixed frame by

X{ X2,

y{ Y2, z{ Z2.

(26a) (26b) (26c)

Finally, we should note that the rotation matrix (18) differs from that found in texts that adopt the passive viewpoint. According to the equivalence (5b), the active rotation that is equivalent to nr,r)(o) is n~a)(_o). Therefore the rotation matrix for a passive rotation through a positive angle 0 about ez is obtained from


( 1 9 ) b y r e p l a c i n g 0 w i t h - 0 , i . e .

(

C O S o s i n o 0 )

R z ( - o ) = - s i n o c o s o 0 .

2 . E u l e r A n g l e R o t a t i o n s i n ~3

( a J D e f i n i t i o n s o f t h e E u l e r A n g l e s

o 0 1

4 7 3

( 2 7 )

I t i s c o n v e n i e n t t o p a r a m e t r i z e r o t a t i o n s i n ~, a n d t h e m o s t w i d e l y u s e d p a r a m -

e t r i z a t i o n i s t h a t o f E u l e r a n g l e s . I n t h i s f o r m u l a t i o n , a n a r b i t r a r y r o t a t i o n i s c a r r i e d

o u t v i a t h r e e s u c c e s s i v e r o t a t i o n s a b o u t p r e s c r i b e d a x e s .

E u l e r a n g l e r o t a t i o n s a r e o f t e n d e s c r i b e d i n t h e p a s s i v e v i e w p o i n t , i n w h i c h

t h e f i x e d c o o r d i n a t e s y s t e m ( e x , e

y

, e

z

) i s r o t a t e d t h r o u g h t h r e e p o s i t i v e a n g l e s , t h e

E u l e r a n g l e s a , ( 3 , a n d ' Y . * E a c h r o t a t i o n p r o d u c e s a n e w s e t o f a x e s , w h i c h a r e

l a b e l l e d b y s u c c e s s i v e l y m o r e c u m b e r s o m e s e t s o f p r i m e s . T h e t h r e e s t a g e s o f a

p a s s i v e E u l e r a n g l e r o t a t i o n a r e

P A S S I V E E U L E R A N G L E R O T A T I O N

S t e p 1 : R o t a t e t h e ( u n p r i m e d ) f i x e d a x e s t h r o u g h a n a n g l e a ( 0 ~ a ~ 2 7 1 - )

a b o u t e

z

, t a k i n g t h e ( e x , e

y

, e

z

) c o o r d i n a t e s y s t e m i n t o t h e ( e

x

"

e

y

l

, e z l ) s y s t e m .

S t e p 2 : R o t a t e t h e p r i m e d a x e s f r o m s t e p 1 t h r o u g h ( 3 ( 0 ~ ( 3 ~ 7 1 " ) a b o u t e y l ,

t a k i n g t h e ( e

x

"

e y l , e z l ) s y s t e m i n t o t h e ( e x l l , e y l l , e z l l ) s y s t e m .

S t e p 3 : R o t a t e t h e d o u b l e p r i m e d a x e s f r o m s t e p 2 t h r o u g h ' Y ( 0 ~ ' Y ~ 2 7 1 " )

a b o u t e

z

l l

, t a k i n g t h e ( e x l I , e y l l , e z l l ) s y s t e m i n t o t h e ( e x I I I , e

y

l l l

, e z l l l ) s y s t e m .

S o t h e p a s s i v e E u l e r a n g l e r o t a t i o n o p e r a t o r i s

' R , ( p ) ( a , ( 3 , ' Y ) = = 'R,~~} ( ' Y ) 'R,~~) ( 3 ) 'R,~p) ( a ) .

( 2 8 )

( 6 J A c t i v e E u l e r A n g l e R o t a t i o n

A s n o t e d a b o v e , w e h a v e a d o p t e d t h e a c t i v e v i e w p o i n t , i n w h i c h t h e p h y s i c a l

s y s t e m , n o t t h e r e f e r e n c e f r a m e , i s r o t a t e d . C o n v e n t i o n a l l y , s u c h a n E u l e r a n g l e

r o t a t i o n i s p e r f o r m e d a s f o l l o w s ( c f . G o l d s t e i n 1 9 8 0 ) :

S t e p 1 : R o t a t e t h e s y s t e m t h r o u g h a n a n g l e a ( 0 ~ a ~ 2 7 1 " ) a b o u t e

z

: 'R,~a) ( a ) .

S t e p 2 : R o t a t e t h e s y s t e m t h r o u g h ( 3 ( 0 ~ ( 3 ~ 2 7 1 " ) a b o u t e

y

l

: 'R,~~) ( 3 ) .

S t e p 3 : R o t a t e t h e s y s t e m t h r o u g h ' Y ( 0 ~ ' Y ~ 2 7 1 " ) a b o u t e

z

l l

: 'R,~~} ( ' Y ) .

T h e c o r r e s p o n d i n g g e o m e t r i c a l r o t a t i o n o p e r a t o r i s

' R , ( a ) ( a , ( 3 , ' Y ) = = 'R,~~}('Y)'R,~~)(3)'R,~a)(a).

( 2 9 )

* T h e p a s s i v e r o t a t i o n o p e r a t o r i s u s e d , f o r e x a m p l e , i n R o s e ( 1 9 5 7 ) . S o m e w h a t c o n f u s i n g l y ,

R o s e d i s c u s s e d E u l e r a n g l e r o t a t i o n s f r o m t h e p a s s i v e v i e w p o i n t a l t h o u g h h i s r o t a t i o n s i n ! R

3

a n d i n s t a t e s p a c e a r e a c t i v e .


But this parametrization mixes the active and passive viewpoints, since steps 2 and 3 entail rotations of the system about rotated axes. So it is convenient, both conceptually and operationally, to rewrite (29) in terms of rotations solely about fixed axes. In Appendix A, we use the class angle relation - which shows how to change the axis with respect to which a rotation operator is defined - to rewrite (29) as

R(a)( a, (3,,) = R~a)( a) R~a)({3) R~a)( ,). (30) This result prescribes the following steps:

ACTIVE EULER ANGLE ROTATION Step 1: Rotate the system through an angle, (0 ~, ~ 271") about ez . Step 2: Rotate the system through {3 (0 ~ (3 ~ 71") about ey Step 3: Rotate the system through a (0 ~ a ~ 271") about ez Notice that when we rotate about the fixed axes, we perform the rotation by, first; when we rotate about the rotated axes, we perform this rotation last.

(c) Passive Euler Angle Rotations The passive Euler angle rotation (28) can also be written as a product ofrotations

about the fixed axes, viz.

R(p)( a, (3,,) = R~p)( a) R~p)({3) R~p)(,). (31) The effect on a vector of the equivalent active rotation operator must be the inverse of what R(p)(a, (3,,) does to the coordinate axes, i.e.

R(p) ( a, (3,,) [R~a)( a)R~a)({3)R~a)(,) r 1 = R~a)-l(,) R~a)-l({3) R~a)-l(a) = R~a)( -,) R~a)( -(3) R~a)(-a).

(d) The Euler Angle Rotation Matrix

(32a) (32b) (32c)

To write down the rotation matrix that corresponds to R(a)( a, (3, ,), we first use Eq. (30) to express this operator in terms of three, single angle, active rotations and write the result in terms of the matrices of II.1(b), viz.

R(a,{3,,) = Rz(a)Ry({3)Rz(/). (33) We can now construct an explicit form for the Euler angle rotation matrix R(a,{3,,) from the single angle rotation matrices (18) as

(COS a - sin a 0) ( cos (3 0 sin (3) (cos, - sin, 0 )

R(a,{3,,) = sin a cos a 0 0 1 0 sin, cos, 0 . o 0 1 - sin {3 0 cos {3 0 0 1

(34) (Notice the location of the minus sign in the second of these matrices.) We need not append a superscript (a) to R(a,{3,,), because the matrix (34) applies to the active rotation R(a)(a, (3,,) and to the equivalent passive rotation. Note that in the


4 7 5

p a s s i v e c o n v e n t i o n , w h i c h i s p r e f e r r e d b y s o m e a u t h o r s ( c f . P a c k a n d H i r s c h f e l d e r

1 9 7 0 ) t h e E u l e r a n g l e r o t a t i o n m a t r i x i s R ( - a , - j 3 , - , ) ,

P a r t I I I . R O T A T I O N O P E R A T O R S I N H I L B E R T S P A C E

1 . S i n g l e A n g l e R o t a t i o n o f a n A r b i t r a r y K e t

C o n s i d e r a n a c t i v e r o t a t i o n R t

a

) ( 8 ) o f a p h y s i c a l s y s t e m . L e t [ : d e n o t e t h e s t a t e

s p a c e o f t h e s y s t e m a n d I ' l j ; ( t ) ) t h e k e t t h a t d e s c r i b e s a s t a t e o f t h e s y s t e m a t t i m e

t . A s a c o n s e q u e n c e o f r o t a t i o n o f t h e s y s t e m i n 1 R

3

, t h e k e t t h a t d e s c r i b e s i t s

s t a t e i n [ : w i l l c h a n g e ; l e t I ' l j ; ' ( t ) ) d e n o t e t h e " r o t a t e d k e t " , i . e . t h e k e t i n [ : t h a t

r e p r e s e n t s t h e s t a t e i m m e d i a t e l y a f t e r r o t a t i o n .

C o r r e s p o n d i n g t o t h e r o t a t i o n R t

a

) ( 8 ) i n 1 R

3

, t h e r e i s a u n i t a r y r o t a t i o n o p e r -

a t o r t h a t a c t s i n [ : t o m a p I ' l j ; ( t ) ) i n t o I ' l j ; ' ( t ) ) :

R u ( 8 ) : I ' l j ; ( t ) ) f - > I ' l j ; ' ( t ) ) ,

I ' l j ; ' ( t ) ) = R u ( 8 ) 1 ' l j ; ( t ) ) .

( 3 5 a )

( 3 5 b )

T h e r o t a t i o n o p e r a t o r t h a t c o r r e s p o n d s t o R t a ) ( 8 ) i s a n e x p o n e n t i a l * i n v o l v i n g

t h e p r o j e c t i o n o n t h e r o t a t i o n a x i s e

u

o f t h e t o t a l a n g u l a r m o m e n t u m o p e r a t o r j ,

l . e .

R t

a

) ( 8 ) = e x p ( - i 8 J

u

) ,

( 3 6 )

w h e r e w e h a v e u s e d a t o m i c u n i t s ( I i = 1 ) . [ T h i s o p e r a t o r i s d e r i v e d f r o m t h e

o p e r a t o r f o r a n i n f i n i t e s i m a l r o t a t i o n b y c o n s i d e r i n g R

t

a ) ( 8 ) a s a n i n f i n i t e s u c c e s s i o n

o f i n f i n i t e s i m a l r o t a t i o n s ( c f . C o h e n - T a n n o u d j i e t a l . 1 9 7 7 ) . J H e n c e t h e b a s i c H i l b e r t

s p a c e t r a n s f o r m a t i o n ( 3 5 b ) i s

I ' l j ; ' ( t ) ) = e x p ( - i 8 J

u

) I ' l j ; ( t ) ) . ( 3 7 )

T h e e q u i v a l e n t p a s s i v e r o t a t i o n i s R~\':'8). F o r t h i s r o t a t i o n , t h e f i n a l k e t i s

I ' l j ; ' ( t ) ) o f ( 3 5 b ) , s o t h e a p p r o p r i a t e r o t a t i o n o p e r a t o r i s j u s t t h e o n e u s e d f o r t h e

a c t i v e r o t a t i o n i n E q . ( 3 6 ) .

F o r a p a s s i v e r o t a t i o n t h r o u g h a p o s i t i v e a n g l e 8 , t h e r o t a t i o n o p e r a t o r i s d e t e r -

m i n e d b y t h e f u n d a m e n t a l c o r r e s p o n d e n c e ( 5 b ) , w h i c h r e q u i r e s t h a t w e c h a n g e t h e

s i g n o f t h e a n g l e i n ( 3 6 ) :

R } ! ) ( 8 ) = e x p ( i 8 J

u

) .

( 3 8 )

T h i s c o n v e n t i o n i s u s e d b y m a n y a u t h o r s .

2 . E u l e r A n g l e R o t a t i o n o f a n A r b i t r a r y K e t

L e t u s n o w w r i t e d o w n t h e H i l b e r t s p a c e o p e r a t o r t h a t c o r r e s p o n d s t o a n a r b i -

t r a r y a c t i v e r o t a t i o n i n t h e E u l e r a n g l e p a r a m e t r i z a t i o n . R e c a l l t h a t f o r ( p o s i t i v e )

E u l e r a n g l e s a , j 3 , a n d I , t h e r o t a t i o n o p e r a t o r i n 1 R

3

i s

R ( a ) ( a , j 3 , I ) = R~a)(a) R~a)(j3) R~a)(,),

( 3 9 )

* T h e e x p o n e n t i a l o f a n o p e r a t o r i s d e f i n e d b y a s e r i e s t h a t h a s t h e s a I n e c o e f f i c i e n t s a s t h e

e x p o n e n t i a l o f a f u n c t i o n , i . e . e x p { A ) = ; : : : : : " = 0 A n / { n ! ) .


The quantum mechanical rotation operator corresponding to R (a) (a, /3, ,) is the product of three, single angle, rotation operators, each of the form (36):

k(a)(a, /3,,) = k~a)(a) k~a)(/3) k~a)(,) (40a) = exp(-iaJz ) exp(-ij3Jy) exp(-hJz )' (40b)

[We cannot write (40b) as a single exponential, because Jz and Jy do not commute.] Using this operator, we can express the effect on a ket I'l/J(t)) in of the rotation (39):

I'l/J'(t)) = k(a)(a, /3,,) I'l/J(t)). ( 41)

3. Euler Angle Rotations of a Position Eigenket

Some of the most important applications of rotation operators in quantum me-chanics are made in the position representation (cf. Weissbluth 1979). It is there-fore important to know the effect on a position eigenket in of an active rotation R(a)(a,/3,,) of a vector Tl in ~3, such as

T2 = R(a)(a,/3,,)Tl. (42) Using the rotation operator (40b), we can exhibit the relationship between the rotation in ~3 and its effect in ; denoting the "rotated ket" by I T2), we have

IT2) = k(a)(a,/3,,) ITt} = IR(a)(a,/3,,)Tl)' (43) In 111.6, we shall need the dual of this result. Since the rotation operator is unitary, i.e.

k(a)-l(a,j3,,) = k(a)t(a,j3,,), the bra corresponding to I T2 ) is

'( )t (T21 = (TlIR a (a,/3,,), whence

(Tli = (T2Ik(a)(a,/3,,).

4. Euler Angle Rotations of an Angular Momentum Eigenket

( 44)

(45a)

(45b)

The mathematical manipulations required to rotate an arbitrary ket in E can be simplified by first expanding the ket in an angular momentum basis. If we denote by J2 and Jz the square and z projection of the total angular momentum J, we can write this basis as { I (k), j, m)}. [Note carefully that Jz is the projection of J on the fixed ez axis.] The quantum numbers j and m are defined by the eigenvalue equations for the basis kets,

J21(k),j,m) = j(j+1)I(k),j,m), Jzl(k),j,m) = ml(k),j,m),

where j and m obey the usual restrictions (Cohen-Tannoudji et al. 1977): . 0 1 1 3 J = , 2' , 2"'"

m -j,-j+1,-j+2, ... ,j-2,j-1,j.

(46a) (46b)


4 7 7

T h e s y m b o l ( k ) r e p r e s e n t s t h e s e t o f q u a n t u m n u m b e r s t h a t , t o g e t h e r w i t h j a n d

m , c o m p l e t e l y s p e c i f y a p a r t i c u l a r s t a t e ; i . e . t h e q u a n t u m n u m b e r s ( k ) c o r r e s p o n d

t o t h e e i g e n v a l u e s o f a s e t o f o p e r a t o r s A t h a t , t o g e t h e r w i t h j 2 a n d j z , c o m p r i s e

a c o m p l e t e s e t o f c o m m u t i n g o p e r a t o r s ( C S C O ) .

T h e e x p a n s i o n o f a k e t I ( t ) ) i n t h i s b a s i s i s

I ( t ) ) = L L L I ( k ) , j , m ) ( k ) , j , m l ( t ) ) .

( 4 7 )

( k ) j m

W e a r e h e r e c o n c e r n e d w i t h t h e a n g u l a r m o m e n t u m p r o p e r t i e s o f q u a n t u m s t a t e s ,

s o f r o m n o w o n w e s h a l l s u p p r e s s t h e i n d i c e s ( k ) , e . g . f o r I ( k ) , j , m ) w e s h a l l w r i t e

I j , m ) .

U s i n g t h e e x p a n s i o n ( 4 7 ) , w e c a n r o t a t e a n a r b i t r a r y k e t i f w e k n o w t h e e f f e c t

o f a r o t a t i o n o p e r a t o r R ( a ) ( a , ( 3 , , ) o n a n e i g e n k e t . T h i s r e s u l t w i l l e v e n t u a l l y l e a d

u s t o t h e W i g n e r D - m a t r i c e s .

T o d e v e l o p a g e n e r a l r e s u l t f o r t h e " r o t a t e d k e t " R ( a ) ( a , ( 3 , , ) I j , m ) , w e u s e

c l o s u r e o f t h e a n g u l a r m o m e n t u m b a s i s { I j , m ) } t o w r i t e

+ j

R ( a ) ( a , ( 3 , , ) l j , m ) = L 1 j , 1 l ) { j , I l I R ( a ) ( a , ( 3 , , ) I j , m ) . ( 4 8 )

J - l = - j

[ I n w r i t i n g E q . ( 4 8 ) w e u s e d t h e l a c t t h a t ! ! J . e e i g e n s u b s p a c e d e f i n e d b y t h e q u a n t u m

n u m b e r j , w h i c h w e d e n o t e b y ( k , j ) = = ( j ) , i s g l o b a l l y i n v a r i a n t u n d e r a n y o p e r -

a t o r f u n c t i o n o f t h e c o m p o n e n t s o f J , a n d h e n c e i s i n v a r i a n t u n d e r R ( a ) ( a , ( 3 , , ) . ]

T h e q u a n t i t y ( j , I l I R ( a ) ( a , ( 3 , , ) I j , m ) i n E q . ( 4 8 ) i s t h e f a m o u s W i g n e r D -

m a t r i x : t h e m a t r i x e l e m e n t o f t h e u n i t a r y r o t a t i o n o p e r a t o r R ( a ) ( a , ( 3 , , ) o n t h e

e i g e n s u b s p a c e l ( j ) w i t h r e s p e c t t o t h e a n g u l a r m o m e n t u m b a s i s : *

v t , m ( a , ( 3 , , ) = = ( j , 1 l 1 R ( a ) ( a , ( 3 , , ) I j , m ) . ( 4 9 )

U s i n g t h e d e f i n i t i o n o f t h e D - m a t r i x , w e c a n w r i t e t h e t r a n s f o r m a t i o n e q u a t i o n ( 4 8 )

a s

+ j

R ( a ) ( a , ( 3 , , ) I j , m ) = L I j , l l ) v t , m ( a , ( 3 , , ) . ( 5 0 )

J - l = - j

T o o b t a i n t h e D - m a t r i x a p p r o p r i a t e t o a p a s s i v e r o t a t i o n , w e c a n u s e t h e r e -

l a t i o n s h i p b e t w e e n a c t i v e a n d p a s s i v e r o t a t i o n o p e r a t o r s , E q s ( 3 6 ) a n d ( 3 8 ) , t o

t r a n s f o r m o u r e q u a t i o n s i n t o t h e p a s s i v e v i e w p o i n t . A c c o r d i n g t o E q . ( 3 8 ) ,

R ( p ) ( a , ( 3 , , ) = R ( a ) t ( a , ( 3 , , ) , ( 5 1 )

s o t h e D - m a t r i x e l e m e n t f o r a p a s s i v e r o t a t i o n , ( j , 1 l 1 R ( p ) ( a , ( 3 , , ) I j , m ) , i s s i m p l y

r e l a t e d t o t h e c o r r e s p o n d i n g e l e m e n t f o r a n a c t i v e r o t a t i o n :

( j , 1 l 1 R ( p ) ( a , ( 3 , , ) I j , m ) ( j , 1 l 1 R ( a ) t ( a , ( 3 , , ) I j , m ) ( 5 2 a )

= ( ( j , m I R ( a ) ( a , ( 3 , , ) I j , I l ) ) * ( 5 2 b )

= V!n~J-l(a,(3,,).

( 5 2 c )

* U s e f u l m e t h o d s f o r c a l c u l a t i o n o f t h e D - m a t r i c e s c a n b e f o u n d i n W a l k e r ( 1 9 7 5 ) a n d C a r i d e

a n d Z a n e t t e ( 1 9 7 9 ) ; s e e a l s o A p p e n d i x B .


Thus, to transform an equation written in the active convention to its counterpart in the passive convention, we merely replace Vi,m(a,(3,,) everywhere by V!n~iJ(a,(3,,). Not only must we take the complex conjugate of the D-matrix element, we must also switch the indices. *

5. Physical Significance of a Rotated Angular Momentum Eigenket

The "rotated" ket k(a)(a,(3,,) !j,m) in (50) is an eigenvector of J2 but not of jz, the component of the total angular momentum along the fixed z axis. But this ket is an eigenket of the component of J along a different quantization axis.

The axis of quantization of the unrotated ket is ez. Let ezl denote the vector in ?R3 into which ez is rotated by n(a)(a,(3,,), i.e.

~ -n(a)( (3 ) ~ ez l ,\- a", ez . (53) To rotate ez is to rotate the axis of quantization of the unrotated ket I j, m). Therefore, the rotated axis ezl is the axis of quantization for the rotated eigenket k( a) (a, (3, ,) I j, m ). If we denote the rotated eigenket by I j, m )', we have

+i Ij,m)' = k(a)(a,(3,,)!j,m) = L Ij,Jl)Vi,m(a,(3,,). (54)

iJ=-j

The eigenvalue equations for the rotated ket are

J2!j, m)' = j (j + 1) I j, m)', jZ/Ij,m)' = m!j,m)'.

(55a) (55b)

Note that the eigenvalues are unchanged by this rotation - the eigenvalue of I j, m) with respect to jz is m which, according to (55b), is also the eigenvalue of I j, m)' with respect to jzl. Equation (54) is a unitary transformation from the basis { I j, m) } of the CSCO {A, j2, jz}, to the basis {I j, m)'} of the CSCO {A, j2, jz/}. 6. Transformation of the Spherical Harmonics

As an application of the D-matrices, we shall transform spherical harmonics between two coordinate systems (Steinborn and Ruedenberg 1973). Such transfor-mations are important in, for example, the theory of electron-molecule scattering (Lane 1980; Morrison 1983, 1987). The relative orientation of the two reference frames is described by the Euler angles a, (3, and ,.

The spherical harmonics are the position representation of the eigenkets of the orbital angular momentum. More precisely, if i is the orbital angular momentum operator, then Ie, m) is a simultaneous eigenket of L2 and Lz with quantum num-bers e and m, respectively. In the position representation, the angular dependence of the scalar product of I e, m) and a position eigenket I r) is given by the spherical harmonic yr (B , r.p). * Authors such as Pack and Hirschfelder (1970) who work in the passive convention use D-

matrices that are the Hermitian adjoint of the matrices in this section; see Appendix B.

R o t a t i o n s i n Q u a n t U I l l M e c h a n i c s

4 7 9

F i r s t w e w r i t e t h e b a s i c r e l a t i o n s h i p s b e t w e e n " r o t a t e d " a n d f i x e d k e t s , E q . ( 5 4 ) ,

i n t e r m s o f t h e e i g e n k e t s 1 C , m ) :

+ l

, A ( )

I C , m ) = R a ( a , ( 3 , ' Y ) I C , m )

E IC,Jl)D~,m(a,(3,'Y).

( 5 6 )

/ - < = - l

T h e k e t I C , m ) i s a n e i g e n k e t o f { A , 2 , z } ( w h e r e A c o m p l e t e s t h e C S C O ) w i t h

e i g e n v a l u e s C ( C + 1 ) a n d m , a n d 1 C , m ) ' i s a n e i g e n k e t o f { A , 2 , L z ' } w i t h t h e s a m e

e i g e n v a l u e s .

N o w , l e t r b e a v e c t o r t o a p o i n t P i n t h e f i x e d f r a m e . C o n s i d e r a s e c o n d

c o o r d i n a t e s y s t e m w h o s e z a x i s i s c o i n c i d e n t w i t h t h e q u a n t i z a t i o n a x i s o f t h e

r o t a t e d k e t 1 C , m ) ' . W e s h a l l d e n o t e t h i s r o t a t e d s y s t e m * b y a s i n g l e s e t o f p r i m e s ,

( e x " e

y

" e

z

' ) ' T h e v e c t o r t o P i n t h e r o t a t e d s y s t e m w i l l b e d e n o t e d r ' .

T h e p o s i t i o n e i g e n k e t s t h a t c o r r e s p o n d t o t h e s e t w o v e c t o r s a r e r e l a t e d b y a

p a s s i v e r o t a t i o n , i . e .

1 r ' ) = k ( p ) ( a , ( 3 , 1 ' ) 1 r ) = k ( a ) - l ( a , ( 3 , 1 ' ) 1 r ) .

( 5 7 )

T h e d u a l o f t h i s r e s u l t i s

( r ' 1 = ( r 1 k ( a ) ( a , ( 3 , 1 ' ) .

( 5 8 )

N o w , t o e x p r e s s E q . ( 5 6 ) i n t h e p o s i t i o n r e p r e s e n t a t i o n , w e o p e r a t e o n t h e l e f t

w i t h t h e b r a ( r I , v i z .

+ l

( r I C , m ) '

E ( r I C,Jl)D~,m(a,(3,'Y).

/ - < = - l

B u t

( r I C , m ) ' = ( r I k ( a ) ( a , ( 3 , 1 ' ) I C , m ) ,

s o w e c a n w r i t e ( 5 9 ) a s

+ l

( r I k ( a ) ( a , ( 3 , 1 ' ) I C , m )

E ( r I C,Jl)D~,m(a,(3,'Y).

/ - < = - l

N o w , u s i n g E q . ( 5 8 ) f o r ( r I k ( a ) ( a , ( 3 , 1 ' ) , w e o b t a i n

+ l

( r ' I C , m ) = E ( r I C,Jl)D~,m ( a , ( 3 , 1 ' ) .

/ - < = - l

( 5 9 )

( 6 0 )

( 6 1 )

( 6 2 )

W r i t i n g t h i s r e s u l t i n t e r m s o f t h e s p h e r i c a l h a r m o n i c s w e o b t a i n t h e b a s i c t r a n s -

f o r m a t i o n r e l a t i o n

+ l

y r ( B ' , r p ' ) = E yt(O,rp)D~,m(a,(3,'Y).

( 6 3 )

/ - < = - l

T h i s e q u a t i o n i s a s p e c i a l c a s e o f t h e g e n e r a l H i l b e r t s p a c e t r a n s f o r m a t i o n ( 5 4 ) .

* T h i s f r a m e i s n o n - i n e r t i a l ; t h i s f a c t n e e d n o t c o n c e r n u s , s i n c e w e c o n s i d e r i t s s p a t i a l o r i e n t a t i o n

o n l y a t f i x e d t i m e .


Part IV. CONCLUDING REMARKS The rotation operators itua)(8) and the Wigner rotation matrices Di,m (a,,8,,)

find application in a wide range of fields, including molecular spectroscopy, the theory of irreducible tensor operators, nuclear structure, and scattering theory. The tables in Appendix B provide most commonly-used formulas involving these operators and matrices, expressed in the notation and conventions of this paper, together with some guidance to other notations used in the vast literature of angular momentum theory.

Acknowledgments The authors are indebted to the students in their graduate quantum mechanics

classes and research groups for providing the stimulus for this paper and for reading and commenting on it in its many drafts. We would particularly like to thank Dr R. T. Pack for carefully reading an earlier version of this paper and vigorously de-fending the passive viewpoint of rotations in quantum mechanics and Dr Thomas L. Gibson for suggesting a valuable example that wound up on the cutting room floor because of length restrictions.

References Bayha, w. T. (1984). Am. J. Phys. 52,370. Biedenharn, L. C., and Louck, J. D. (1981). 'Angular Momentum in Quantum Mechanics: Theory

and Application, Encyclopedia of Mathematics and its Applications, Vol. 8' (Cambridge Univ. Press: New York), Chaps 2, 3, and 6.

Bohm, A. (1979). 'Quantum Mechanics' (Springer-Verlag: New York), 111.3. Bohr, A., and Mottelson, B. R. (1969). 'Nuclear Structure. Vol. I: Single-Particle Motion' (Ben-

jamin: New York). Brink, D. M., and Satchler, G. R. (1962). 'Angular Momentum' (Oxford Univ. Press: London). Burke, P. G., and Chandra, N. (1972). J. Phys. B 5, 1696. Caride, A. 0., and Zanette, S. I. (1979). J. Comput. Phys. 33, 44l. Cohen-Tannoudji, C., Diu, B., and Laloe, F. (1977). 'Quantum Mechanics' (Wiley: New York),

Chap. VI; especially Complement B. Condon, E. U., and Short ley, G. H. (1953). 'The Theory of Atomic Spectra' (Cambridge Univ.

Press: New York). Cushing, J. T. (1975). 'Applied Analytical Mathematics for Physical Sciences' (Wiley: New York),

2.6 and Chap. 9. Davydov, A. S. (1976). 'Quantum Mechanics, Second Edition' (Pergamon: New York), Chap. VI. Dicke, R. H., and Wittke, J. P. (1960). 'Introduction to Quantum Mechanics' (Addison-Wesley:

Reading, Mass.), Chap. 9. Edmonds, A. R. (1960). 'Angular Momentum in Quantum Mechanics, Second Edition' (Princeton

Univ. Press). Fano, U., and Racah, G. (1959). 'Irreducible Tensorial Sets' (Academic Press: New York). Goldstein, H. (1980). 'Classical Mechanics, Second Edition' (Addison-Wesley: Reading, Mass.),

4-l. Gottfried, K. (1966). 'Quantum Mechanics Volume I: Fundamentals' (Benjamin/Cummings: Read-

ing, Mass.), 10. Hammermesh, M. (1962). 'Group Theory and Its Applications to Physical Problems' (Addison-

Wesley: Reading, Mass.), Chap. 9. Harter, W. G., Patterson, C. W., and Da Paixao, F. J. (1978). Rev. Mod. Phys. 50, 37. Judd, B. R. (1975). 'Angular Momentum Theory for Diatomic Molecules' (Academic: New York),

Chaps 2 and 3. Lane, N. F. (1980). Rev. Mod. Phys. 52, 29. Leubner, C. (1980). Am. J. Phys. 48,650. Liboff, R. L. (1980). 'Introductory Quantum Mechanics' (Holden-Day: San Francisco), Chap. 9. Merzbacher, E. (1970). 'Quantum Mechanics, Second Edition' (John Wiley: New York), Chap. 9.


M e s s i a h , A . ( 1 9 6 6 ) . ' Q u a n t u m M e c h a n i c s ' ( W i l e y : N e w Y o r k ) , C h a p . X V I I I .

M o r r i s o n , M . A . ( 1 9 8 3 ) . A u s t . J . P h y s . 3 6 , 2 3 9 .

M o r r i s o n , M . A . ( 1 9 8 7 ) . A d v . A t . M o l . P h y s . 2 3 ( i n p r e s s ) .

4 8 1

N o r m a n d , J e a n - M a r i e ( 1 9 8 0 ) . ' A L i e G r o u p : R o t a t i o n s i n Q u a n t u m M e c h a n i c s ' ( N o r t h H o l l a n d :

N e w Y o r k ) .

P a c k , R . T . , a n d H i r s c h f e l d e r , J . O . ( 1 9 7 0 ) . J . C h e r n . P h y s . 5 2 , 5 2 1 .

P o w e l l , J . L . , a n d C r a s e m a n n , B . ( 1 9 6 1 ) . ' Q u a n t u m M e c h a n i c s ' ( A d d i s o n - W e s l e y : R e a d i n g , M a s s . ) ,

C h a p . 1 0 .

R o s e , M . E . ( 1 9 5 7 ) . ' E l e m e n t a r y T h e o r y o f A n g u l a r M o m e n t u m ' ( W i l e y : N e w Y o r k ) , P a r t A .

S a k u r a i , J . J . ( 1 9 8 5 ) . ' M o d e r n Q u a n t u m M e c h a n i c s ' ( B e n j a m i n / C u m m i n g s : R e a d i n g , M a s s . ) ,

C h a p . 3 .

S a x o n , D . S . ( 1 9 6 8 ) . ' E l e m e n t a r y Q u a n t u m M e c h a n i c s ' ( H o l d e n - D a y : S a n F r a n c i s c o ) , C h a p . X .

S c h i f f , L . I . ( 1 9 6 8 ) . ' Q u a n t u m M e c h a n i c s , T h i r d E d i t i o n ' ( M c G r a w - H i l l : N e w Y o r k ) , C h a p . 2 7 .

S h a n k a r , R . ( 1 9 8 0 ) . ' P r i n c i p l e s o f Q u a n t u m M e c h a n i c s ' ( P l e n u m : N e w Y o r k ) , C h a p . 1 2 .

S i l v e r , B . L . ( 1 9 7 6 ) . ' I r r e d u c i b l e T e n s o r M e t h o d s : A n I n t r o d u c t i o n f o r C h e m i s t s ' ( A c a d e m i c : N e w

Y o r k ) , C h a p s 1 a n d 2 .

S t e i n b o r n , E . 0 . , a n d R u e d e n b e r g , K . ( 1 9 7 3 ) . A d v . Q u a n t u m C h e r n . 7 , 2 .

T i n k h a m , M . ( 1 9 6 4 ) . ' G r o u p T h e o r y a n d Q u a n t u m M e c h a n i c s ' ( M c G r a w - H i l l : N e w Y o r k ) , C h a p . 5 .

W a l k e r , R . B . ( 1 9 7 5 ) . J . C o m p u t . P h y s . 1 7 , 4 3 7 .

W e i s s b l u t h , M . ( 1 9 7 9 ) . ' A t o m s a n d M o l e c u l e s ' ( A c a d e m i c : N e w Y o r k ) , C h a p s 1 a n d 2 .

W i g n e r , E . P . ( 1 9 5 9 ) . ' G r o u p T h e o r y ' ( A c a d e m i c P r e s s : N e w Y o r k ) .

W o l f , A . A . ( 1 9 6 9 ) . A m . J . P h y s . 3 7 , 5 3 1 .

W y b o u r n e , B . G . ( 1 9 7 0 ) . ' S y m m e t r y P r i n c i p l e s a n d A t o m i c S p e c t r o s c o p y ' ( W i l e y - I n t e r s c i e n c e :

N e w Y o r k ) .

A p p e n d i x A : T h e C l a s s A n g l e P r o o f

L e t R~a) ( 8 ) r e p r e s e n t a r o t a t i o n i n ~3 t h r o u g h 8 a b o u t f u . W e c a n u s e t h e

c l a s s a n g l e r e l a t i o n ( B i e d e n h a r n a n d L o u c k 1 9 8 1 ) t o d e t e r m i n e t h e o p e r a t o r f o r a

r o t a t i o n t h r o u g h t h e s a m e a n g l e a b o u t a d i f f e r e n t a x i s , s a y f v '

T h e r e e x i s t s a n a c t i v e r o t a t i o n t h a t c a r r i e s a v e c t o r t h a t l i e s a l o n g f u i n t o a

v e c t o r t h a t l i e s a l o n g f v . L e t b e t h e a n g l e a n d n t h e a x i s f o r t h i s r o t a t i o n , i . e .

R~a)() f u 1 - + f v .

( A I )

A c c o r d i n g t o t h e c l a s s a n g l e r e l a t i o n , t h e o p e r a t o r f o r r o t a t i o n t h r o u g h 8 a b o u t t h e

r o t a t e d a x i s f v i s g i v e n i n t e r m s ofR~a)(8) b y

R~a)(8) = R~a)()R~a)(8)R~a)-1(). ( A 2 )

W e w a n t t o u s e t h i s r e s u l t t o e x p r e s s t h e a c t i v e E u l e r a n g l e r o t a t i o n ( 2 9 ) ,

R ( a ) ( a , f 3 , ! ) = R~~!b)R~~)(f3)R~a)(a),

( A 3 )

i n t e r m s o f r o t a t i o n s a b o u t t h e f i x e d a x e s . T o d o s o , w e u s e t h e f o l l o w i n g t h e o r e m

( B i e d e n h a r n a n d L o u c k 1 9 8 1 ) , w h i c h f o l l o w s f r o m E q . ( A 2 ) :

T h e o r e m : L e t R k = R n k ( h ) , w h e r e k = 1 , 2 , 3 , . . . , N l a b e l s a s e t o f N r o t a t i o n s

t h r o u g h a n g l e s k a b o u t a x e s n k i n ~3. D e f i n e N n e w r o t a t i o n a x e s b y

n~ = [ R k - 1 R k - 2 . . . R 2 R 1 r

1

n k , k = 2 , 3 , . . . N ,

A /

n

1

n 1 .

T h e n

R n R

n

-

1

R

n

-

2

. . . R 2 R 1

R~ R~ . . . R~_l R~.

( A 4 a )

( A 4 b )

( A 5 )

482- M. A. Morrison and G. A. Parker

To apply this theorem to the Euler angle rotation (A3), we must consider three rotations:

R1 = R~a)(a) ~


L i s t o f S y m b o l s

e x , e

y

, e

z

U n i t v e c t o r s i n t h e f i x e d f r a m e

e x ' , e y l , e z l U n i t v e c t o r s i n t h e r o t a t e d f r a m e

a , f 3 , ' Y E u l e r r o t a t i o n a n g l e s

j x , j y , j z S p a c e - f i x e d a n g u l a r m o m e n t u m o p e r a t o r s

j S p a c e - f i x e d r a i s i n g a n d l o w e r i n g o p e r a t o r s

j x

l

, j y l , j z l B o d y - f i x e d a n g u l a r m o m e n t u m o p e r a t o r s

j 1 B o d y - f i x e d r a i s i n g a n d l o w e r i n g o p e r a t o r s

A ( j ) A ( j ) A ( j ) .

: J x , : J y , : J z S p a c e - f i x e d a n g u l a r m o m e n t u m m a t r I X o p e r a t o r s

I j m ) E i g e n k e t o f t h e a n g u l a r m o m e n t u m o p e r a t o r s

4 8 3

~m ( e , ) S p h e r i c a l h a r m o n i c ( C o n d o n - S h o r t l e y p h a s e c o n v e n t i o n )

T

k q

S p h e r i c a l t e n s o r o p e r a t o r

2 F l H y p e r g e o m e t r i c f u n c t i o n

R ( a ) A c t i v e r o t a t i o n o p e r a t o r

R ( p ) P a s s i v e r o t a t i o n o p e r a t o r

R u ( 8 ) R o t a t i o n m a t r i x

i / ( a , f 3 , ' Y ) W i g n e r r o t a t i o n m a t r i x

d j R e d u c e d W i g n e r r o t a t i o n m a t r i x o p e r a t o r

p~nl,n2) J a c o b i p o l y n o m i a l

p

l

m

A s s o c i a t e d L e g e n d r e p o l y n o m i a l

P I L e g e n d r e p o l y n o m i a l

I

j

I d e n t i t y m a t r i x

C ( j l , i 2 , j ; m l , m 2 , m ) C l e b s c h - G o r d a n c o e f f i c i e n t

C O r t h o g o n a l C l e b s c h - G o r d a n m a t r i x

X

j

C h a r a c t e r o r t r a c e o f t h e W i g n e r r o t a t i o n m a t r i x

a o , a l , a 2 , a 3 E u l e r - R o d r i g u e s p a r a m e t e r s

8 ( x - x ' ) D i r a c d e l t a f u n c t i o n

D . ( j d 2 j 3 ) T r i a n g l e r e l a t i o n

p ~ A n g u l a r m o m e n t u m m a t r i x p r o j e c t i o n o p e r a t o r

I x , I

y

, I z P r i n c i p a l m o m e n t s o f i n e r t i a


Definitions

The Wigner rotation matrix i/ (ex, ;3, ,) transforms a set of kets Ij Jl) according to Eq. (54),

Ij, m)' = k(a)ljm) = exp( -iexjz) exp( -i;3jy) exp( -if jz) Ijm) = exp( -ioju)ljm)

j = L vtm(ex,;3,,)ljJl).

,,=-j

(B1)

The ranges of the Euler angles are

0:Sex:S271', 0:S;3:S 71', O:S ,:S271'. (B2) The projection quantum numbers in (B1) must satisfy

- j:Sm:Sj, (B3) - j:SJl:Sj.

The angular momentum operators satisfy the usual commutation relations j x j = ij, (B4)

l.e. [ji, jj] = ijkl (B5)

where the subscripts i, j, k are any cyclic permutation of x, y, z.

Relations between Commonly-used Notations

vtm(ex,;3,,) = Rtm(


4 8 5

E x p l i c i t E x p r e s s i o n s

T h e e l e m e n t s o f t h e W i g n e r r o t a t i o n m a t r i x a r e m a t r i x e l e m e n t s o f t h e r o t a t i o n

o p e r a t o r ( R o s e 1 9 5 7 ) ( s e e I l I A )

V~ m ( a , ( 3 , , ) = ( j l - ' I e x p ( - i a j z ) e x p ( - i { 3 j y ) e x p ( - i , j z ) l i m )

= e x p ( - i l - ' a ) ( i l - ' I e x p ( - i { 3 j y ) I i m ) e x p ( - i m , ) ( B 6 )

= e x p ( - i l - ' a ) d i m ( { 3 ) e x p ( - i m r ) ,

w h e r e

d

j

( ( 3 ) = [ U - m ) ! U + I - ' ) ! ] 1 / 2 ( c o s ( 3 / 2 ) 2 j + m -

l 1

( - s i n { 3 / 2 ) I 1 -

m

1 1

m

U + m ) ! U - I - ' ) ! ( I - ' - m ) !

X 2 F 1 ( I - ' - i , - m - i j I - ' - m + I j - t a n

2

~) ,

T h e h y p e r g e o m e t r i c f u n c t i o n 2 F 1 ( a , b , c j z ) i s

m ' ~ m .

a b a ( a + l ) b ( b + 1 ) 2

2

F

1 ( a , b , c j z ) = I + - ; - z + 2 ! c ( c + l ) z + . . .

( B 7 )

( B 8 )

I n ( B 8 ) , a a n d b a r e n e g a t i v e i n t e g e r s o r z e r o , s o t h e h y p e r g e o m e t r i c f u n c t i o n i s

a f i n i t e p o l y n o m i a l o f d e g r e e m i n ( 1 1 - ' - i l , . l m + i l ) . T h e W i g n e r r o t a t i o n m a t r i x

e l e m e n t s c a n a l s o b e e x p r e s s e d i n t e r m s o f t h e J a c o b i p o l y n o m i a l s ( B i e d e n h a r n a n d

L o u c k 1 9 8 1 ) p f

m

, I 1 ) , v i z .

J _ J m . J m . ! : . . ! : .

. [ ( . _ ) 1 ( ' + ) 1 ] 1 / 2 ( f . I ) m + l 1 ( f . I ) m - 1 1

d

l 1 m

( { 3 ) - ( j + I - ' ) ! ( j - I - ' ) ! c o s 2 s m 2

X pf~;;;I1,m+I1)(cos{3).

T h e J a c o b i p o l y n o m i a l s i n ( B 9 ) c a n b e w r i t t e n

p~nl,n2)(cos{3) = ( n + n 1 ) ! ( n + n 2 ) !

1 ( . 2 ( 3 ) n - 8 ( 2 ( 3 ) 8

X - s m - c o s -

L s ! ( n + n 1 - s ) ! ( n 2 + s ) ! ( n - s ) ! 2 2 '

8

( B 9 )

( B I 0 )

w h e r e n t , n 2 , a n d n a r e i n t e g e r s a n d t h e s u m m a t i o n o v e r s e x t e n d s o v e r a l l i n t e g e r s

f o r w h i c h t h e a r g u m e n t s o f t h e f a c t o r i a l s a r e n o n n e g a t i v e . T h e J a c o b i p o l y n o m i a l s

a r e o r t h o g o n a l o n t h e i n t e r v a l -1~ c o s {3~1 w i t h w e i g h t i n g f a c t o r ( 1 - c o s ( 3 ) n l ( 1 +

c o s ( 3 ) n

2

.

S y m m e t r y R e l a t i o n s

T h e r e d u c e d r o t a t i o n m a t r i c e s d j a r e r e a l a n d s a t i s f y t h e s y m m e t r y r e l a t i o n s

( R o s e 1 9 5 7 )

d i m ( - { 3 ) = ( - I ) I 1 -

m d

i m ( { 3 ) = d ! n 1 1 ( { 3 ) = d~11 - m ( { 3 ) = (-lr-l1d~m - 1 1 ( { 3 )

T h e r e f o r e t h e i J \ a , ( 3 , , ) m a t r i c e s s a t i s f y

( B 1 1 )

V~ m ( - , , - { 3 , - a ) = V ! : l 1 ( a , { 3 , , ) ,

( B I 2 )


Vfm(a,{3,,) = (_1)"-mv~,, _m(a,{3,,), Vim (a + 7rnw {3 + 27rnfJ" + 7rnl') = (-1 )2i n,i+"n",+mn-YV i m (a, {3,,),

(BI3) (BI4)

where nOll nfJ and nl' are integers. Equation (BI4) shows that i/ (a,{3,,) has periodicity 47r in {3 and 27r in a and, for all half integral j, and periodicity 27r in {3 and 7r in a and, for all integral j. Special Values

The matrix elements d I are related to the associated Legendre polynomials p lm by (Rose 1957)

I m -m. m [(I )1] 1/2 dmo ({3) = (-1) (I + m)! PI (cos {3) (BI5)

and to the Legendre polynomials PI by d~o ({3) = PI ( cos {3). (BI6)

For m = J1. = j or m = J1. = -j Eq. (Bl6rbecomes d~ .({3) = di . . ({3) = cos2i ~. 13 -J-J 2 (BI7)

A I The matrix elements']) (a, {3, ,) are also related to the spherical harmonics Yim by

( 4 ) 1/2 V!n o( a, {3,,) = 21: 1 Y,;"({3, a),

( 4 )1/2 V~ m(a,{3,,) = 21: 1 Yi-m({3,,), where we have used the phase convention of Condon and Short ley (1953). Unitarity Relations

The Wigner rotation matrices are unitary (Brink and Satchler 1962), i.e. i L Vi"m(a,{3,,)Vt m/(a,{3,,) = Om m/,

/J=-i i L Vi m(a,{3,,)vt: m(a,{3,,) = 0" ,,,.

m=-i In matrix notation, Eqs (B20) and (B21) are written

(BI8)

(BI9)

(B20)

(B21)

A it A i A i A it . ']) (a,{3,,)']) (a,{3,,) = ']) (a,{3,,)']) (a,{3,,) = I J (B22)

or A it [ Ai] -1 .

']) (a, {3,,) = ']) (a,{3,,) , = '])J(-,,-{3,-a), (B23) where Ii is the (2j + 1) by (2j + 1) identity matrix. The determinant of a rotation matrix is (Biedenharn and Louck 1981)

det [Vi (a,{3,,)] = 1. (B24)


4 8 7

R e l a t i o n t o I r r e d u c i b l e T e n s o r O p e r a t o r s

A s p h e r i c a l t e n s o r o p e r a t o r ( o r i r r e d u c i b l e t e n s o r o p e r a t o r ) t r a n s f o r m s u n d e r a

r o t a t i o n a s ( B i e d e n h a r n a n d L o u c k 1 9 8 1 )

j

f l ( a ) 1 j m [ f l ( a ) r

l

= = L V i m ( a , ( 3 , , ) 1 j j J ,

( B 2 5 )

j J = - j

w h e r e 1 j m i s a s p h e r i c a l t e n s o r o p e r a t o r . T h i s t e n s o r h a s r a n k j a n d 2 j + 1 c o m -

p o n e n t s . T h e p r o d u c t o f t w o r o t a t i o n m a t r i x e l e m e n t s c a n b e w r i t t e n a s a l i n e a r

c o m b i n a t i o n o f t h e s e m a t r i x e l e m e n t s ( t h e C l e b s c h - G o r d a n s e r i e s ) ( R o s e 1 9 5 7 ) :

h + h

. ~j2 = =

~Jl V j J 2 m 2 . . I

V / J 1 m 1 j = i J 1 - 3 2

L

C ( j l , h , j ; l l l , 1 l 2 , l l l + 1 l 2 ) C ( j l , h , j ; m l , m 2 , m l + m 2 )

X V~1+/J2 m 1 + m 2 '

( B 2 6 )

T h e i n v e r s e o f t h i s s e r i e s i s

j j

V i m = = L L C ( j l , h , j ; l l l , l l - I l l , I l ) C ( i t , h , j ; m l , m - m l , m )

/ J 1 = - j

m

1 = - j

( B 2 7 )

x V i i v h

1 ' 1 m 1 / J - j J 1 m - m 1 '

w h e r e C ( j l , h , j ; I l l > 1 l 2 , I l ) i s a C l e b s c h - G o r d a n c o e f f i c i e n t i n t h e n o t a t i o n o f R o s e

( 1 9 5 7 ) . I n m a t r i x n o t a t i o n t h e s e t r a n s f o r m a t i o n s b e c o m e

C [ i >

h

0 i > h ] d = = i > h + h E B i > h + h - l E B E B i > l h - h l

[ V

i

: + ; '

o

i > h + h -

l

o

o 1

o

. '

i > l j ; - h l

( B 2 8 )

w h e r e 0 s t a n d s f o r a d i r e c t p r o d u c t a n d E B f o r a d i r e c t s u m . T h e d i m e n s i o n s o f

t h e m a t r i c e s i n ( B 2 8 ) a r e ( 2 j l + 1 ) ( 2 h + 1 ) b y ( 2 i t + 1 ) ( 2 h + 1 ) , a n d t h e m a t r i x

o n r i g h t - h a n d s i d e o f t h i s e q u a t i o n i s b l o c k d i a g o n a l , w i t h b l o c k s c o n s i s t i n g o f t h e

W i g n e r r o t a t i o n m a t r i c e s . F i n a l l y , t h e e l e m e n t s o f t h e o r t h o g o n a l t r a n s f o r m a t i o n

m a t r i x C a r e

C j m ; m 1 m 2 = = C ( j l , h , j ; m l , m 2 , m ) ,

( B 2 9 )

w h e r e t h e p a i r s o f i n d i c e s j , m a n d m l , m 2 l a b e l r o w s a n d c o l u m n s r e s p e c t i v e l y .

T h e s e r e l a t i o n s s h o w t h a t t h e W i g n e r r o t a t i o n m a t r i c e s c o n s t i t u t e a n i r r e d u c i b l e

r e p r e s e n t a t i o n o f t h e g r o u p s 0 + ( 3 ) a n d S U ( 2 ) ( c r . T i n k h a m 1 9 6 4 ) . T h e c h a r a c t e r

o r t r a c e o f t h e s e m a t r i c e s i s ( W e i s s b l u t h 1 9 7 9 )

x

j

( 8 ) = = T r [ i > j ( a , ( 3 , , ) ] = = ~ V

j

( a ( 3 ) = = s i n ( j + 1 / 2 ) 8

L . . . J m m , " / '

m = - j s m 8 . 2

( B 3 0 )


where 0 is a rotation about an arbitrary axis. For 0 = 0 the character ofthe Wigner rotation matrix must equal its dimension, i.e.

xj (0) = 2j + 1 = lim sin(j + 1/2)0 0 ...... 0 sin 0/2 (B31)

Using explicit expressions for the rotation matrix elements and Eq. (B30), we find a simple relation between the rotation angle 0 about an arbitrary axis and the Euler angles:

cos (~) = cos [a; ~] cos (~) . The product of two rotations is a rotation, so (Biedenharn and Louck 1981)

Aj Aj Aj 'D (a',~',,,y')'D (a,~,,) = 'D (a",~",,"). (B32)

The Euler angles are related to the Euler-Rodrigues parameters (ao, a) where a == (aI, a2, (3). These parameters, which are the Cartesian components of a point on a unit sphere in four space, are related to the Euler angles by

ao = cos (~) cos ( I ; a) = cos x, (B33) . (~) . (,-a) . eA..' a1 = sm 2" sm -2- = sm cos 'I' SIn X, (B34) . (~) (' -a) .' e . A.. a2 = SIn 2" cos -2- = sm sm'l'smx, (B35)

(~) . (,+a) e . a3 = cos 2" SIn -2- = cos smX, (B36) where 0:S;:S;27T, O:s;e:S;7T and O:S;X:S;7T are the spherical polar coordinates in four space. The Euler angles in Eq. (B32) satisfy the quaternionic composition rule

" , A' A ao = aoa - a . a,

a" = aoa' + a~a + a' x a. For small ~ the reduced rotation matrix elements are

dtn p.(~) ~ Omp. - i~(jmljyljJl) ~ Om p. + ~{-Op.,m+dj(j + 1) - m(m + 1)]1/2

+ Op.,m-djU + 1) - m(m - 1)]1/2}.

Orthogonality and Normalization Relations

(B37) (B38)

(B39)

The Wigner rotation matrix elements are orthogonal but not normalized (Rose 1957):

J .. . 87T 2 V~', m,(a,~,,)V~~ m,(a,~,,)dQ = 2j1 + 1 OP.,p.2 0m ,m,Oj,j" (B40)


4 8 9

w h e r e

J 1

2 ' 1 1 " 1 ' 1 1 " 1 2 ' 1 1 "

d Q = 0 < h 0 q B s i n , 8 0 d y = 8 7 r

2

( B 4 1 )

i s t h e i n t e g r a l o v e r t h e s u r f a c e o f a f o u r s p h e r e .

I n t e g r a l s

T h e i n t e g r a l o f t h e p r o d u c t o f t h r e e W i g n e r r o t a t i o n m a t r i x e l e m e n t s i s ( R o s e

1 9 5 7 )

J 1Jt~ m3(0:,,8'i)1Jt~ m2(0:,,8'i)1Jt~ m l ( o : , , 8 , i ) d i l

8 7 r

2

C ( ' . . ) C ( ' . . )

2 i s + 1 J l , J 2 , J 3 j 1 ' 1 , 1 ' 2 , 1 ' 3 J 1 , J 2 , J 3 j

m

l , m 2 , m 3

( B 4 2 )

T h i s i n t e g r a l i s z e r o u n l e s s m l + m 2 = m 3 , 1 ' 1 + 1 ' 2 = 1 ' 3 a n d t h e t r i a n g l e r u l e

~(ithj3) i s s a t i s f i e d . E q u a t i o n ( B 4 2 ) l e a d s t o t h e G a u n t f o r m u l a f o r s p h e r i c a l

h a r m o n i c s :

J Yi~; ( , 8 , 0 : ) Y i : n

2

( , 8 , 0 : ) Y i : r ' l ( , 8 , o : ) < h s i n , 8 d , B

[

( 2

h

+ 1 ) ( 2 / 2 + 1 ) ] 1 / 2

= 4 7 r (

2 1

3 + 1 ) C ( l t , 1

2

, 1

3

j

m

1 , m 2 , m 3 ) C ( h , 1 2 , /

3

j O , O , O ) ,

( B 4 3 )

w h i c h i s z e r o u n l e s s I t + 1 2 + 1 3 i s e v e n , m 1 + m 2 = m 3 , a n d ~(l11213) i s s a t i s f i e d .

M a t r i x E l e m e n t s o f t h e A n g u l a r M o m e n t u m O p e r a t o r s

T h e n o n v a n i s h i n g m a t r i x e l e m e n t s o f t h e r a i s i n g a n d l o w e r i n g o p e r a t o r s j =

j x i j y a r e ( R o s e 1 9 5 7 )

( j m 1 1 J l j m ) = [ j ( j + 1 ) - m ( m 1 ) ] 1 / 2

= [ ( j = r = m ) ( j m + 1 ) ] 1 / 2 .

( B 4 4 )

T h e r a i s i n g a n d l o w e r i n g o p e r a t o r s a r e n o t H e r m i t i a n j i n s t e a d , J~ = j _ . B y

i n v e r t i n g t h e d e f i n i t i o n s o f t h e s e o p e r a t o r s , w e o b t a i n j x = t ( J + + j _ ) a n d j y =

- t ( J + - J - ) a n d t h e c o r r e s p o n d i n g m a t r i x e l e m e n t s

[.7~)Lm = ( j J . l I J x l j m ) = ~{8/J,m+1[j(j + 1 ) - m ( m + 1 ) ] 1 / 2

+ 8 / J , m - 1 [ j ( j + 1 ) - m ( m - 1 ) P / 2 } ,

[.7~j)Lm = ( j J . l I J y l j m ) = ~{-8/J,m+1[j(j + 1 ) - m ( m + 1 ) P / 2

+ 8 / J , m - t [ j ( j + 1 ) - m ( m - 1 ) ] 1 / 2 } .

T h e W i g n e r - E c k h a r t T h e o r e m

( B 4 5 )

( B 4 6 )

T h e W i g n e r r o t a t i o n m a t r i c e s c a n b e u s e d t o d e v e l o p a n e l e g a n t p r o o f o f t h e

W i g n e r - E c k h a r t t h e o r e m ,

( j ' J . l I T k q l j m ) = C ( j , k , j ' j m , q , J . l ) ( j ' I I 7 ( k ) l l j } , ( B 4 7 )

w h e r e ( j ' I I 1 ( k ) l l i ) i s a r e d u c e d m a t r i x e l e m e n t j f o r d e t a i l s , s e e C u s h i n g ( 1 9 7 5 ) .


Differential Equations

The operators

, a ., ~-i~~, Jx = i cos a cot,B aa + z sm a a,B sin,B or /j ., f.I a . a. sin a a '" y = z sm a cot f.J aa - z cos a a,B - Z sin ,B or '

, a Jz = -i-

or

satisfy the relation (Biedenharn and Louck 1981) , (j) , j , 'j

:J i 1) (a,,B,r) = -Ji 1) (a,,B,r)

(B48)

(B49)

(B50)

(B51) The the operator on the left-hand side of Eq. (B50) is a matrix operator, but the operator on the right-hand side is a differential operator. The minus sign in Eq. (B 50) is essential so that the differential and matrix operators satisfy the same commutation relations j x j = ij and j(j) x j(j) = ij(j). The operators (B49)-(B51) are the angular momentum operators associated with a space-fixed coordinate system. It should be noted that

}; = ei' j and };, = ii .j. The differential operators satisfy the commutation relations

[};, jj] = ijk. The space-fixed raising j+ and j_ operators are

j+ = jx + ijy, j- = jx - ijy.

(B52)

(B53) (B54)

The corresponding operators associated with the body-fixed coordinate system are

jx' = (cos a cos,B cos r - sin a sin r)jx + (sin a cos,B cos r + cos a sin r)jy - sin ,B cos r jz , (B55)

jy' = - ( cos a cos ,B sin r + sin a cos r )jx + ( - sin a cos ,B sin r + cos a cos r )jy + sin ,B sin r jz ,

jz' = sin,B cos ajx + sin,B sin o:Jy + cos ,Bjz, or, equivalently,

(B56) (B57)

jj' = LRij'(a,,B,r)ji = LjiRij'(a,,B,r) { Z=x,y,Z; (B58) ., '" J =x,y,z.

These body-fixed operators satisfy an equation analogous to Eq. (B49), , j , (j) " j

1) (a,,B,r):Ji' = -:1;,1) (a,,B,r) (B59)


4 9 1

a n d t h e c o m m u t a t i o n r e l a t i o n s

[ J i l , j j l ] = - i j k l . ( B 6 0 )

[ N o t e : T h e c o m m u t a t i o n r e l a t i o n f o r t h e b o d y - f i x e d o p e r a t o r s h a v e a f a c t o r o f - i ,

b u t t h e s p a c e - f i x e d o p e r a t o r s ( E q . B 5 3 ) h a v e a f a c t o r o f + i . ] T h e b o d y - f i x e d r a i s i n g

a n d l o w e r i n g o p e r a t o r s a r e

j + 1 = j x l - i j y l ,

( B 6 1 )

j _ 1 = j x l + i j y l . ( B 6 2 )

[ N o t e : A s a r e s u l t o f t h e m i n u s s i g n i n E q . ( B 6 1 ) t h e e q u a t i o n s t h a t d e f i n e t h e

b o d y - f i x e d r a i s i n g a n d l o w e r i n g o p e r a t o r s d i f f e r f r o m t h e e q u a t i o n s t h a t d e f i n e t h e

s p a c e - f i x e d r a i s i n g a n d l o w e r i n g o p e r a t o r s ; s e e E q s ( B 5 4 ) a n d ( B 5 5 ) ] . T h e b o d y -

f i x e d a n d s p a c e - f i x e d c o m p o n e n t s o f t h e a n g u l a r m o m e n t u m o p e r a t o r s c o m m u t e ,

l . e .

[ J i , j d = O . ( B 6 3 )

T h e t o t a l a n g u l a r m o m e n t u m o f t h e s y s t e m i s a c o n s t a n t o f t h e m o t i o n , s o

j 2 = j ; 1 + j ; 1 + j } 1 = j ; + j ; + j } . ( B 6 4 )

T h e o p e r a t o r s j 2 , j z a n d j z l f o r m a c o m p l e t e s e t o f c o m m u t i n g o p e r a t o r s . T h e i r

e i g e n v a l u e e q u a t i o n s a r e

A 2 j * . . j *

J V I " m e a , ( 3 , , ) = J ( J + 1 ) V I " m e a , ( 3 , , ) ,

j z V i " m ( a , ( 3 , , ) = J - l

V

i " m ( a , { 3 , , ) ,

j

z

I

V i " m ( a , { 3 , , ) = m V i " m ( a , { 3 , , ) .

( B 6 5 )

( B 6 6 )

( B 6 7 )

I n ( B 6 6 ) j z i s t h e c o m p o n e n t o f t h e a n g u l a r m o m e n t u m o p e r a t o r a l o n g t h e s p a c e -

f i x e d z a x i s ; j z l i n ( B 6 7 ) i s t h e c o m p o n e n t o f t h e a n g u l a r m o m e n t u m o p e r a t o r

a l o n g t h e b o d y - f i x e d z a x i s ( s y m m e t r y a x i s ) .

T h e r a i s i n g a n d l o w e r i n g o p e r a t o r s s a t i s f y t h e f o l l o w i n g r e l a t i o n s :

A j * . . 1 / 2 j *

J V l " m ( a , { 3 , , ) = [ ( r f J - l ) ( J J - l + 1 ) ] V I " 1 m ( a , { 3 , , ) ,

A r . . 1 / 2 r

J I V I " m ( a , { 3 , , ) = [ ( J = f m ) ( J m + 1 ) ] V I " m 1 ( a , { 3 , , ) .

( B 6 8 )

( B 6 9 )

T h e f o l l o w i n g r e l a t i o n i s u s e f u l f o r d e t e r m i n i n g t h e e l e m e n t s o f t h e W i g n e r r o t a t i o n

m a t r i x ( B i e d e n h a r n a n d L o u c k 1 9 8 1 ) :

V r ( a ( 3 ) _ [ ( j + J - l ) ! ( j + m ) ! ] 1 / 2 ( A ) j - I " ( A ) j - m ' *

I " m , " - ( 2 j ) ! ( j - J - l ) ! ( 2 j ) ! ( j - m ) ! J - J - I V ; j ( a , { 3 , , ) ,

w h e r e

2 j

V } ; ( a , { 3 , , ) = e x p ( i j a ) (cos~) e x p ( i h ) .

( B 7 0 )

( B 7 1 )

T h e s p h e r i c a l c o m p o n e n t s o f t h e a n g u l a r m o m e n t u m o p e r a t o r s a r e r e l a t e d t o t h e

r a i s i n g a n d l o w e r i n g o p e r a t o r s ( W e i s s b l u t h 1 9 7 9 ) b y

A 1 A

J + 1 = - h J + ,

1 A

: r - r . ; J - ,

" - 1 - v 2

a n d

J O = J z .

( B 7 2 )


Hence the square of the total angular momentum operator is

1

j2 = L (-1)qjqj_q q=-1

= -2j+1j-1 + jo(jo -1) = -2j-1j+1 + jo(jo + 1).

Eigenfunctions of the Symmetric Top

(B73)

The Wigner rotation matrix elements are proportional to the orthonormal eigen-functions Rj/Jm(a, (3, I) of the symmetric top Hamiltonian (Biedenharn and Louck 1981)

[ 1 A2 A2 1 A2] . 21x' (:lx l + :lyl) + 2Izl :lzl V~ m(a,(3,/)

- [_1 .(. 1) ~ (J:... _ J:...) 2] Vi" ( (3 ) - 21x, J J + + 2 lzl lx' m /J m a, 'I' (B74)

where lx'= lyl, and lzl are the principal moments of inertia, Jl is the component of angular momentum along the space-fixed z axis and m is the component of angular momentum along the body-fixed symmetry axis. In particular, the normalized symmetric top eigenfunctions are (Burke and Chandra 1972)

( 2i+1)1/2 . Rj/Jm(a, (3, I) = --s,;:2. V~ m (a, (3, I)

Because the Wigner rotation matrix in the passive convention (Steinborn and Rue-denberg 1973) is the Hermitian conjugate of the Wigner rotation matrix in this appendix, the matrix elements Vi m (a, (3, I) are (unnormalized) eigenfunctions of the symmetric top Hamiltonian when the passive convention is used.

Recursion Relations

The following recursion relations, which can be derived from the Clebsch-Gordan series (B26), are useful for calculating Wigner rotation matrices (Biedenharn and Louck 1981):

[(j - m)(j + m + 1)]1/2 sin(3d~,m+1(3) +[(j + m)(j - m + 1)P/2 sin(3d~,m_1(3) = 2(m cos (3 - Jl)dt,m (3) , (B75)

[(j + m)(j - m + 1)]1/2d~,m_1(3) +[(j + Jl)(j - Jl + 1)]1/2d~_1,m(3) = (m - Jl) cot ~dt,m(3). (B76)


P r o j e c t i o n O p e r a t o r s

A j

T h e o p e r a t o r s ' P m d e f i n e d b y ( B i e d e n h a r n a n d L o u c k 1 9 8 1 )

, j ; ' " ( j ) - I j

P~ = = I I . . . , y J . t

/ J = - j m - J . t

/ J T - m

a r e i d e m p o t e n t , i , e ,

A j A j A j

' P m ' P m = ' P m '

T h e y a r e a l s o m u t u a l l y e x c l u s i v e ,

A j A j

' P m ' P m ' =

f o r

m - : f m / ,

a n d f o r m a r e s o l u t i o n o f t h e i d e n t i t y , i , e ,

j ,

L P~ = I j ,

m = - j

T h e r e f o r e t h e m a t r i c e s P ~ a r e p r o j e c t i o n o p e r a t o r s ,

4 9 3

( B 7 7 )

( B 7 8 )

( B 7 9 )

( B 8 0 )

U s i n g t h e s e o p e r a t o r s , w e c a n w r i t e t h e s p e c t r a l r e s o l u t i o n ( C u s h i n g 1 9 7 5 ) o f

I T ~j) a s

A ( j ) _ ~ A j

: T y - L . . . J m ' P m '

m = - j

F o r a n y w e l l - d e f i n e d f u n c t i o n F , o n e h a s

F[IT~)l = t F(m)P~,

A j

T h e r e f o r e o n e c a n d e t e r m i n e d ( f J ) a s

m = - j

, C ' ) j ,

A J A J " " ' A J

d ( f J ) = e x p ( - i f J : T y ) = L . . . J e x p ( - i J . t f J ) ' P m '

/ J = - j

S o l u t i o n o f L a p l a c e ' s E q u a t i o n i n F o u r S p a c e

( B 8 1 )

( B 8 2 )

( B 8 3 )

T h e e l e m e n t s o f t h e W i g n e r r o t a t i o n m a t r i x e l e m e n t s a r e e i g e n f u n c t i o n s o f t h e

L a p l a c i a n i n f o u r s p a c e ( J u d d 1 9 7 5 )

\l~V~ m ( a , f J , , ) = 0 ,

( B 8 4 )

w h e r e

{ ) 2 { ) 2 { ) 2 { ) 2

\l~ = ' 2 i 2 " + ' 2 i 2 " + ! : l 2 + ' 2 i 2 " '

u X o u X

1

u X

2

u X

3

( B 8 5 )

i(1 + cos (3)2 ~ sin (3(1 + cos (3)

dZ((3) = j'isinZ (3 ~ sin (3(1 - cos (3)

i(1- cos (3)2

Wigner rotation matrices for j = 1/2, j = 1, j = 3/2, j = 2

( I!.' I!.) 1/2 cos 2 -sm 2 d ((3) = . sm% cos %

-~ cos (3

-V3 cos % sin % cos3 % - 2 cos % sin2 % 2 cosz % sin % - sin3 %

V3 cos % sinz %

V3 cos % sin2 % sin3 % - 2 cos2 % sin % cos3 % - 2 cos % sin Z %

V3 cos2 % sin %

-~ sin (3(1 + cos (3) j'isin2 (3 -~ sin(3(1- cos (3) ~(2cos2 (3 + cos (3 -1) -.fi sin (3 cos (3 -~(2 cos2 (3 - cos (3 - 1)

.fi sin (3 cos (3 ~ cosZ (3 - ~ -.fi sin (3 cos (3 _~(2cos2 (3 - cos (3 - 1) -.fi sin (3 cos (3 ~(2 cos2 (3 + cos (3 - 1)

~ sin (3( 1 - cos (3) j'isin2 (3 ~ sin(3(1 + cos (3)

i(1- cos (3)2 -~ sin(3(1- cos (3)

j'isin2 (3

-~ sin(3(1 + cos (3) i(1+ cos (3)2


Calculation of the Wigner Rotation Matrices

There are several methods that one can use to obtain dim (,8), including (i) explicit expressions in terms of the hyper geometric function or Jacobi poly-

nomials (B7 or B9); (ii) differential equations (B 71);

(iii) recursion relations (B76 and B77); (iv) projection operators (B84); (v) boson operator techniques (Biedenharn and Louck 1981, Chapter 5).

Appendix C: The Passive Convention In the text of this paper and in Appendix B, we used predominantly the active convention for rotations. For the convenience of readers who prefer the passive convention, we here express some of the most important results from this paper in this convention. The equation numbers in this appendix refer to the corresponding equation (in the active convention) in the text or in Appendix B.

The passive Euler angle rotation matrix:

( COS a sin a 0) (cos /3 0 R (p ) ( a, /3 , 'Y) = . - sin a cos a 0 . 0 1 o 0 1 sin/3 0

- sin (3) ( co~ 'Y o - Slll'Y

cos /3 0

Definition of the Wigner rotation matrix:

1Jt m(a,/3,'Y) = Upl exp(i'Yjzll) exp(i/3jyl)exp(iajz)ljm) = exp(ip'Y)Upl exp(i/3jy)ljm) exp(ima) = exp( ip'Y )dtm (/3) exp( ima).

Slll'Y COS'Y

o D (34) (B6)

Note that dim(/3) in the passive convention can be determined from Eqs (B7) or (B8) by replacing /3 by -/3 in the right-hand side of these equations.

Effect of the passive rotation operator on an angular momentum eigenket: +j

Jj,m)2 = k(p)(a,/3,'Y)lj,m)l = L Ij,ph 1Jt,m(a,/3,'Y). (54) JJ=-j

Rotation of spherical harmonics:

+l Yt(B2,


E i g e n v a l u e e q u a t i o n s f o r t h e W i g n e r r o t a t i o n m a t r i c e s :

j 2 V t m ( o : , { 3 , ' Y ) = j ( j + 1 ) V t m ( o : , { 3 , ' Y ) ,

jzv~ m ( o : , { 3 , ' Y ) = mV~ m ( o : , { 3 , ' Y )

jz'v~ m ( o : , { 3 , ' Y ) = f-tV~ m ( o : , { 3 , ' Y )

s p a c e - f i x e d z a x i s ,

b o d y - f i x e d z a x i s .

E f f e c t o f t h e r a i s i n g a n d l o w e r i n g o p e r a t o r s o n t h e r o t a t i o n m a t r i c e s :

jv~ m ( o : , { 3 , ' Y )

j,v~ m ( o : , { 3 , ' Y )

[ ( j = f f - t ) ( j f - t + 1 ) ] 1 / 2 V~1m ( 0 : , ( 3 , ' Y ) ,

[ ( j = f m ) ( j m + 1 ) ] 1 / 2 V~m1(0:, ( 3 , ' Y ) .

T h e t i m e - i n d e p e n d e n t S c h r o d i n g e r e q u a t i o n f o r a s y m m e t r i c t o p :

[

1 ' 2 ' 2 1 ' 2 ] .

2 I x ' ( J

x

' + J

y

' ) + 2 I

z

' J z ' V~ m ( o : , { 3 , ' Y )

= [2Lj(j+1)+~(L - L)m2]V~m(0:,{3,'Y)'

N o r m a l i z e d e i g e n f u n c t i o n s o f t h e s y m m e t r i c t o p :

(

2 j + 1 ) 1 / 2 .

R j j J m ( o : , { 3 , ' Y ) = ~ . V~ j J ( o : , { 3 , ' Y ) .

4 9 7

( B 6 5 )

( B 6 6 )

( B 6 7 )

( B 6 8 )

( B 6 9 )

( B 7 4 )

M a n u s c r i p t r e c e i v e d 3 0 J u n e 1 9 8 6 , a c c e p t e d 1 3 F e b r u a r y 1 9 8 7

ph 870465

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