IL NUOV0 CIMENTO VOL. 22 B, N. 2 11 Agosto 1974

Quantum Mechanics of the Rigid Rotator.

J. DENi~EAD SlVs

University of Leeds - ;Seeds College o/ the Resurrection - Mir]ield, Yorks. (*)

(ricevnto il 4 Settembre 1973)

S u m m a r y . - - A representation of the position and angular-momentum variables of a quantum-mechanical rigid rotator is obtained from general symmetry principles. The method used is similar to the treatment of spin in the case of the quantum particle.

1 . - I n t r o d u c t i o n .

The apparen t accidental degeneracy of the Hami l ton ian in the case of certain symmet r ica l quantum-mechanica l rigid ro ta tors is well known to have an explanat ion in the fac t tha t thei r s y m m e t r y groups are larger t h a n the spat ia l ro ta t ion group. The principle behind this is t h a t in addit ion to the group of rota t ions of the ro ta to r abou t fixed space axes~ the s ta te space is also t r ans formed under the group of rota t ions about a set of axes t ha t m a y be con- sidered as fixed in the ro ta to r itself (1).

I n this pape r the representat ions of the posit ion and angu l a r -momen tum variables of the q u a n t u m ro ta to r are obta ined b y means of s y m m e t r y principles alone. B y considering the ro ta t ion groups about two sets of axes, a descript ion of the ro ta to r is produced in a similar manner to t ha t used for establishing the s tandard (SchrSdinger) representa t ion for particles with spin. I t is also shown how this <~ axiomat ic )> approach is re la ted to the description of a sys tem of spinning q u a n t u m particles which are held together in the shape of a rigid molecule. Before beginning the quan tum-mechanica l t r ea tmen t a brief r6sum6 of the classical dynamics of the ro ta to r is given.

(*) Present address. (1) H. V. ~C]~N~COSE: Amer. Yourn. Phys. , 27, 620 (1959).

337

3 3 8 J. DENMEAD SMITH

2 . - T h e c l a s s i c a l r o t a t o r .

Classically, the rigid ro ta to r is conceived as a sys tem of particles which are constrained to move in space in such a way t h a t the distances between the particles remain invar iant , and for which one particle 0 has a fixed position, the centre of rotat ion. The posit ion of the ro ta to r at any t ime m a y be described with the aid of a f r ame of reference 2: which is fixed in the body and has its origin at O. I f Zo is a f r ame of reference fixed in space, also with its origin a t O, there is a unique ro ta t ion x satisfying Z---- xZo which carries Zo into Z. Thus the configuration space of the ro ta to r can be considered as the set {x) of rota- t ions abou t O, i.e. the group S03 of or thogonal matr ices with posi t ive deter- minant .

Now suppose t h a t z is a ]ixed element of S03 and x is a var iable element which represents the posi t ion of the body. Since zxZo = z(XZo), i t is seen t h a t zx represents the posit ion of the rigid body af ter i t has undergone a rota- t ion z about the space axes. Thus left mult ipl icat ion of x b y z moves the body th rough the ro ta t ion z abou t the <( fixed )> f r ame Zo. Similarly, the t ransforma- t ions t h a t are represented b y r ight mul t ip l icat ion are rota t ions relat ive to the body axes. I f z ~ SOa is fixed, the t rans format ion x - + xz -1 represents the ro- t a t ion of the body th rough the fixed ro ta t ion z about the (( body )~ f rame Z.

~7ow the classical phase space of a mechanical sys tem which has a manifold M for its configuration space is the cotangent bundle T*(M)----{(q, p)), the co- t angent vector p (( si t t ing over )> the configuration point q ~ M being the set of generalized m o m e n t a of the syste~n in posit ion q (see, for example, ref. (~), Sect. 1.3).

I f G is a / - p a r a m e t e r group of t ransformat ions act ing on M which has the vector field ~A~(q)8/Sq ~ for its infinitesimal generator, where qt, ..., q~ are

local co-ordinates, there is canonically associated with G the m o m e n t u m ob- servable ~AJ(q)pr This observable has the impor t an t p rope r ty t h a t it is

conserved when the dynamica l act ion (equivalently, the t t ami l tou ian) is in- va r i an t under G. For example , for a classical part icle moving in /~3 each 1 -pa ramete r group of t ranslat ions gives rise to the component of the linear m o m e n t u m of the part icle in the direction of t ranslat ion.

When the configuration manifold M is a Lie group H the t angen t bundle T(M) is trivial, being isomorphic to H x h , where h is the Lie algebra of H ; likewise, T*(M) ~ H • where h* is the set of invar ian t differential forms on H. Thus for the ro ta to r the classical phase space is isomorphic to SOa x ~ a.

(2) G . W . MACKEY: The Mathematical •oundations o] Quantum Mechanics (New York, :N. Y., 1963). (a) G. W. MACKEY: Induced Representations o] Groups and Quantum Mechanics (:New York, N. u 1968).

QUA~NTUM MECHANICS O1~ THE I~IGID Ir162 ~ 9

The left- and right-sided actions of the group S03 on the configuration space (als o S03) described above thus give rise to classical observables via their in- finitesimal generators. The generators of the ]eft rotat ions give the components (Ml(Io), M~(!o), M~(10)) of the rota tor ' s angular momen tum about O, measured relat ive to 270, and the generators of the right rotat ions give the components (MI(1), M2(1), M3(Z)) of the rota tor ' s angular momentum relat ive to 27.

3 . - T h e q u a n t u m r o t a t o r .

Our purpose is to obtain for the quan tum ro ta to r a representat ion of the left and right actions of the ro ta t ion group as described in Sect. 2, and to determine their relat ion to the position observables of the rotator . To achieve this we employ a procedure exact ly similar to tha t of MAC~EY (3) for rep- resenting the position and momentum observables of a single particle in space.

I t will be supposed tha t the pure states of the ro ta tor are defined b y unit vectors in a separable Hilber t space ~%f with inner product ( . , . ) . F r o m general considerations there is then associated with each Borel subset E of S03 (endowed with the usual topology) an orthogonal project ion PE on ~ , with the p roper ty tha t when the ro ta tor is in the state y~ the probabil i ty tha t its configuration is described by an element of the subset E is (~p, P ~ p ) . The map E - ~ P ~ must also define a project ion-valued measure (PVlVl), t ha t is to say

i) P~ and PF are mutua l ly orthogonal if E and /7 are disjoint;

if) P~ = ~ -P~z when E is the dis}oint union of Borel subsets Ea, ~ = 1, 2,... ; g=i

iii) P~o. : I .

As in the classical ease, the phase space (here ~ ) is t ransformed under the left and right actions of SOa. Specifically, there is associated with a pair of elements g, h of #03 a un i ta ry operator Ug.a on 9~ which represents a ro ta t ion g of the ro ta tor about the space axes 27o, together with a ro ta t ion h about the body axes I . I t is not necessary to assume tha t the map (g, h) -> Ug.h is a homo- morphism of the group SOn • S03 into the un i ta ry group on ~ (i.e. it is a rep- resentation), bu t U must be a (continuous) ray representa t ion (a). This means tha t there is associated with each pair (g~, h~), i = 1, 2, a complex number ~(gl, hi; g~, h2) of modulus un i ty for which

is the multiplier of the ray representation. The relation between the ray representations of S03 and the ordinary

representations of the covering group S U2 is well known, bu t it will be useful

340 J. DENM]~AD SMITII

to give an outline in order to fix the notat ion. I f s denotes the 2 • 2 ma t r i x

s = �89 ( 1 - - so s~-t- is~ t

\Sl -- is~ 1 q- so]

it is readily seen t h a t the points (so, s~, s~) which belong to S ~, the uni t sphere in/~3, m a y be described b y the set of t t e rmi t i an matr ices s satisfying det (s) = 0, Tr ( s ) = 1. Now when u is an element of SU~ the ma t r ix u-~su is of this same class as s, and the m a p s--~u-~su therefore describes un or thogonal t ransformut ion of S ~. :Furthermore, since it is or ientat ion-preserving u represents a rotat ion, whilst any element of SO~ can be thus represented in two ways, b y elements • for some u~SU2. We therefore have a 2-1 ep imorphism g:SU~-->SO~. The kernel is ~he subgroup N---- (e, - - e}, where e is the iden- t i t y in SU2, and hence SOs ~ SU~/N. I t will be convenient also to define an inverse m a p fi:SO~--~SU2 which satisfies f i a = i d e n t i t y ; this m a y be ar- b i t rary , except in so fur as we require it to be measureable with respect ~0 the

Haur measures. Thus fi(gl)fi(g~)= • when g~, g~SO~. Clearly the m a p g--->fl(g) is a 2-dimensional r ay representa t ion of S03, with mult ipl ier ffl(gl, g~) satisfying

/~(gl)fi(g2) = (rl(g~, g~)fl(glg2).

:Now any ray representa t ion of SO~ gives a r a y representa t ion of S U2 via the

m a p a, and the converse is also t rue via ft. Since S U2 is bo th semi-simple and s imply connected all its r ay representat ions are ordinary representat ions (4) (or, more precisely, are equivalent to ordinary representat ions). Thus the mni- tip]ier group of SOs has the two representat ives ao(g~, g2)-~1 and r g~)~ defined above. The group SU2 • SU~ is also semi-simple and s imply connected: therefore its r ay representat ions are ordinary representat ions, and so we m a y assume tha t the mult ipl ier of U is given b y a-= (r,• i.e.

a(gl, hl; g:, h~) ---- a,(gl, g~)aj(h~, h2)

for some i, j = 0 or 1. There are thus four distinct cases in all. :Now f rom physical considerations the following relat ion between the rep-

resentat ion U and the PVS~ P will be required to hold:

(3.1)

This means t h a t P is a sys tem of impr imi t iv i ty for the representa t ion U of SOs • S03 with respect to the t ransi t ive action x -+ g-lxh of S03 • SOa on the

(4) V. BAt~GMA~" Ann. Math., 59, 1 (1954).

Q U A ~ N T U M M E C H A N I C S O F T H E R I G I D R O T A ~ T O R 341

base space SOs. The stability subgroup tha t leaves invariant a selected (arbitrary) element of the base space, here taken to be the ident i ty element,

is the diagonal subgroup G---- {(g, g): g e SOs} ~ SOs. Thus if U is a ray rep- resentation of SOs • with multiplier ~ - - a s • U is unitari ly equivalent to an induced representation U L, where L is a a~aj representation of SOs, whilst /)~ is equivalent to multiplication by the characteristic funct ion of E on the base space S03: this is g direct consequence of the imprimit ivi ty theorem

(ref. (s), Sect. 2.4). An invariant measure on the base space is Haar measure, and the Hilbert

space of the representation is accordingly ~ - = ~f(Z) | Zs(S03), the space of square-integrable vector-valued wave functions on SOs, where ~ ( Z ) is the

I t i lbert space on which L is defined. The representation of P~ is then given by the multiplication operator defined by

(3.2) (P~q~)(k)~--I q~(k), k e E , ! O, k ~ E .

There are two possible realizations of Ug.~, as follows:

(3.3) (U~.~)(k) : L~(g-~kh)Tj ,

where the number ~ satisfies fl(g)-~fi(k)fl(h)~--T, fl(g-~kh), tha t is ~ , = a~(k, h). .(~(g, g-lkh). I t is clear tha t both reduce to the representation L of G on the

(~ fibre ~ above e ~ SO~; moreover, they each define a~ • representations of

SOs • S03 and are therefore realizations of U ~. Their equivalence is given explicitly by the uni tary (( twisting ~> operator S on ~ , defined by (S~0)(k)~-- ~--Lk~(k ). We now fix the notation, and define U by eq. (3.3).

I t will suit our purposes to effect a uni ta ry t ransformation of ~ which will make the calculation of infinitesimal generators easier. To do this we need

to make some observations about the group SU~. Each element of SU~ can be written

u : ( uo + iul u~ + iu~ I

\-- u~ ~- iu~ u o - - iul!

~ 2 § u~ = 1, and hence the group manifold of S U~ in which det (u) ~ u o ~- u 1 ~- u~ can be identified with S s, the unit sphere in R ~. Purthermore~ in a similar

manner to tha t used previously to describe SOs any rotat ion of S s may be written u--~aub -~ where a, b ~ SU~, so tha t Hagr measure on SU~ corresponds

to spherical Lebesgue measure on S s. Finally, since

~ o , ~ ~ • ~ / { ( e , ~), ( - e, - ~)},

U m a y be considered as a ray representation of SO~.

2 2 - I I Nuovo Cimenlo B.

342 J . D E N M E A D S M I T H

Under the map fl wave functions defined on SOs give rise to wave functions defined on hall of S~; they m a y then be extended as either odd or even func- tions to the whole of S 3 with respect to the inversion x - + - x. Thus there are two un i ta ry maps q)o and q)~:

where r takes the values 0 and 1, and the subscripts 0 and 1 denote the even and odd subspaees of L*(S3), the space of square-integrable wave functions on S ~.

The un i ta ry map r162 t ransforms P s and U~.~ into operators Qs and V~j, which act on ~ , and these are defined b y

(3.4)

und

(3.5)

= ] ~j(x) for fl(x) c E , (Q~w~)(z)

[ 0 otherwise,

(V~.~ ~zj)(x) = L~.~j(fi(g)-ixfi(h) )

for any F j + ~ j . Equat ions (3.4) and (3.5) are easily shown to define a PVM and a a~ • a~ r ep resen tagon of SOa• SOs which satisfies the impr imi t iv i ty rela- tion (3.1), and the equivalence with (3.2) and (3.3) is pract ical ly immediate .

4. - A n g u l a r m o m e n t u m .

The components of the angu l a r -momen tum operators M(X,) and M(Z) are (taking units in which ~ = 1) the infinitesimal generators of the left- and r ight-rotat ion subgroups of S03 • S03 in the representa t ion U, and the ma t r i x

[Xr~ ] of position operators is defined b y

( x , ~ q J ) ( x ) - x , c f ( z ) ,

where x ~ S03, qJ ~ ~ . By taking the inverse funct ion fi :SOs ~ SU~ to be differentiable in a neigh-

bourhood of the ident i ty element and using (3.4) and (3.5) the following ident i ty

can be verified:

3

(4.~) M~(r) = m~ + ~ X~M~(2o),

where ml, m~, m3 are the infinitesimal generators of the r ay representa-

t ion L of SO~.

Q U A N T U M M E C H A N I C S OF T H E R I G I D ~OTATOI~ 343

Thus, in the ~ scalar ~> case, where ~L is the identi~y representat ion and

m = 0, these equations reduce to a s tatement tha t the angular momen tum transforms as a vector between frames Z0 and X, as for the classical rotator.

Although M(Zo) and M(2:) are operators tha t do not commute with X = IX,,], the order of the factors in the right-haald side of (4.1) is un impor tan t because

of the equation ~ X ~ M ~ ( X o ) = M,,,(-~o)X~. I t is therefore suggested on

physical grounds tha t only the cases where L is the identity, or a multiple of the identity, will oecnr for the rigid molecule, and this implies tha t U is either a % • ao or a ~1 • ~ representation. These facts will be confirmed in the next

Section.

5. - R e l a t i o n t o a s y s t e m o f p a r t i c l e s .

Ldt 1tl, ..., H,~ denote the spin sp~ces of n p~rticles which are, for con-

venience, assumed to be nonidentical. The Hilbert space of the k-th particle is H~ | L2(R~), and the Hilbert space of the whole system is thus 9 i (N) | L2(5~),

in which 9i(N) = H1 | ... | H~, N denotes the representation of S03 in 9 i ( h r) ~nd S z = R 3 • ~.

~0 c 5 ~ is the set of configurations in 9 ~ for which the p~rticles are all col-

linear with the origin. These <~ singular ~> configurations have measure zero, and hence if 5 p_ = S t - 5Zo, L2(Sz_)= L2(Sz). S03 acts in the obvious (i.e. com-

ponentwise) manner on 90_ and ~-" is defined to be Zz_/S08, the class of non- equivalent configurations of the system, in which two configurations are counted

equivalent if they may be rotated into each other by a rotat ion about the origin. F rom ~his the ident i ty ~ _ ~ S03 • Y follows immediately. Now the me~sttre

on 5 p_ factors into the product of Haar me~sure on SO~ and another measure

on J ' . Thus L2(~_) ~ L~(S03) | L2(3-), and the t t i lbert space of the system can be writ ten 9i(h;) | L2(S03) | L2(J'). This state space is the direct in- tegral of the Hilbert space ~ g ' = 9 i (N) | J52(S03) integrated over the <~ base space ,> ~-'. Ideally, for a perfect rigid body which is maintained in the same configuration, the wave functions belong to a single , fibre ~> 9 i over one point

of ~--, but actually, of course, the wave functions are spread over ~-" (otherwise they would not be normalizable).

In concrete terms the identification 5p_ ~ SO3 • depends on selecting representative s~ ~ in 5 ~_ for each element o f f . Thus the elements x, s~ ~ E S03 • define s_ : x s ~ b ~ Right multiplication on 203 corresponds to making a different choice of a <( fixed ,) frame (represented by s~); al ternatively it m a y

be regarded as motion of the rota tor through a rotat ion which is fixed relative to the body.

I t is now convenient to consider the state space of the system as the direct integral of 9if(N) | L~(~ -') integrated over the t~ base ~> SO~. Since this Hilbert

3 4 4 j . DENMEAI) SMITH

bundle is trivial, the fibres over the poin%s of S03 are identified in the obvious way. The f b r e over the point xge S03 is t r ans formed into the fibre above x when a ro ta t ion g is performed about the body axes Z, and the t rans format ion of the fibre-spaces (each identitied with .~(2v') ~ L~-(.Y-)) under the body tra.ns- format ion g is given b y the linear t r ans format ion N(x)b'(xg) -~. This is ex- plained as follows. When the configuration is rota ted f rom its position a t xg to the s tandard position where 2: and Zo coincide, the fibre-space is t ransformed under ~Y(xg) -~, and N(x) is the corresponding transformat.ion when the sys tem is re turned to its fina.l posit.ion at x.

In the ((idealized ,) s i tuat ion with a configuration localized in one point of J our a t ten t ion m a y be restr icted to i F - - ~5~f(N) | L2(SO~), now t h a t L~ ~) has been <, fac tored away ~>. The representa t ion U of S03 • SOa in off has left (of) and right (~ ) restrict ions (i.e. SOz-actions), where U - - ~ ( ~ • ' ~ and

(~ ,~ ) (x ) = N(g) ~ (g - ,x ) ,

( ~ ~;,)(x) = N(x)hT(xg)-~ ~f(xg).

I t can be seen tha t .~f,~ 22~ is the ident i ty on the (( fibre )> over c ~ S03, and hence U ,~ U z -0 U z +0 -.. ~ U*. The mult ipl ic i ty of this representa t ion is equal to the dimension of the spin space ~f~(N). I f the to ta l spin is inte~ 'a l , U is an ordinary rq)resentat ion, and, if the to ta l spin is no~ integral, U is a al • representat ion. These facts m a y be seen otherwise by performing the un i t a ry t rans format ion T on ~ , where Ty(x) = N(x)~,(x). Therefore, apar t f rom mul- tiplicity, the si tuat ion described in the previous Section obtains, in which Z is the i d e m i t y representat ion.

6 . - C o n c l u s i o n .

W h a t is suggested b y eq. (4.1), namely t h a t to correspond with the physieal model of a rigid molecule L mus t be the ident i ty representa t ion of SO,~, was confirmed in the previous Section. As far as the posit ion and angu la r -momen tum x, ariables are concerned, the s ta te vectors of a rigid molecule m a y be r ep re sen t ed to within un i ta ry equivalence b y K wave functions on 23 (the unit sphere in/P,), where K is the dimension of the spin space, and the wave functions are all even or odd according as the to ta l spin of the sys tem is integral or not. The representa t ion of the position and angular m o m e n t u m variables proceeds via the na tura l 2-1 identification of S 3 with 803.

I t is hoped tha t the more general t r e a t m e n t of the problem, apar t f rom its interest as an axiomat ic derivation, m a y have applicat ion to the question of s y m m e t r y .ffroups for particles.

QU&NTUM ~r OF T~IE RIGID I~OT~TOR 345

�9 R I A S S U N T 0 (*)

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(*) T~'aduzione a cura della Redazione.

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