Orbital angular momentum and group representations

Orbital angular momentum and group representations

ANNALS OF PHYSICS: Orbital 47, 232-214 (1968) Angular Momentum and Group Representations CLASINE VAN WINTER Afdeling voor Theoretische Natuur...

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ANNALS

OF PHYSICS:

Orbital

47,

232-214 (1968)

Angular

Momentum

and Group

Representations

CLASINE VAN WINTER Afdeling voor Theoretische Natuurkunde, Natuurkundig Laboratorirrnl Groningen, The Netherlands

der Rijksuniversiteit,

A new approach is presented to the question why in quantum mechanics the orbital angular momentum has integral eigenvalues only. The problem is formulated in terms of linear operators on the Hilbert space Jj of square-integrable functions of the angular variables 9 and cos B. The components of the orbital angular momentum are represented by self-adjoint operators Lj (j = x, y, z) on $j which are constructed starting from the familiar differential operators in terms of 8 and y. The construction is possible in infinitely many ways. These are labeled by a continuous parameter 6, the choice 6 = 0 corresponding to integral eigenvalues. For every particular 6, the operator Zj Lj2 has one and only one extension L2 which is self-adjoint on 5j and such that, for each j, the domain of L2 is contained in the domain of Lj _This L2 is the only operator which may represent the square of the orbital angular momentum. It is only in the case 6 = 0 that L2 commutes with each Lj . Also, 6 = 0 is the only case in which the spectra of L, , L, , and L, are the same. These are results which refer directly to the outcome of measurements. It is suggested that this explains why nonintegral eigenvalues for the orbital angular momentum are not found in nature. It is observed that orbital angular momenta with nonintegral eigenvalues do not fit in the framework of group theory. This is discussed from a general point of view. For a set of self-adjoint operators Lj on some Hilbert space R to generate a unitary representation of a simply connected, compact, semisimple Lie group, it is shown to be necessary and sufficient that the operators Li satisfy suitable commutation relations and commute with a self-adjoint Casirnir operator L2. If, in addition, the group is of rank 1, it is necessary that the generators all have the same spectrum, which must be symmetric with respect to the origin. There is thus a general connection between the occurrence of group representations and the properties of observables.

I. INTRODUCTION I. 1. THE ORBITAL-ANGULAR-MOMENTUM

PROBLEM

In quantum mechanics, it is a good procedure to discuss orbital angular momentum with the help of the spherical harmonics Yrm. These are eigenfunctions of a well-known differential operator T2 acting on the polar angles t!J and 9~ They refer to integral angular momenta in the sense that I and m take integral values only. If T2 did not have other eigenfunctions, the theory would be straight-

232

ORBITAL

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233

forward. It has been known for a long time, however, that there are also eigenfunctions Z,p characterized by nonintegral quantum numbers h and p. The question therefore arises why nonintegral eigenvalues for the orbital angular momentum do not occur in nature. The function ZAp contains a factor exp(ipp), and so it changes its phase when the azimuthal angle v is increased by 2~. The possibility of nonintegral eigenvalues can thus be ruled out by requiring that an acceptable wavefunction be singlevalued. This argument is, in fact, presented in many textbooks, but most students reject it as being unsatisfactory. It is true that some time ago Merzbacher (1) argued that the single-valuedness requirement is altogether reasonable, yet this condition does certainly not have the status of a fundamental principle of quantum mechanics. Years before Merzbacher’s paper, Blatt and Weisskopf [(2), Appendix A] called the single-valuedness condition fallacious, and they cited an argument due to Nordsieck, according to which half-integral orbital angular momenta give rise to probability currents with sources and sinks. It appears that this idea remained unpublished until it was worked out by Whippman (3). An entirely different approach was put forward by Buchdahl(4). This was based on commutation relations. Green [(5), Section 5.51 developed an argument starting from the observation that the orbital angular momentum of a particle is perpendicular to the position vector. The most elaborate paper is due to Pauli (6), who gave a detailed discussion of differential operators and their eigenvalues and also commented on certain group-theoretical aspects of the problem. If Ti (j = x, y, z) stands for the differential operator associated with the j-component of the orbital angular momentum, each Tj transforms Ylffb into a linear combination of functions YT’ with the original 1. The ladder operators T* = T, f iT, can thus be applied to YClnany number of times. They give finite sums of spherical harmonics with the same 1. In the case of nonintegral values h and p, this is no longer true. Given any ZAp with p > 0, repeated application of Tultimately yields functions which are not even square-integrable. Likewise, if p < 0 repeated application of T+ yields functions which are not square-integrable. There is thus a substantial difference between the integral and nonintegral cases. Essentially it was this difference which caused Pauli (6) to reject nonintegral eigenvalues for the orbital angular momentum. Pauli’s paper was discussed by Merzbacher (I), who pointed out that applying the operators T* to functions ZA* ultimately yields functions with respect to which T2 is not Hermitian. In this form, the observation is not entirely correct, because an operator is either Hermitian or not Hermitian, and not Hermitian with respect to a set of functions is not a meaningful concept. Merzbacher’s remark is valuable, however, in that it suggests that there are differentiable functions to which it may not be appropriate to apply the operator T2. In the

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present paper, this notion plays an important role. It is shown to result in an entirely new approach to the problem. 1.2. METHOD

AND RESULTS

Becausethe angular-momentum operators are not bounded, one must carefully indicate their domains, that is, the sets of functions on which they are allowed to act. In order that any particular operator represents an observable quantity, it must be a self-adjoint operator on a Hilbert space, and so its domain must coincide with the domain of its adjoint. In Sections III and IV this point is investigated for the formal differential operator T, . It is shown that there are infinitely many ways of turning this into a self-adjoint operator L, on the Hilbert space of square-integrable functions of F and cos 8. The possible choices for L, are labeled by a parameter 6 such that 0 < 6 < 1, the case 6 = 0 corresponding to integral eigenvaluesfor L, . Now let L, be characterized by any particular 6, let its domain be B(L,), and write 6 for the set of functions f E D(L,) which are sufficiently smooth for all operators TjTk to be applicable. On 6, the differential operators Tj(j = x, y, z) are not self-adjoint, but they have unique self-adjoint extensions Lj . These are constructed in Section VII, the new operator L, being the sameas the original one. We assumethat the operators Lj represent the components of the orbital angular momentum. Because L, and L, depend on 6, different values of 6 give rise to different angular momenta. Given the three operators L, , the domains D(LjL,) are known, and so one can consider the intersection fij,lc D(LjL,). It follows from Section VIII that this coincides with K. It is the largest set on which the commutation relations can be applied. On Cc,it is also meaningful to apply the operator ~j Li2. It is shown in Sections X-XII that this has one and only one self-adjoint extension L2 such that the domain D(L2) is in nj 7D(LJ. We assume that this L2 represents the square of the total orbital angular momentum. It depends on 6. Given 6, it is the only self-adjoint extension of zj Lj2 which has self-adjoint components Lj , and so it appears to be the only operator L2 which might be of physical interest. Once the operators Lj and L2 have been constructed, their properties can be investigated. The following facts are then observed. 1. Given 6, the eigenvalues of L, are equal to p=W+,

m = 0, *l,

*2 ,... .

(1.1)

Given p, the eigenvalues of L2 are equal to h(h + I), with h = I CLI>I p I + I,... .

(1.2)

ORBITAL ANGULAR

MOMENTUM

235

The case6 = 0 thus corresponds to integral eigenvalues.If S = 0, so that h and TV reduce to integers 1 and m, the spherical harmonics YLmprovide a complete set of simultaneouseigenfunctions of L2 and L, . If 6 f 0, the functions ZApdiscussed in Section II provide a complete set of simultaneous eigenfunctions of L2 and L, . 2. The eigenvalues of L, and L, are integral, irrespective of 6. If 6 f 0, they are thus not equal to the eigenvaluesof Lz . If 6 = 0, the operators L, and L, are unitarily equivalent to Lz . If 6 ;t 0, this is not so. 3.

If8=0,

L2f = Lc2f + Lv2f + Lz2f

(1.3)

for every f E ID(L2). If 6 # 0, this is not true. 4.

If 6 = 0, L,L,f

- L,L=f = iL,j

(1.4)

for every f E D(L”). If 6 f 0, this is not true. In the latter case there are eigenfunctions of L2 to which neither L,L, nor L,L, can be applied. 5. If 6 = 0, the operator L2 commutes with each Li . If 6 # 0, the operator L2 still commutes with L, , but it does not commute with L, and L, . In this case, L2 and L, do not have simultaneous eigenfunctions. The sameappliesto L2 and L, . For physical purposes,Point 5 is of particular importance. It showsthat between the integral and nonintegral casesthere is a fundamental difference which can be expressedin terms of observables which do or do not commute. Point 5 thus refers to properties which are directly accessibleto measurements.We feel that it offers the most reasonable explanation as to why nonintegral eigenvalues for the orbital angular momentum do not occur in nature. In addition to Point 5, Point 2 reveals an undesirable property of the nonintegral casewhich would also show up in experiments. This appearsto be another reasonwhy integral eigenvalues for the orbital angular momentum are preferred. 1.3. SURVEY OF THE LITERATURE In a mathematical analysis, one must be very careful becauseof Point 4. This says, essentially, that in the nonintegral case the domain Cs on which the commutation relations can be applied is too small for these relations to be safe. As a rest& of this, arguments based on commutation relations are inconclusive. It is not legitimate to reject an alternative on the basis of a relation which does not apply to it. It would be another thing if the validity of the commutation relations was used as a criterion, but this is not what is done in the literature. It is always taken for granted that the commutation relations can be applied

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WINTER

to any eigenfunction of L2. Thus Buchdahl’s paper (4) is incomplete, and it would require a detailed analysis to see if his ideas can be made reliable. The same applies to Green’s argument (5), because this also makes use of commutation relations. Whippman’s approach (3) is invalid for a different reason. Here, it is argued that there are wavefunctions for which the &component of the probability current has undesirable properties. But this is obvious. For instance, one may have a wavefunction which cannot be differentiated with respect to 9 but is well-behaved otherwise. Apparently, however, it is Whippman’s point that in his example the undesirable wavefunction # is an eigenfunction of the orbital angular momentum L2. Now, firstly, there does not seem to be a compelling reason why the probability current should be well-behaved whenever one is in an eigenstate of L2. And, secondly, it is not true that # is an eigenfunction of L2. It follows from the present paper that # is not even in the domain D(L2). There is thus no connection between $ and L2, hence one cannot draw a conclusion on L2 on the basis of an argument about $. It may be helpful to clarify our remark on # not being in ID(L2), because this will illustrate the point of view of the present paper. In the notation of our Section II, Whippman’s function t/ is of the form * = zy2 + cq,

(1.5)

with some complex number 01f 0. The functions 2:‘2 and Xi” are formal eigenfunctions of the differential operator T2, both with eigenvalue X(X + 1). According to Section XII, Z, 1’2 is also an eigenfunction of the orbital-angularmomentum operator L2. Now suppose that I+/Jis also an eigenfunction of L2. Then so is Xi’2. However, the set {Z,““} with h = 4, g,... is already complete in the space of all functions of the form exp($$)f(9)), where f(S) is a squareintegrable function of cos 6. Thus L2 would have more than a complete set of eigenfunctions. As a result it would not be self-adjoint, and so it would not represent an observable. This shows that it is not possible for L2 to have both Z,1” and X,1’2 as eigenfunctions. Conversely, given T2, it is not obvious that this is associated with a self-adjoint L2. It is rather a result of the present paper that a self-adjoint L2 does in fact exist. It is not obvious either that Z:‘2 is an eigenfunction, and not Xi’2. This is due to our requirement that L2 should have self-adjoint components Lj . Being an eigenfunction, Zt’2 is, of course, in D(L2). It follows from Section XII that Xi” is not in ID(L2). In the last few years, the functions Zt’2 and Xii2 were also discussed by Pandres (7) and Sannikov (8). Because, in addition, these authors considered functions which are not square-integrable, it is impossible to compare their work with ours. Pandres used the idea of “a vector which is a representative of the

ORBITAL

ANGULAR

MOMENTUM

237

vanishing vector even though it is not of vanishing functional form”. This is incompatible with our point of view. Let us now turn to Pauli’s paper (6). Here, there is an inkling that in the nonintegral case there is something wrong with the commutation relations, but the idea is not worked out. In fact, it is stated several times that Lz and Lj commute. The paper does not indicate the methods to analyse this point should doubt arise. In particular, the question of the domains of the various operators is not considered. One might thus raise several objections, but we want to emphasize that Pauli’s observations are basically correct. It is discussed in Section XIX how Pauli’s criterion is related to the requirement that L3 and Li should commute. 1.4. CONNECTION WITH GROUP REPRESENTATIONS Pauli demanded that each operator Lj should turn an eigenfunction of LL into an eigenfunction with the original eigenvalue. He remarked himself that this criterion is closely related to the requirement that under rotations the eigenfunctions of L2 with any particular eigenvalue transform among themselves. It is well known that this requirement is fulfilled by the spherical harmonics. These are characterized by 6 = 0, the corresponding operators Lj being the generators of a unitary representation of SU(2). If 6 f 0, the relation between the operators Lj and the group SU(2) is destroyed. According to Points 2 and 5, the role of L, and Lu is then different from L, , and so there is no longer an equivalence between all directions in space. Because the notion of a transformation group appears to be fundamental in quantum mechanics, it is an attractive idea that certain classesof observables should be represented by the generators of group representations. As it stands, however, this cannot be checked directly by an experiment. We have therefore examined the relation between group representations and the requirement that L2 should commute with each Lj . The results are presented in Sections XV-XVII. Section XV gives a summary of someknown facts about unitary representations of Lie groups on Hilbert spaces.In particular, it is discussedhow a unitary group representation is associatedwith a set of self-adjoint generators Lj . In Sections XVI and XVII conditions are examined under which a given set of self-adjoint operators Lj generatesa unitary representation of a group. In the case of a simply connected, compact, semisimpleLie group, it is found to be necessaryand sufficient that the operators Lj satisfy suitable commutation relations and commute with a self-adjoint Casimir operator L2. The proof makes use of a paper by Nelson (9) on analytic vectors. In the context of high-energy physics, this was recently reviewed by Stein (IO). It appears that the orbital angular momentum with nonintegral eigenvalues provides one of the more sophisticated examples of certain dangers against which Nelson and Stein warn their readers.

VAN WINTER

238

Whereas Sections XVI and XVII establish a relation between our Point 5 and the theory of group representations, Section XV111 is devoted to the unitary equivaIence of the operators Lj and thus refers to Point 2. For a connected, compact, semisimple Lie group of rank 1, it is shown that the generators of a unitary representation are unitarily equivalent and thus have the same spectrum. Also, Lj is unitarily equivalent to -Lj , so that the spectrum is symmetric with respect to the origin. With a view to the orbital-angular-momentum problem, these results are discussed in Section XIX. The paper concludes with two appendices on symmetric operators and commuting operators, respectively. NOTATION. In the present paper Hilbert spaces are considered throughout. If R is a Hilbert space andfand g are any two elements of R, their inner product is written as (g,f), the notation being such that (1.6)


for every pair of complex numbers 01, /3. Here p is the complex conjugate of ,& The norm off is denoted by Ilfi/. If A is a linear operator on 53, then A stands for the closure of A and A* for the adjoint of A. The domain of A is denoted by ID(A). The operator A is called symmetric if B(A) is dense in R and (5%m

(1.7)

= (&f)

for every f and g in B(A). The operator A is called self-adjoint if A = A*. It is called essentially self-adjoint if A is self-adjoint. II. ASSOCIATED LEGENDRE

FUNCTIONS

We start with a summary of some known facts about associated Legendre functions of general order [MacRobert (II), Ch. XVIII]. Let differential operators be defined by T, = i(sin y) a/a8 + i(cos y cot a) a/Zv, TV = --i(cos T, = -i

(2.1)

y) a/31? + i(sin F cot 8) a/aq,,

a/&p

(2.2) (2.3)

T2 = Tz2 + TV2 + Tz2 = -cY2/a82 - (cot 8) 8188 - (sin 9.)-2 a2/aq2

(2.4)

and consider the functions evW) (2x + l) Qh + CL+ l) I” (sin $>-o (d/dcos $)A+2 (sin $,)2A @f(8) = p&j + 1) [ UT-p+ 1) (2.5;

1

ORBITAL

ANGULAR

MOMENTUM

239

TVand X taking the values p > -1;

x = p, p + I,... .

(2.6)

The functions Zp(9,

y) = (2~)-‘/~ = (2~)-l/~

exp(&)

O,@(8)

t&L 3 0; h = pL,p + L...)

exp(&g, + ipr) OTfi(8)

(2.71

(p < 0; h = -p, -p + I,...)

then satisfy the equations PZ,f

= h(X + 1) ZAP,

T,Z,p = pz,\*.

WV

They are of the general form Z,@(8, y) = exp(&v)(sin 8)lpl$@-,r,(cos a),

(2.9)

where Pf-, p, is a polynomial of degree h - I TVI. If h and p are integers I and m, the function ZApis nothing but the spherical harmonic Yrm,and is thus appropriate for the description of an integral orbital angular momentum. The present paper is concerned with the question why orbital angular momenta with nonintegral quantum numbers h and p do not occur in nature. If a fixed 6 is chosen such that 0 ,< 6 < 1, and h and p are restricted to the values p=a+m,

I?2= 0, il,

&2,...;

x = I p I, I p I + I,...,

(2.10)

a set of functions is obtained which is of special interest. It is henceforth denoted by (Z,p}. The members of this set are orthonormal,

and the set is complete in the spaceof square-integrable functions of q~and cos 6. Indeed, if p runs through all the values (2.10), the set ((2~)~; exp(ipy)} is complete in q. Also, if p is fixed and A runs through the values (2.10), each function of the set {@pi} is a polynomial in cos 8 times the square root of the weight function (sin 8)“lp’. The set {@k’} is therefore complete in cos 8, by the completeness properties of sets of orthogonal polynomials. Summarizing, the set {Z,p} is complete in v and cos 8, as was assertedabove. lff(8, y) is any square-integrable function of y and cos 8, it can be developed in terms of this set. The corresponding seriesis henceforth denoted by s = c (ZAP,f) z.iw,

(2.12)

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it being understood that 6 is fixed and that X and p run through the values (2.10). In the cases 0 < p < 1 and --I < p < 0, the equations (2.8) admit a second solution, which is Xf(8,

y) = (2r)-l/% exp(&g, + &n) O;“(G) = (2~r~‘i~ exp(ipv)

O,,p(8)

(0


< 1; X = -/L, -I* + I,...) (2.13)
It is of the general form X,@(8, CJI)= exp(@)(sin

6)-l~lP A+,@,(cos 6).

(2.14)

The function X,,p is not finite as sin 9 vanishes, but it is square-integrable. Indeed, the set (XA”} with fixed TVis orthonormal and complete in the space of all functions of the form exp(ipv)f(G)), where f(8) is a square-integrable function of cos 9. For larger nonintegral values of / p 1, there is also a second solution. This is not square-integrable, however, and need therefore not be considered in the following. On applying the ladder operators G = T, I!Z ii”, = exp(iiv)[fa/a6 the following

+ i(cot 8) a/+],

(2.15)

chains are obtained.

T+Z,y = [(A - p)@ + ,u + I)]“”

Zy+l

(p<--I,p>O)

= [(A - p)(h c p + l)]‘/”

x:+1

(-1

T-Z,,fi = [(A + p)(h - p + l)]“”

Z;-l

(p < 0, p 3 1)

= [(A + p)(X - p + l)]“”

X:-l

(0 < p < 1).

If 6 = 0, there are no vice versa. If 6 f 0, Applying the operator not square-integrable. of many difficulties.

III.



(2.16)

(2.17)

functions XAp, and so the chain runs from ZAA to Z;‘, and however, the chain breaks into two disconnected parts. T, to XAs, or T- to XAg’ then yields functions which are It will become clear in the following that this is the root

A DIFFERENTIAL

OPERATOR

ON

A HILBERT

SPACE

The previous section was purely formal, being a mere collection of useful formulas. From now on, we study differential operators as operators on a Hilbert space of square-integrable functions. By way of introduction, we start with the operator -iajt.$.

ORBlTAL

Let e(y)

ANGULAR

241

MOMENTUM

consist of all measurable functionsf(y)

such that

and let Z(Q) be the set of functions f E 4j(y) which are absolutely continuous, whose derivative df/dp, is in e(y), and which satisfy the boundary conditions f(2Tr) =f(O)

= 0.

(3.2)

Let Q be an operator on $5(y) with domain D(Q), and let it act according to Qf = --i dfidp,

(3.3)

for every f E D(Q). Then Q is closed and symmetric, but it is not self-adjoint. The domain D(Q*) of the adjoint Q* consists of all functions f E Js(v) which are absolutely continuous and whose derivative df/dg, is in 5(~) [Achieser and Glasmann (22), Section 491. The operator Q has infinitely many self-adjoint extensions. These can be distinguished with the help of a parameter 6 such that 0 < 6 < 1. The domain of the self-adjoint extension Qs consists of all functions f E B(Q*) which satisfy the boundary condition f(2r)

= exp(2&)

f(0).

(3.4)

The eigenfunctions of Qs are of the form (27r-+ exp(ipy), with TV= 8 + m, m = 0, fl, 12 )... . The eigenvalues thus differ by integers [(12), Section 491. In order to carry the foregoing facts over to the theory of orbital angular momentum, we now consider the space !$r, 9, q~) consisting of all measurable functions f(r, 9, F) such that jR12d~/[sin9d~jflf(r;4,~)12dplr

co.

(3.5)

In an obvious notation, we also consider the space $j(r, 8). In fi(u, 9, y), it is not true that the domain of a self-adjoint operator -ia/+ is necessarily characterized by a boundary condition of the form (3.4). In fact, let a,@, 8) and u2(r, 8) be any two orthogonal functions, and let pI and p2 be real numbers. Then there are many ways to construct a self-adjoint operator --ia/+ which has the two functions exp(ip.,F) a,(r, 8) among its eigenfunctions (n = 1, 2). The numbers pI and p2 need not differ by an integer, and so it can be arranged, for instance, that -ia/+ has both integral and half-integral eigenvalues. It is intuitively clear that this case is not relevant for the theory of orbital

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VAN WINTER

angular momentum. It is automatically excluded if one only admits operators -ia/+ which commute with every projection operator of the form Rf(r, 6, F) = a(r, 19)1,” r’2 dr’ 1: G(r’, 8’) f(r’, 8, CJI)sin 8’ da’,

(3.6)

a(r, 8) being some function of norm 1 in $(r, 8). THEOREM 3.1. Let Q be a symmetric operator on $(r, 8, y), let its domain be B(Q), and let it commute I+ith every projection operator of the form R. Let every function f E B(Q) be absolutely continuous in q~for almost every r, 8, such that aflap, is in Jj(r, 6, q), and let Qf be equal to -iaf/lap, for every f E D(Q). Then there is a constatlt 6 in the interoaE0 < 6 < 1 such that, for every f E D(Q),

f(r, 6, 2~) = exp(2&) f(r, 6, 0)

(3.7)

for almost every r, 8. Proof. For every f E B(Q), the function af/lav is in $J(r, 6, p), and so it is easily seen that f(r, 8, v) is actually in $(r, 6) for every fixed v. If {a,(r, S)} (n = 1, 2,...) is any complete orthonormal set in JS(r, a), the function f can therefore be developed according to f(r, 8, y) = f

4dr, $1 fn(v),

(3.8)

n=1

the seriesconverging in the norm of sj(r, 6) for every fixed CJJ. The assumption that Q commutes with every projection operator of the form R implies that every term unfn is in B(Q) [(Z2), Section 401. If f and g are any two functions in B(Q), integrating by parts yields

(3.9) The symmetry of Q requires that the second term on the right vanishes, and so there is a 6, in the interval 0 < 6, < 1 such that fn(24 = exp(%4f,(O).

(3.10)

Here 6, does not depend onf. If B(Q) is such that fn(0) vanishes for every f and n, every 6, may be chosen arbitrarily, and so it can obviously be arranged that all parameters 6, are equal. This can still be arranged if there is only one n for which there are nonvanishing quantities fn(0).

ORBITAL ANGULAR

Now let bothf,(O) andf,(O)

MOMENTUM

243

differ from 0, and consider the projection operator

[Qr’, 8’) + &Jr’, a’)] g(r), 8, q~)sin 0’ d8’.

(3.11)

This is associated with a certain 6, , in analogy to Eq. (3.10). Applying yields

R, to

x {,‘Pdr’/i

a,f, and to a,f,

f;(2~r) = exp(2ia,r) h(O)

(j = n, m).

(3.12)

Hence 6, = 6, = 6, )

(3.13)

owing to Eq. (3.10). The domain B(Q) is thus characterized by one single parameter 6. With Eq. (3.8), it is now easily shown that the desired relation (3.7) holds true throughout D(Q). This proves the theorem. From now on, it is assumed that the z-component of the orbital angular momentum is represented by a differential operator --ia/+ which commutes with every projection operator of the form (3.6). This yields the relation (3.7). It is further assumed that each component of the orbital angular momentum commutes with every projection operator of the form Rf(r, 8, y) = u(r) J’,” C(r’) f (r’, 6, v) r12dr’.

(3.14)

If this is so, the following theorem is useful. THEOREM 3.2. Let A be a closed linear operator on the Hilbert spacesi, let its domain be D(A), and let {R,} (n = 1, 2,...) be a complete set of mutually orthogonal projection operators on Ji which commute with A. Thenf is in D(A) if and only if

(3.15) Also, Af = 2 ARnf

(3.16)

n=l

for every f E ID(A).

A proof of this theorem can be found in the book by Achieser and Glasmann [(12), Section 401.

244

VAN

Since the right-hand

WINTER

side of Eq. (3.14) is of the form a(r)f(6,

v), where

(3.17) Theorem 3.2 shows that the orbital-angular-momentum problem reduces to a problem concerning operators on the space $j(S, y) consisting of all measurable functions which satisfy Eq. (3.17). This space is henceforth denoted by 9. The present reduction is, of course, the usual one. The last few paragraphs will clarify the assumptions on which it is based.

IV.

THE

z-COMPONENT

OF

THE

ORBITAL

ANGULAR

MOMENTUM

The present section is devoted to the self-adjoint operator L, on 9 which is going to represent the z-component of the orbital angular momentum. THEOREM 4. Let D(L,) be the set of all functions f E !$ which are absolutely continuous in y for almost every 8, whose derivative aflag, is in 5, and which satisfy the boundary condition

f(S, 2~) = exp(2i&r)f(9,

0)

(4.1)

for almost every 6 and someJixed 6 in the interval 0 < 6 < 1. Let L, be an operator on $ with domain B(L,), and let it act according to

Lzf = -iafl+ for every f E ID(L,). Then L, is self-adjoint, f E Sy,satisfying

(4.2)

D(L,)

is equal to the set of allfunctions

(4.3) and Lzf = c /GA”,

P.A

for every f E B(L,).

f) Z,,r

(4.4)

Here h and p run through the values (2.10).

Proof. Let {b,(8)} (n = 1, 2,...) be any set of functions and orthonormal in cos 6, write

a,,(o, ~1 = CW-1/2 exp(4.v) b,(@

which

is complete

(4.5)

ORBITAL

ANGULAR

and consider the set D(Q) consisting

245

MOMENTUM

of all finite sums of the form

SCM

N

(4.6) where p = 6 + nz, nz = 0, 1;1,..., iM. If Q denotes the restriction of L, to a(Q), every u,,, is an eigenfunction of Q. Since the set (a,,} is complete, it follows from Appendix I that the closure Q is self-adjoint, the domain a(Q) consisting of all functions f E 9 for which ,cn I AG, * 0”

< a.

(4.7)

It is obvious that L, is symmetric, and so Q* = QC

LzC LT.

(4.8)

Taking adjoints yields Q > Lz, hence Q = Lz > Lz . Now suppose that we can show that a(Q) C n(L,). Then it follows that 0 = L, , hence that L, is self-adjoint. To prove that a(@) C n(L,), choose any f~ S(Q). This can be approximated by a finite linear combination of functions a,, , in the spirit of Eq. (4.6). Denoting the finite sum by fMN yields (4.9) Now let h4 and N tend to co. The function fMN(8, ‘p) then tends to f(8, CJJ). The right-hand side of Eq. (4.9) tends to i c Qf dq~', owing to Eq. (4.7). Hencef,,(B, 0) also tends to a limit. and &.j&

(if(a2

p?> -

From this it follows thatf(8, Also

fMdsT

O> -

iJ’Ic

t?fc8? ddq'iI

= O-

(4. IO)

y) is absolutely continuous in v for almost every 8. -iaf/iap

= (if,

(4.11)

hence af//ap is in Jj. If Eq. (4.11) is now integrated from 0 to v, it follows that f(8,O) is in 5, with f(s, ~1 - f(a, 0) = i 1: Qf<~, 54 do’.

(4.12)

With Eq. (4.10), this yields

lim llfMN@, 0) - f(s, W = 0.

M,N+m

There is a similar relation for

f(8, 2~).

(4.13)

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It is obvious that (4.14) for any finite M, N. If in this relation M and N tend to co, Eq. (4.1) is obtained. This shows that B(Q) C D(L,), hence that L, is self-adjoint. If instead of the set {a,,} one chooses the set {Z,p>, the relation (4.7) for B(L.,) goes over into Eq. (4.3). The relation (4.4) then follows from Appendix I. This proves the theorem. According to the present section, there are infinitely many self-adjoint operators L, . These are distinguished by the parameter 6, but it will not be necessary to bring this out in the notation. It is the purpose of the present paper to explain why only the case 6 = 0 is of physical significance.

V. A SET

OF

SMOOTH

FUNCTIONS

In order to make further progress, it is useful to study the subset D C D(L,) consisting of all functions f E B(L,) which are absolutely continuous in 8 for almost every v, and such that af/lai? and (sin 8)-l af/ap, are in @. The set {Z,J‘} is in 3, hence B is dense in !$. On 9 one can apply the differential operators Ti (j = X, y, z). It is from this property that II) derives its usefulness. In the next few sections the connection between the operators Tj and the orbital angular momentum is discussed. In order to prepare for this, we first list some properties of functions in D. Let .f be in 3 and define (5.1)

Then PJsatisfies

Eq. (4.1). It is not difficult to show that

ap,fl+

apdjas = P,afyas.

= p,afi+,

(5.2)

Also, aP,f/a8 and (sin 8)-l aP,f/ap, are in rj. Hence P,f is in a. Iffis in 9, it satisfies jr (sin 8)-l d8 jzw 1j: @f/&p’) dy’ 1’ dp, 0

0

,< 4372jT (sin 8)-l dt9 /“2n J af/+’ 0

“0

I2 d$ < co.

(5.3)

ORBITAL

ANGULAR

247

MOMENTUM

Hence (5.4) Now suppose that 6 # 0, and choose first # = 0, next $ = 27~. This yields two independent results, from which it follows by linear combination that (sin 8)-lf is in jj. If f and g are any two functions in 3 and 6 # 0, then (sin 8)-l f is in & and so gf is an integrable function of 8 for almost every v. Also, a(gf sin 9)/&Y is an integrable function of 6 for almost every q. Hence @sin 6 is continuous in 8 for almost every v. We want to show that lim gf sin 9 = 0

(6 f 0)

9+0

(5.5)

for almost every y. Suppose that Eq. (5.5) is not correct, and denote the limit on the left by C(F). For fixed v, there is then a positive c such that I gfsin

6 j 3 4 I c(y)1 > 0

(0 < 8 < c).

(5.6)

Hence gfis not integrable. This is a contradiction, and so Eq. (5.5) is true. If 6 = 0, the foregoing argument fails. In this case, however, it follows from Eq. (5.4) that (sin 8)-l (f - P,f) is in sj. Hence g(f - P,f) sin 8 tends to 0 if 9 tends to 0.

VI.

RESTRICTIONS OF

THE

OF ORBITAL

THE

x- AND

ANGULAR

y-COMPONENTS MOMENTUM

From now on, Ji (j = x, y, z) denotes the restriction of Tj to 3. It is clear that J, C L, , hence JL is symmetric. Now choosefand g in a, and suppose that 6 f 0. For almost every F, one has

jr g(af /a@ sin 8 da= - jv (ag/M) f sin 6 d$ -jr 0

0

If cos 6 d8 + jf sin 6 1:::. 0

(6.1) The last term on the right vanishes owing by sin 9) and integrating over v thus yields (g, i[sin ?,I

to the previous

section. Multiplying

afaiq = (i[sin 9)] agla8, f) - i(g,f sin v cot 8).

(6.2)

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Likewise, (g, i[cos F cot a] afl/ap) = (i[cos q cot i+] ag/+,,f)

+ i(g,fsin

q cot a),

(6.3)

the integrated term in this relation vanishing due to Eq. (4.1). From the last two equations, it follows that .Z, is symmetric, but the argument only applies to the case 6 f 0. If 6 = 0, the above reasoning can be used to show that

(8, Jrlf - ~0.n) = (J&f - Pd.

(6.4)

This relation remains true if g is replaced by P,,g. From the form of J, it is obvious that (6.5) (P&3 JrP0.f) = (J,Pog, Pof) = 0. Hence J, is also symmetric in the case 6 = 0. Likewise, J, is symmetric. For a further development of the theory, it is convenient to consider the set of all functions f E D for which Jjf is again in a (j = x, y, z). Let this set be denoted by 6. On (5, one can apply the differential operator T2. The restriction of T2 to (5. is henceforth denoted by J2. Hence J”f=

Jz2f+

J,“j- + J,“f=

Ty

(6.6)

for every f E &. The operator J2 is going to be a restriction of the square of the orbital-angular-momentum operator, but this point will not be taken up until Section IX. We first discuss the restriction of Jj to (5, which is denoted by Zj . It is clear that Z, CL, . If in Section IV the set {b,(6)} is chosen in such a way that the functions (sin 9)-2 b,(B), (sin 6)-l &,(6)/d6, and d2b,(@)ldiP exist and belong to $5, then the set {a,,} of Eq. (4.5) is in K The operator Z, thus has a complete set of eigenfunctions. From this it follows with Appendix I that Zz is essentially self-adjoint, hence I, = L, . In the next section, it is shown that Z, and Z, are also essentially self-adjoint. The self-adjoint extensions are denoted by L, and L, , and it is assumed that these represent the x- and y-components of the orbital angular momentum. It is shown in Section VIII that D = nj D(Lj), where D(Lj) is the domain of Li . From this it follows that 0. = nj.k D(LjLk). VII.

THE

x- AND

y-COMPONENTS

OF

THE

ORBITAL

ANGULAR

MOMENTUM

In the present section, it is shown that 1, and I, are self-adjoint. These operators are denoted by L, and L, , and some of their properties are discussed. The operator Z, is the easier one. It can be investigated with the help of the function E = E(E, 9, v) = exp[--/(sin 9 sin ?)“I, (7.1)

ORBITAL ANGULAR

249

MOMENTUM

where 0 < E < 1. It is obvious that 0 < E < 1. Also, for fixed E and 8 lii

E(E, 8, y) = iii/k

8. 9) = 0.

(7.2)

If p(8, y) is any polynomial in cos 6, sin 8 COST,and sin 6 sin v, it is not difficult to show that

As long as E is nonzero, Ep vanishes for 91= 0 and q~= 2~. The function Ep thus satisfies Eq. (4.1). It can be differentiated any number of times, and from the properties of the derivatives it is clear that Ep is in 6. The operator I, can therefore be applied to Ep. It acts as the differential operator T, . Now let p be a spherical harmonic YZVZ.This yields Z,EYln’ = ET,Y,”

= E(-

;i[(/ - m)(l + m f

l)]‘/” Yr+l

+ $i[(l + m)(l - I??+ 1)11/Zry-‘}.

(7.4)

If E tends to 0, the function EY,l” tends to YLm,by Eq. (7.3). Also, Z,EYlm tends to T,Ytm. Hence Yy is in the domain %(I,), and il, Yz”, = TYy 1.“1

(7.5)

From the theory of spherical harmonics, it is well known that there exists a unitary operator U, which does not depend on 8 and y, such that

The set {CJYlm} with m = 0, &I, *2 ,...; I = 1m 1, / nz / + I ,... is complete and orthonormal, and by Eq. (7.5) its members are eigenfunctions of i, . Hence I, is self-adjoint, by Appendix I. The self-adjoint operator i, is henceforth denoted by L, . Its properties can be summarized as follows. THEOREM 7.1. The restriction of T, to 0 has one and only one self-adjoint extension L, . The domain D(L,) consistsof all functions f E @for which

,c, I m(UY,“, f>l” < a,

(7.7)

every f E D(L,) satisfying

(7.8)

250

VAN

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It is to be noted that this result is valid for every 6. The operator L, has integral eigenvalues and it does not depend on 6. We now turn to the operator I,. Since L, is self-adjoint, one can consider the unitary operator V = exp(&l,). (7.9) For every f E 6, it acts according ~‘(8,

v) = (2+l

to

C exp(ipp, + &p> P

jr

exp(--id)

f(a,

$I&‘,

(7.10)

and so, for almost every 9 and v, VW,

y,) = f(8,

(0 < rp < $77)

F + tn)

= exp(2i 8,) f(S, v - &f)

($77< $c< 24.

(7.11)

Now consider the function VEY,rn = E’(E, 6, ($2)YzyQ) ql + $T)

(0 < qJ < $n)

= exp(2i 67~)E’(r, 6, y) Ylm(6, 9 + $r)

(7.12)

($7~< F < 2~r),

where E’(E, 8, v) = exp[--/(sin

(7.13)

6 cos v)“].

The function VEYlm satisfies Eq. (4.1) and can be differentiated any number of times. In particular, both the function itself and all its derivatives vanish at y = &T, and so this is not a discontinuity. It is not difficult to seethat VEYCnl is in 0. Also, I,vEYp

= E’(E, 6, cp)T,Y,“(B,

y + &r)

= exp(2i 6~) E’(E, 6, y) T,Y,“(6,

(0 < v < Qd F + 4~)

(7.14) (&i- < 92< 27T).

On comparing the expressions for T, and T, , it follows that ZczVEY,“l = VET 2/Y1. m

(7.15)

If E now tends to 0, then VEYtm tends to VYtm and Z, VEYcm tends to VT,Y,“‘. Hence VYtm is in D(r,), and i,VUYlnf

= VT,UYlm = mVUY,“.

(7.16)

With the help of Appendix I it follows that i, is self-adjoint. The properties of the self-adjoint operator i, = L, can be summarized as follows.

ORBITAL ANGULAR

251

MOMENTUM

THEOREM 7.2. The restriction of T, to 6 has one and only one self-adjoint extension L, . The domain ‘I)(L,) consistsof all functions f E $ for which

(7.17)

C I m(vUYP, f>l* < co,

711.1

every f 6 B(L,) satisfying L,f

= 2 m( VUY,“, f) VUY1”. m.z

The operator L, depends on 6 through VIII.

FURTHER

PROPERTIES

(7.18)

V.

OF THE SET OF SMOOTH

FUNCTIONS

The intersection ni B(LJ is the largest set on which one can apply each operator Lj, and it is therefore an important physical concept. Since Jj _CLj and each Jj has domain CD,it is clear that a is not larger then nj B(LJ. Actually, there is the following theorem. THEOREM 8. Let 33 be tke set of all functions f E 5 which are absolutely continuous in v for almost every 8, absolutely continuous in 19for almost every y, sucfzthat (sin 19)-l af/Lkpand aflat? are in 5, andsuch that the boundary condition (4.1) is satisfied. Let Lj (j = x, y, z) be the operators discussedin Theorems 4, 7.1, and 7.2. Then (8.1)

9 = w,) n w,) n w,),

the set CDconsistingof all functions f 6 8 which satisfy 1 X(X + 1) KZA’“,f>l” < 00.

(8.2)

Ir,A

Proof. Assume that 6 f 0 and choose a function f satisfying Eq. (8.2). We first show that this is in nj D(L.,). It follows from Eq. (4.3) that f is in D(L,). If fA is defined by

then fA is in 3, and so it is in nj B(Lj). If J+ stands for J, * iJ, , J+fA = c c

VT, f)Kx - /4(X + CL+

fififti-1A
1>11’” Zy+l

+ A& <-y, f)[O - 6 + 1)(X+ S)l”” XA6.

(8.4)

252

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This is an expansion in terms of the complete set consisting of the functions ZAp with p # 6, together with the functions X,, s. Since h is restricted to the values I CLI, I P I + lY.3 the coefficients in the series expansion satisfy (A ~ p)(X + p + 1) < h(h + 1).

(8.5)

From this it follows with Eq. (8.2) that J+fA tends to a limit if (1 tends to co. Likewise, J-f* tends to a limit. Hence J,& and .J,fA tend to limits. At the same time, fA tends to J Hence ,f is in D(J,) and in CD(J,). This means that f is in D(L,) n a(&). Since f is also in a(&), it follows thatf is in nj ID(&). In an obvious notation, J+fA tends to L+$ Owing to Eq. (8.4),

L&f = 1 (Z,?. f) -&ZAP

(8.6)

LL,A

whenever f satisfies Eq. (8.2). Now suppose that f is in nj CD(&). The next part of the proof shows that satisfies Eq. (8.2). indeed, L, and L- can then be applied to f, and

f then (8.7)

Since ZA@is in 3, it is in nj Z(Lj). follows from Eq. (8.7) that

Owing to the symmetry

2 I(L+ZP, Omitting

the terms with

p = 6-

f)l’

of L, and L, , it thus

< CQ.

(8.8)

1 yields

c @ -PI@ + P + 1) l(z~+1,f>12< m, P.A iif6-1

(8.9)

and replacing L- by L, in Eq. (8.7) gives

1 0 + A0 - p + 1) K-q-l, f>l” < co.

(8.10)

P.A wf6

If in this expression the sum over p is restricted to p = 6 + 1, and it is remembered that the series xA ~(ZAs,f)lz certainly converges, it follows that 1 ho + 1) l(ZA6, 0” The corresponding

inequality for Zff

follows

< *.

from Eq. (8.9). Sincefis

(8.11) in nj a(&),

ORBITAL

ANGULAR

253

MOMENTUM

it must also satisfy Eq. (4.3). If this is combined with the above relations, Eq. (8.2) follows. This is therefore a necessary and sufficient condition for ,f to belong to (lj D(Lj). Since nj ID(&) is contained in a(&)? it is obvious that every f E 0, B(L.,) is absolutely continuous in y for almost every 19 and satisfies Eq. (4.1). With the help of Eq. (8.2), it is now shown that every fg nj B(L)) is also absolutely continuous in fi for almost every v, and that its derivatives are such that it is in 2. Since it is already known that 3 is not larger than n, B(Lj), the desired relation (8.1) then follows. Since fA is in 3, it follows with Eq. (2.15) that

Iff is in nj If)(&), Eq. (8.2) holds true, and from this it was deduced above that J&J, tends to L&f. The relation (8.12) thus yields &L-,f(6’, This shows that

v)] d& j/ = 0. (8.13)

f is absolutely continuous in 6 for almost every q~, with i3f/la8 = 3 exp(- iv) L, f - * exp(iqJ) L-f.

(8.14)

Likewise, (cot 9) aflay

= - ii exp(-iy)

L,

f - ii exp(iF) L-f.

(8.15)

The right-hand sides of these expressions are in ~j$,and so it follows that f is in 3. This means that Eq. (8.1) holds true, and that the condition (8.2) is necessary and sufficient for f to be in B. This completes the argument for 6 # 0. If 6 = 0, there are no functions XAs, and so Eq. (8.4) must be modified. The modification is straightforward, however, and it is not difficult to justify the assertions (8.1) and (8.2) also for the case 6 = 0.

IX.

A RESTRICTION

OF

THE

SQUARE

OF

THE

ORBITAL

ANGULAR

MOMENTUM

We are now prepared to investigate the operator J2 which was defined in Eq. (6.6). If 6 = 0, every function Ytm is in Q. The set (Ylm} is complete and orthonormal, and so j2 is self-adjoint, by Appendix I. If the self-adjoint operator J2 is denoted by L2, then D(L2) is the set of all functions f E fi which satisfy n;c I 41 + l)(Yl”, 595/47/2-4

f>l” < a.

(9.1)

254

VAN

WINTER

Since 3 is characterized by Eq. (8.2), Iz)(L2) C 3. If the operators Lj are applied to f E D(L2) according to Eqs. (4.4) and (8.6), functions Lf are obtained which again satisfy Eq. (8.2) and therefore are in 3,. Hence f E D(L2) is in fact in 0. On the other hand, L2 = I2 > J2. Hence L2 = J2, and P is self-adjoint. This argument applies only to the case 6 = 0. In the case 6 f 0, the functions Z,p are still eigenfunctions of the differential operator T2, but ZAs and Zff are no longer in a. As a result of this, J2 is not self-adjoint. In fact, it follows from Section XIII that J2 has more than one self-adjoint extension. Among all self-adjoint extensions, however, there is only one operator L2 whose domain is entirely in a. To see how this comes about, denote D(J2*) n 3 by D(N2), and write N2 for the restriction of J2* to D(Nz). In order that there be a self-adjoint extension L2 3 J2 such that B(L2) C 3, it is necessary that L2 C N2. Now it follows from Sections X and XI that N2 is selfadjoint. Hence, in order that L2 be self-adjoint, it must be equal to N2. There is thus one and only one self-adjoint extension L2 of J2 such that D(L2) _C9. It is assumed that this represents the square of the total orbital angular momentum. It is the only self-adjoint extension of J2 with self-adjoint components Li , and so it appears to be the only operator L2 which might be of physical interest. X. A REDUCTION

FORMULA

The difficulties which arise for 6 # 0 are due to the fact that Z,” and Zf-l are not in 0. All the other functions Z,p do belong to 6, and so it is convenient to discuss these separately. This can be done with the help of the projection operators P, defined in Eq. (5.1). According to Section V, P,f is in a whenever f is in ID. Likewise, P,f is in 6 whenever f is in 0. It is not difficult to see that P, commutes with J2. Now consider the projections P8, Psel, and P,, = 1 - Ps - Psel . Let these be denoted by P, (a = 6, 6 - 1, r). Write 6, for 0: n P&, and let J,” be the restriction of J2 to 6, . Then f is in a if and only if each P, f is in KU. For every f e CE,

J”f = 1 J,2P,f,

(10.1)

infanalogy to Eq. (3.16). There are corresponding reductions for the operators J2* and N2. The functions Z*p with / p [ > 1 together form a set which is complete and orthonormal in P,,!$ Hence J,,2 is self-adjoint on P,!& the domain a@,“) consisting of all functions P,,f E P$j which satisfy F* I w

+ lWA@, P,f)12 < a.

(10.2)

ORBITAL

ANGULAR

255

MOMENTUM

Now let Eq. (10.2) be fulfilled. Then so is Eq. (8.2); hence P,fis in a. If LjP,f is determined with the help of Eqs. (4.4) and (8.6), it is easily seen that LjP,.f again satisfies Eq. (8.2). Hence P,fis in 6, , the operator J,” is self-adjoint on P,..C,. and J," = J,"* = Ny2.

(10.3)

Here NY2 is the restriction of N2 to D(N”) 0 P$. The self-adjoint operator J.,” is henceforth denoted by Ly2. Its domain is in 9.

Xl. THE DIFFICULT

PART OF THE ORBITAL

ANGULAR

MOMENTUM

If f is in D(N2), it is clear that fe”” sin 8 is in CD. We assert that fei’ sin B is even in D(N2). To prove this, choosefg 9(N2) and g E 0. This yields (fe”’ sin 6, J2g) = (f, J2[geci” sin 81) + 2(feiq cos 6, ag/as) - 2i(j@, [sin 19-l ag/av) - 2(fe+ sin 6, g).

(11.1)

Since f is in B(J2*) and ge+ sin 6 in 0, the first term on the right is equal to (eQ[sin 61 J2*f, g). Since f is in 9, integration by parts shows that the remaining terms on the right are each of the form (kn , g), with functions k, in $j. This means thatye@’ sin 6 is in D(J2*). Since it is also in 9, it follows that it is in D(N?). This proves the assertion. Now let P,f be in D(Ns2). By the foregoing paragraph, e@(sin a) P,f is then in D(NY2), and so it is in (r. It is not difficult to see that a2g/aa2 exists and is in 5 whenever g is in (5’. Hence, a2P,f/&Y2 exists and a2[(sin 9) P,f ]/as2 is in $5. Since P,f is in D, it follows that (sin 8) a2Psf/M2 is in &. Again, let P&f be in 3 and let (sin 8) a2P&S2 be in $j. Integration by parts then shows that (P& J2g) = ([sin a] T2P,f, [sin 91-l g)

(11.2)

for every g E 6;. Here, T2 is the differentia1 operator (2.4). The relation is true, in particular, if Psf is in ID(Ns2). In this case, one also has (Psf, J2g) = ([sin 61 N,2P,f, [sin 91-l g).

(11.3)

Now let g run through all the functions (sin 9) ZAs, with h = 6, 6 + I,... . It is easily seen that these are in CC:.Since the set {Z,s) is complete in P&j, it follows from Eqs. (11.2) and (11.3) that (sin 8) Na2P,f = (sin 19)T”P,S

(11.4)

256

VAN

WINTER

for every P,f E 3(Ns2). Since Ns2Psf is in $ by assumption,

T2Psf is also in 33, and

Na2P,f = T2Psf.

(11.5)

Suppose now that P,f is in 9 and that T2P8f is in 5. Then (sin 9) a2P,fli362 is in -5. Hence Eq. (11.2) is true for every g E E. Since T2Psf is in 5, Eq. (11.2) can be simplified to (11.6)

(Pss, J2d = (T2P,L 8).

From this it follows that P,f is in D(Ns2). It was shown above that from P,f E D(Ns2) it follows that T2P,f is in 9. Hence P,f is in lr)(NE2) if and only if P8f is in D and T2P8f is in J3. From the above results it follows, in particular, that the functions ZA6 are in D(NS2) and that they are eigenfunctions of Ns 2. If Ns2 is written as a differential operator according to Eq. (11.5) integration by parts shows that Ns2 is symmetric. Hence IVES is self-adjoint on P&j, by Appendix I. The domain D(ms2) consists of all functions P,f E Pssj for which c Iw A

(11.7)

+ l)(ZA6, P,f )I” < a.

Now let P,f be in 3(m,2). Then Eq. (8.2) is fulfilled, hence P&f is in II). If L, is applied to P,f according to Eq. (8.6), a function L+P,f is obtained which is again in CD. To this L- can be applied. Likewise, L, and Lz2 can be applied to Psf. Since they act on functions in 9, all the operators Li behave as differential operators. But for differential operators it is well known that T2 = T-T,

+ T, + Tz2.

(11.8)

Hence T” can be applied to P,f, and T2P8f is in !$ But from this it follows that P,f is in D(Na2). Hence D(m,2) is not larger than D(Ns2), the operator Ns2 is self-adjoint on P&j, and P,f is in D(Ns2) if and only if Eq. (11.7) holds true. The operator Ns2 is henceforth denoted by L8 2. It is the desired extension of Js2. By a similar argument, it can be shown that N& is self-adjoint on P6--1$3. This operator is henceforth denoted by Lie_, .

XII. THE SQUARE

OF THE ORBITAL

ANGULAR

MOMENTUM

As was already remarked in Section X, N2 satisfies a reduction formula of the form (10.1). This is due to the fact that N2 commutes with each P, . Since NW2 is self-adjoint on P&Y, it follows that N2 is self-adjoint on 6. From now on N2 is

ORBITAL

ANGULAR

257

MOMENTUM

denoted by L2. By Theorem 3.2 and Eqs. (10.2) and (11.7), f E J3 is in D(L2) if and only if ?A I w According

+ l)(Z,~‘, f)l”

< CfJ.

(12.1)

f> ZAfi

(12.2)

to Appendix I, Lff = c w

u.A

+ l)(Zf,

for every fE B(L2). Since LY2 = J,” and J,” is a differential whenever f is in D(L2), and

L2f=

operator,

so is LY2.In fact, T2f is in sj

T2f,

(12.3)

owing to Eq. (11.5). Given f6 B(L2), it follows from Eqs. (4.4) (8.2), and (12.1) that f and Zf/ap,are in 3. Now suppose thatf and af/ia(p are in 9, and that T2fis in 5. Then (sin S)-r a2f&p2 and (sin 3) a2f/M2 are in 5. Integration by parts can therefore be used to show that f is in D(J”*). Hence, since f is in 3, it follows that f is in I)(L2). The results of the last few sections thus yield the following theorem.

12. The restriction of T2 to CXhas one and only one se&adjoint extension L2 such that D(L2) C ID. The domain D(L2) consists of all functions f E$ which satisfy Eq. (12.1). Also, a function f E 5 is in II)(L2) if and only iff is in 3, af//arp is in a, and TZfis in &. For every f E D(L2), the function L2fsatisfies Eq. (12.2) and is equal to T2f. THEOREM

XIII.

MISCELLANEOUS

REMARKS

To get more insight into our problem, it is useful to examine the functions X>r. If P,f is in C&, it is not difficult to see that a2P8fJa62, (cot 9) aP,f/M, and (sin 6))” P,f are in 8. Integration by parts can therefore be used to show that

VA62JW = KS, J2f’d = W + l)Wn”,f>

(13.1)

for every f 6 K This shows that XAs is an eigenfunction of J2” with eigenvalue X(h + 1). The same applies to Xf-‘. Now consider the set I)(M2) of all functions f 6 9 such that

c ,$l

(13.2)

258

VAN

WINTER

and define M2 by M2f =

c

W + ~)(ZA~, f) Z.P +

i fi=6-1

C A@ + l)(x,fi, f) x,,fi,

(13.3)

A

,g1

for everyfe D(M2). Owing to the completeness properties of the sets (Z,p} and (XAp), the operator M2 is self-adjoint. It is not difficult to see that M2 C P-*, and so A4* is a self-adjoint extension of J*. Clearly X,,” is in ID(Ma), hence D(M2) contains functions which are not in 3. In the last few sections the requirement that D(L2) should be in D was thus essential to single out the operator L2. According to Eqs. (7.7) and (7.17), f is in D(L,) if and only if Vf is in ~(L.,). In other words, (13.4) VTqL,) = D(L,). By Eqs. (7.8) and (7.18), v-lL,Vf

(13.5)

= L,f

for everyfE D(L,). This shows that L, and L, are unitarily equivalent. It is clear that XAs is in a(&) but not in 3. Suppose now that X,* is in n(L,). The function VXAs would then be in a(&.). But (13.6)

by Eq. (7.9). Hence X,* would be in D(L,), and therefore in a. Since this is a contradiction, it follows that X,* is not in D(L,). By a similar argument, X,* is not in a(&) either. The same applies to X:-l. It was shown in Section VII that the eigenvalues of L, and L, are integral irrespective of 6. If 8 f 0, the eigenvalues of L, are not integral. Hence, for 6 # 0, L, is certainly not unitarily equivalent to L, and L, . If 6 = 0, the set { Yrm} is a complete set of eigenfunctions for L, In the relations (4.3) and (4.4) for L, , the functions ZAr must then be replaced by Ycm. Given the operator U introduced in Eq. (7.6), this yields UW,) For everyfE D(L,),

= ‘I)(L),

VUII)(L,)

= ID(L,)

(6 = 0).

(13.7)

one has

iFL,Uf

= u-1 V-IL, VUf = L,f

(6 = 0).

(13.8)

This expresses the well-known fact that the operators L, , L, , and L, are unitarily equivalent if 6 = 0.

ORBITAL

ANGULAR

XIV.

MOMENTUM

259

DISCUSSION

There is now enough information to compare the cases 6 = 0 and 6 # 0. The major differences have already been summarized in Section 1. Point 1 and Eqs. (1.I) and (1.2) for the eigenvalues of L, and L2 are by now obvious. Point 2 refers to the unitary equivalence or nonequivalence between L, , L, , and L, , and thus follows from the previous section. Now consider Points 3 and 4. If 6 = 0, then 1D(L2)= 0, and so each operator Lj2 can be applied to any f~ ID(L2). It is easily checked that Eq. (1.3) holds true in this case. If 6 f 0, however, .ZAsis in Iz)(L2) but not in 6. The function L,Z,s is a linear combination of Ztfl and Xy. But Xi-’ is not in II( and so ZA6is not in D(Lz2). Likewise, Z,s is not in D(LU2). This is the reason why Eq. (1.3) fails for 6 f 0. The relation (1.4) fails for 6 # 0 because ZA6 is neither in ID(L,Lv) nor in D(L,L,). We now come to Point 5, which states whether L2 and Lj commute. If A and B are bounded operators whose domains are the whole Hilbert space &, they are said to commute if ABf equals BAf for every f E Js. For unbounded operators, the concept of commutativity is not so straightforward. In the special case of self-adjoint A and B, however, it is meaningful to say that A and B commute if their spectral resolutions commute [seeVon Neumann (13), Ch. II, Section lo]. This is discussedin Appendix II. If A and B are self-adjoint and have purely discrete spectra, Appendix II showsthat they commute if and only if they have a complete set of simultaneous eigenfunctions. If 6 = 0, the set {UYIFn} is a suitable one to show that L2 commutes with L, . It can be seensimilarly that L2 commutes with L, and L, . If 6 f 0, the set {Z,fi} is a complete set of simultaneous eigenfunctions for L2 and L, . However, let 6 f 0 and suppose that L2 and L, have a simultaneous eigenfunction f. This must be of the form C, fpZ,p, with some set of coefficients f, . BecauseL,f must be equal to af, with some real oc,the function PsLsf must be equal to aP,jI Hence afEZ, 6 = ifs+l[(A + 6 + I)(h - S)]“” ZAS+ frfapl[(h - 6 + 1)(X + S)]‘/” X,6. (14.1) Since Z,,8 and X,” are linearly independent, this relation can only be satisfied if fspl = 0. Likewise, 4-IZ, 6-1 = &,[(h

- 6 + 2)(/l + 6 - I)]‘/” zI\“-’ + if&(X + 6)(h - 6 + l)]“” Xi-l’-‘; (14.2)

hence fs = 0. From fs = fs-l = 0 it now follows that fs+l = fsp2 = 0. Next, it follows from a2f = Lz2f that fs+2 = fsp3 = 0, and so on. Summarizing, there is a contradiction, and so L2 and L, do not have simultaneous eigenfunctions. There is a similar argument for L, . This explains Point 5.

260

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In papers on the orbital angular momentum, it is usually taken for granted that the commutation relations can be applied to any eigenfunction of L2. According to the foregoing, however, this assumption is not correct if 8 f 0. This explains many ambiguities and apparent contradictions in the literature. It appears that the possibility of general nonintegral values of 6 has not been envisaged before. In published papers, there is only the alternative of integral or half-integral 6. Right from the start, the discussion is confined to these two cases by an argument based on commutation relations. But since the commutation relations do not apply generally, this procedure is not correct. This is the reason for not having restricted 6 in an early stage of the present paper. Let us add here a remark on the functions Z,J‘ with 6 f 0, a. If 6 = 4, it is clear from Eq. (2.10) that X takes half-integral values only. If 6 = 4, however, h takes the values 14, 543”’ if p > 0, and the values 2, s ,... if /* < 0. There is thus a very complicated situation, which arises whenever 6 is neither 0 nor 4.. It was emphasized in Section 1 that our Points 2 and 5 derive their importance from the fact that they refer directly to the outcome of measurements. It is apparent from these points that the case 6 f 0 leads to an inequivalence between L, on the one hand and L, , L, on the other. In fact, it was already remarked by Pauli (6) that, for 6 = 4, the operators Lj do not fit in the usual framework of group theory. We come back to this question in Section XIX. Before it can be discussed, an analysis is required of some aspects of group representations.

XV. GROUP

REPRESENTATIONS

AND LIE ALGEBRAS

It is well known that the elements w of the group SU(2) can be expressed in terms of Euler angles oi, /I, y. If 6 = 0 and Lj (j = x, y, z) stands for the components of the orbital angular momentum, as discussedin the foregoing, it is useful to consider the operators G(w) = exp(--iolL,) exp(-$L,)

exp(--iyL,).

(15.1)

These are unitary operators on 5. They provide a unitary representation of the group SU(2). Now consider, more generally, a group 52with elementsw and a Hilbert spaces. A mapping w ---f G(w) such that to each w E !CJthere corresponds a bounded operator G(w) with domain si is called a unitary representation of Sz provided the following conditions are satisfied. 1. The operators G(w) form a group such that G(w’w) = G(d) G(w).

(15.2)

ORBITAL

ANGULAR

MOMENTUM

261

2. If L stands for the identity element of Q, then G(L) is the identity operator on 52. 3. The mapping w ---f G(w) is continuous in w in the sense that lim Il[G(w’) - G(w)]fll = 0 w’%u

(15.3)

for every f E 9. 4.

For every w E Q, the corresponding

operator G(w) is unitary.

In the following the group with elements G(w) is denoted by G(Q). Now let w -+ G(w) be a unitary representation of .Q, and consider a oneparameter subgroup of G(Q) which again satisfies the conditions 1 to 4. Let this subgroup consist of operators U(a) (- cc < 01< 00) such that 01 = 0 corresponds to w = L and U(CX’+ a) is equal to U(a’) U(a). In the example of SU(2), the operators exp(--ial,) form such a subgroup. Even in the most general case, Stone’s theorem [see Riesz and Sz.-Nagy (14), Section 1371 says that there is a self-adjoint operator Lj such that U(a) = exp( -i&Q. The domain a(&) consists of all elementsfc

(15.4)

9 for which the limit

lii [UC4 - 11f/a

(15.5)

exists. For every f~ B(&), l$

II -iLjf

-

[U(a) - I] f/a 11= 0.

(15.6)

If a self-adjoint operator Lj can be obtained in this way from a unitary group representation, we say that Lj is a generator of the representation in question. Suppose now that 9 is a Lie group, and consider the corresponding Lie algebra fl with basis I1 ,..., I, and Lie product (15.7) k

As 01varies and j is held fixed, the operators G(expaQ form a one-parameter subgroup of G(Q). This is generated by a self-adjoint operator Lj according to Eq. (15.6). Conversely, each element w E 9 in the neighborhood of w = L can be written in the form exp(C, a&.), with some set of real numbers C+,..., (Y, .

262

VAN

It was shown by Girding the property that $q+ 1’ 1I -i for everyfE

CC,.lffis

WINTER

(15) that there is a set 6 which is dense in fi and has

1 ajLj.f - IG [exp (P C ail,,)] - 11f /fi / = 0 j

(15.8)

j

in 8, so is L,S. Furthermore, LjLlef - LfCLif = i C cikLlf 1

(15.9)

for everyfg 6. The Lie algebra (1 thus gives rise to an algebra of operators Lj on 6, the element Ii ~/l being represented by the skew-symmetric operator -iLj . In Garding’s paper, the set 8 was constructed explicitly. It was pointed out by Harish-Chandra [(16), Section 71, however, that for many applications it is more convenient to consider the set 91 consisting of all elements f 6 R for which the mapping o + G(o)f of Q into 3 is analytic. For a discussion of this concept, see the book by Helgason [(17), p. 4401. The set ‘u is dense in R. Unlike 8, it has the important property that G(w)f is in PI whenever f is in a. If f is in 2l, then so is LjJ The commutation relations (15.9) and Eq. (15.8) hold true for every f 6 2l. The set 81 was investigated in detail by Nelson (9), who called an element f E ‘3 an analytic vector for G(Q). Given an operator B on R, Nelson called an element f E Ji an analytic vector for B in case (15.10)

f. P/l BPf H/P! < 00 for some fl > 0. If B is self-adjoint, so that the operator for real /?, and if Eq. (15.10) holds true,

exp(-iPB)

is defined

either side tending to 0 as P tends to co. According to Nelson’s Lemma 7.1, every f E 91 is an analytic vector for each generator Lj . In the context of the present paper, Nelson’s Theorem 3 is particularly important. This can be rephrased as follows. THEOREM 15.1. Let u + G(w) be a unitary representation of the Lie group !S on the Hilbert space R. Let 1, ,..., IS be a basis for the Lie algebra A of Q, and let L, ,..., L, be the corresponding self-adjoint generators on A. Then the restriction of Cj Lj2 to nj,k JI)(LjLk) is essentially self-adjoint. If La denotes the self-adjoint extension, any analytic vector for L2 is an analytic vector for G(Q) and is thus in 2X.

ORBITAL

ANGULAR

263

MOMENTUM

Proof. This is Nelson’s Theorem 3 except for the fact that in Nelson’s paper the sum xj Lj2 was first restricted to a domain which is smaller than nj,k D(LjL,). The last few paragraphs may be summarized by saying that a unitary representation of a Lie group gives rise to an algebra of operators with satisfactory properties, Nelson’s Theorem 5 is devoted to the converse problem whether a given Lie algebra of operators on R generates a representation of the corresponding Lie group a. The result reads as follows. THEOREM 15.2. Let R be a Hilbert space. Let G(A) be a Lie algebra of skewsymmetric operators -iK, on a domain CZC 9~such that Kjf is in C5wheneverf is in E. Let -iK, ,..., -iK,, be a basis for G(A). Suppose that G(A) represents the Lie algebra A, and let 8 be the simply connected Lie group corresponding to A. If Cj Kj2 is essentially self-adjoint, then there is on R a unique unitary representation of Sz generated by the operators Ri .

XVI.

SUFFICIENT

CONDITION

FOR

GROUP

REPRESENTATION

In the present section, a new condition is derived which is sufficient for a set of operators Lj to generate a unitary group representation. The condition demands that each Lj commutes with a certain operator L2 and thus fits in with the orbitalangular-momentum problem. THEOREM 16.1. space A. Write

Let L, ,..., L, be a set of self-adjoint

a = n a(~,), j

operators

Q = n D(LjLJ j,k

on a Hilbert (16.1)

and denote the restriction of Cj Lj2 to Cl by J2. Suppose that there is a self-adjoint operator L2 such that L2 2 J2 and D(L2) C B. Let the families of spectral-resolution operators for Li and L2 be {E,(T)} and {F(T)} (-co < T < CO), so that Li =

m T d&(T), s --m

L.2 =

m 7 dF(T). -cc

(16.2)

Suppose that, for every f E 6, L,Lkf

- LkLjf

where the numbers c:, are the structure

= i C &L,f, 1

(16.3)

constants of the Lie algebra A. Let Q be

264

VAN

WINTER

the simply connected Lie group corresponding with each Lj , in the sense that

to A. Suppose that L2 commutes (16.4)

F(T) Ed4 f = Ed4 F(T) f

for every f E A and every real o and r. Then there is on si a unique unitary representation of Q which is generated by the operators Lj . Proof.

Iff is in R, it is obvious that F(T) f is in ID(L2). From this it follows that

F(T) f is in 3, and so

F(T)

f is in a(&). LjF(T)

Hence

f= F(T) LjF(T)fi

(16.5)

by Eq. (16.4) and Appendix II. Now let ($ be the set of all elements g E A which are of the form F(T)f, with some f E fi and some real 7. The relation (16.5) then shows that Ljg is in @ whenever g is in @. This implies that @ C CC, hence the commutation relations (16.3) hold true on C?. Since the restriction of L2 to E is essentially self-adjoint, the theorem now follows with Theorem 15.2. The above argument also yields a slightly more special result. THEOREM 16.2. Let L, ,..., L, be a set of self-adjoint operators on a Hilbert space A. Define 6 by Eq. (16.1) and suppose that the restriction of Cj Li2 to 0: has a self-adjoint extension L 2. Let {F(T)} (-CO < T < W) stand for the family of spectral-resolution operators for L 2. Let Eq. (16.5) hold true for every f E H, every j, and every finite T . Suppose that, for every f of the form F(T) g, there are commutation relations (16.3), the numbers c:,; being the structure constants of the Lie algebra A. Let r;L) be the simply connected Lie group corresponding to A. Then there is on fi a unique unitary representation of Q which is generated by the operators Lj .

XVII.

REPRESENTATIONS

OF

COMPACT

SEMISIMPLE

LIE

GROUPS

In the case of a Lie group which is compact and semisimple, Theorem 16.1 has a converse which says that for the generators of a unitary group representation it is in fact necessary to commute with a certain operator L2. In order to prepare for this, consider a compact semisimple Lie group Q and its Lie algebra (1. Let 11 ,*-*, In be a basis for (1, let the Lie product take the form (15.7), and consider the Killing form B(Zi ) Zj) = c c:,ci”, . k.1 This is real and symmetric,

and so it can be diagonalized

(17.1) by an orthogonal

ORBITAL ANGULAR

MOMENTUM

265

transformation among the elements Ii. Owing to the assumptions on 9, the Killing form is negative definite [Helgason (27) Ch. II, Proposition 6.61. By a suitable choice of the basis, it can therefore be arranged that B(li ) I,) = -L&j .

(17.2)

If this is done, the quantity -Cj lj” is a Casimir operator. It is not in /1, but it is an element of the universal enveloping algebra of 0. Here (Cj Ij2) I,< equals ,,c(Cj Zj”) for every k. Now consider a unitary representation w * G(w) of Sz on A. It was discussed already that this gives rise to an algebra of generators Lj . Owing to the particular choice of the basis, (1 Lj’)

i

L,f

= LI, @ Lj2] f

i

(17.3)

for every f E ‘?X,that is, for every f which is an analytic vector for G(Q). The special properties of the Killing form now enable the following theorem to be proved. THEOREM 17.1. Let o + G(w) be a unitary representation of the compact semisimple Lie group lCl on the Hilbert space 53. Let II ,,.., I, be a basis for the Lie algebra A of LR, and let it be chosen in such a way that the Killing form takes the form (17.2). Let L, ,..., L, be the corresponding self-adjoint generators on 53. Let the domains 3 and C be defined by Eq. (16.1). Then the restriction of xi Li2 to c(: is essentially self-adjoint. Its self-adjoint extension L2 commutes with every Li . The domain B(L2) is in a. The operators Lj satisfy the commutation relations (16.3) for every f E C.

Proof. Let J2 be the restriction of Cj Lj2 to 6;. Then J2 is essentially self-adjoint by Theorem 15.1. The operator L2 is thus uniquely defined, and is equal to J2. As a result, every f E 3(L2) is the limit of some sequence(fn} (n = 1,2,...) in (Z such that J2fn tends to L2f as n tends to co. Obviously, this sequencehas the property that L2(fn - fm) tends to 0 as 11,m tend to co. Now let g be any element in a. Then (g, Lk2g) is nonnegative, and so

II Ljg II2 G (g3 L2g) G ll(L2 + 1) g l12.

(17.4)

The particular choice g = fn -f, shows that Li(fn - fm) tends to 0 as n, m tend to co. Now, Lj is self-adjoint, and so it is closed. Hence f is in 3(Lj). Since this argument applies to every j, it follows that 9(L2) C 9. If {F(T)} (- co < 7 < co) denotes the family of spectral-resolution operators for L2, every g E R which is of the form F(T) f is an analytic vector for L2. Hence

266

VAN

WINTER

it is an MI, by Theorem 15.1. Since 2l is not larger than 6, Eq. (17.4) applies. With the help of Eq. (17.3), induction thus shows that ll(L2)Q (Lj)pF(~),fl~

< il(L2)q (L2 f- l)“F(~)fll

< ~“(7 f

I)>’ 11F(~)fll

(17.5)

for p, q = 0, I,... . Now choose positive numbers E, T, 01, and p and determine P and Q in such a way that i

E exp(--olT

(01~ + @YP ! < E exp(--iBT),

- a).

(17.6)

p=P+l

This yields

Combining

these relations gives, in an obvious

notation,

I~[exp(-WY exp(--G) - p=oi o=o f ] F(~)fii < Since there is an analogous equation for exp(--iolL,) eXp(-$L2)

eXp(-idj)

F(T)f

=

eXp(-id+)

2~ IIF(7)fll.

exp(-$?12), eXp(-ifiL2)

it follows F(T)f.

Letting 7 tend to cc now shows that exp(-k&) and exp(-$L2) From this it follows with Appendix II that Lj and L2 commute. Since F(‘(7)fis in 81, LjLkF(7)f’ for every f E R. Hence, iff

- LkLjF(T)f

= i C C:kLiF(T)f

(17.8) that (17.9) commute.

(17.10)

is in B,

F(T) LjLkf

- F(T) LkLjf

= iF(T) C

c:&lf,

(17.11)

owing to Appendix II and the fact that L2 and Lj commute. If 7 now tends to co, it follows that the commutation relations (16.3) hold true for every f E Q;. This proves the theorem.

ORBITAL

On combining

Theorems

ANGULAR

267

MOMENTUM

16.1, 16.2, and 17.1, the following

result is obtained.

THEOREM 17.2. Let Q be a simply connected, compact, semisimple Lie group. Let A be the Lie algebra of Sz, and let its basis I, ,..., I, be chosen in such a way that the Killing form takes the form (17.2). Let L, ,..., L,, be a set of self-adjoint operators on a Hilbert space53.In order that L, ,..., L, be the generators of a unitary representation of Q on R, the operator -iLj representing Ii, the conditions of Theorem 16.1 are necessaryand su#icient. Likewise, the conditions of Theorem 16.2 are necessary and suficient. If they are satisfied, the restriction of Cj Lj2 to nj,lc Z(LjL,) is essentially self-adjoint.

The main point of this theorem can be summarized by saying that it is necessary and sufficient that L, ,..., L, satisfy suitable commutation relations on r)j,k 3(LjL,) and commute with L2.

XVIII.

UNITARY

EQUIVALENCE

OF

GENERATORS

It is well known that the orbital-angular-momentum operators Lj (j = X, y, z) characterized by 6 = 0 generate a unitary representation of the group SU(2) on the Hilbert space 9. It was pointed out in Section XIII that these operators are unitarily equivalent. This result can be generalized as follows. THEOREM 18. Let w -+ G(w) be a unitary representation of the connected, compact, semisimpleLie group Q on the Hilbert space 33. Let Sz be of rank 1. Let I, ,..., 1, be a basisfor the Lie algebra A of Q, and let it be chosenin such a bcay that the Killing form takes the form (17.2). Let L, ,..., L, be the corresponding self-adjoint generators on 52. Then the operators L, ,..., L, , -L, ,..., -L, are unitarily equivalent. For eachpair Lj , LI, , there are elementswjk , xj E Q such that

G(wik) exp(-iiaLJ G(xJ exp(-ial,)

G-l(o+J = exp(--iaL,),

(18.1)

G-l(xj) = exp(iotLj)

(18.2)

for every real 01. Proof. To prove Eq. (18.2), it is convenient to extend A, which is an algebra over the real numbers, to its complexification (1 + irl. BecauseQ is semisimple and of rank 1, each lj EA is associated with a set of positive roots p and a set of elements k, , k-, EA + iA such that [-i& , &,I

= f&,

,

[k, , k-J = -iplj ,

(18.3)

268

VAN

WINTER

the elements k, and k-, being determined uniquely except for constants [see Racah (18), Part II, Section 11. Given an element k, of the form xi ajlj, let I;, denote Cj Gjlj . Then it follows from Eq. (18.3) and the uniqueness properties of k, and k-, that there is a real number y such that I?,, = yk-, . If the normalization is chosen in such a way that y = -1, then k, - k-, and i(k, + k-,) are in fl. These elements can thus be denoted by 2lY, and 2l/“Z, . This yields [Iu

, I,

*

i/j]

=

Fip(lu

*

(18.4)

iZj).

For real values of p it follows that

zzz

exp(

f

$p)(I,

k

ilj).

(18.5)

Here adl, is the operator which transforms k into [I, , k], for every k E A + icl. Now select any particular root p and the corresponding element I,. Choose ,B = rr/p and write xj for exp(&,/p). This gives Xjl&’

= -Ii ,

(18.6)

and so, for every real (Y, Xj

exp(aZJ XT’ = exp( - a,,).

(18.7)

If this equation is represented on R, the desired relation (18.2) is obtained. As 01 varies, either side of Eq. (18.2) forms a one-parameter subgroup of G(Q). If the generator for this is determined according to Section XV, it is readily seen that Lj and -Lj are unitarily equivalent. This completes the first part of the proof. Note that for this it is not required that Szbe connected and compact. We proceed to Eq. (18.1). Let flj and (1, be two maximal Abelian subalgebras of A. Since Q is connected and compact, there is an element wjlc EQ such that qrcfljcIJ;; = A,

(18.8)

(Helgason (17), Ch. V, Theorem 6.4). Since 52 is of rank 1, each lj is the basis of a maximal Abelian subalgebra. Given a pair Zi , Z, , there is therefore an element wjlc E Szand a real number 5 such that

ORBITAL ANGULAR

269

MOMENTUM

Since the Killing form is invariant under the transformation fl ---f w.4wp1, - 1 = B(lj ) /j) = B(
(18.10)

Hence 5 = &I. Becauseit is already known that Eq. (18.7) holds true, it can be assumedwithout loss of generality that [ = 1. The relation (18.9) then yields wjk exp(cufJ 0~2 = exp(oll,).

(18.11)

The corresponding relation on R is Eq. (18. I). From this it follows that Lj and L, are unitarily equivalent. This proves the theorem. It is a consequence of this theorem that the operators L, ,..., L, have identical spectra, and that these spectra are symmetric with respect to the origin. XIX.

DISCUSSION

We can now continue the discussionof the orbital-angular-momentum operators which was begun in Section XIV. If 6 = 0, the operators Lj (j = 5, y, z) generate a representation of SU(2). This is in complete agreement with the previous sections.In particular, L2 commutes with each Lj in this case.Also, the operators Lj all have the same spectrum. If 6 f 0, however, L2 does not commute with L, and L, . The spectra of L, , L, , and L, are no longer the same.There is therefore no unitary group representation generated by the operators Lj . In this case there is also the difficulty that the restriction of Cj Lj2 to (5 is not essentially self-adjoint. According to Theorem 17.1, this is an extra reason why the operators Lj do not generate a unitary group representation. It will be observed that the conditions of Theorems 16.1, 17.1, and 18 exactly match the data which are available in the orbital-angular-momentum problem. In case6 # 0, 4, the above result is not surprising. Indeed, the unitary representations of SU(2) are all known. There only exist integral and half-integral ones. If 6 f 0, + there is the additional point that the spectrum of L, is not symmetric with respect to the origin. Hence L, is prevented from being a generator by Theorem 18. Now consider the case 8 = fr. It is worth noticing that in this case there does exist a unitary representation of SU(2) for which Lz is a generator and L” the Casimir operator. In fact, define (R, * iR,) 2,~ = [(A F p)(h 5 p + I)]“” Zy*l,

R,Z,,” = pZ,p.

(19.1)

Let %(RJ (j = x, y, z) consist of all functions f E !?Jfor which (19.2)

595/47/W

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WINTER

and define

W” = 1 G’, f) fijz~~

(19.3)

u,n

for every f E D(Rj). Then each R, is self-adjoint. The operator R, equals L, . The intersection fij D(R,) is nothing but the set a of Eq. (8.2). Also, nj,k: B(RjR,) is the set ID(L2) of Eq. (12.1). The restriction of ‘& Rj2 to ll)(L2) equals L2 and is thus self-adjoint. With the help of Theorem 15.2 it can be shown that there is a unique unitary representation of N(2) generated by the operators Rj . It is not difficult to see that this is such that G(w) ZAP = c G+‘(o) u’

Zf,

(19.4)

where the quantities G;‘@(w) form the unitary (2X + 1) x (2X + 1) matrix which is familiar from the general theory of the group SU(2). It is obvious, however, that R, # L, and R, # L, . The operators R, and R, do not derive from differential operators and so they have no bearing on the orbital angular momentum. In his paper on the problem, Pauli (6) demanded that each operator Li should transform an eigenfunction of L2 into an eigenfunction with the original eigenvalue. It is a slightly milder criterion to say that L#‘(~)fmust be equal to F(r) LjF(T)f for every fE $ and every finite T. This is the relation (16.5). If Eq. (16.5) is given and it is assumed that L, is essentially self-adjoint on the set @ of all functions of the form F(T)~, it is not difficult to prove that L2 and Lj commute. Alternatively, Eq. (16.5) may be combined with the requirement that the operators -iLi represent a Lie algebra on Cc. Then Theorem 17.2 shows that a condition is obtained which is necessary and sufficient for the operators Lj to generate a unitary representation of S(/(2). Under this assumption, it also follows that L2 and Li commute, again by Theorem 17.2. Pauli’s paper and the discussion of Section XIV made us find experimental criteria to see if a set of observables is represented by the generators of a unitary group representation. In the case of a compact semisimple Lie group, Theorem 17.1 demands that each generator commutes with the Casimir operator L2. If, in addition, the group is connected and of rank 1, the generators must all have the same spectrum, which must be symmetric with respect to the origin, by Theorem 18. These are exactly the points by which the orbital angular momentum with integral eigenvalues differs from all non-integral ones. It is suggested that this explains why nonintegral eigenvalues for the orbital angular momentum are not found in nature.

ORBITAL ANGULAR

271

MOMENTUM

ACKNOWLEDGMENTS

The author’s interest in the orbital-angular-momentum problem was aroused by a seminar talk by Professor P. W. Kasteleyn, who gave an approach based on commutation relations. This paper originated from numerous discussions with S. G. van der Wal and M. Winnink. It is a pleasure to acknowledge their contribution to the present result.

APPENDIX

I. A THEOREM

ON SYMMETRIC

OPERATORS

THEOREM Al. Let A be a symmetric operator on a Hilbert space 5%.Suppose that A has a complete orthonormal set of eigenvectors {a,> (n = 1,2,...) such that

(A1.1)

Aa, = cy,a, .

Then the closure A is self-adjoint. The domain D(A) consistsof all elementsf for which C I 4an,f)12

?I

ER

(A1.2)

< 00,

every f E D(A) satisfying

(A1.3)

Proof.

Choose any f E R which satisfies Eq. (A1.2). Clearly the element &

defined by (A1.4)

fN = 2 (any f> 0, ?I=1

is in D(A). If N tends to 00, then fN tends to J Owing to Eq. (A1.2), Afh; tends to the right-hand side of Eq. (A1.3). Hence f is in a(A) and Af satisfies Eq. (A1.3). Now let B be the restriction of A to the domain (A1.2). Then B*3A*3A3

(A1.5)

B.

For every f in D(B*) (B*f, 4

= (f, Ban) = 4f,

In order that B*f be in a, it is thus necessary that shows that D(B*) is not larger than ID(B), hence Eq. (A1.5) it now follows that A = B. The closure by Eqs. (A1.2) and (A1.3) and it is self-adjoint. This

4.

(A 1.6)

Eq. (A1.2) is satisfied. This that B is self-adjoint. With A is thus fully characterized proves the theorem.

272

VAN WINTER

APPENDIX

II. COMMUTING

OPERATORS

Let 2 be a Hilbert space. If A and 3 are bounded operators with domain si, they are said to commute if ABf is equal to BAf for every f E R. For unbounded operators, the concept of commutativity is much more complicated. As long as B is bounded and has domain si, it is meaningful to say that A and B commute if Bf is in 3(A) whenever f is in D(A), and ABf = BAf

(A2.1)

for every f E D(A). Now let A and B be self-adjoint operators and let their domains be ID(A) and ‘I)(B). Let {EA(~)} and {E&))) (-co < Q, /3 < co) denote the families of spectral-resolution operators for A and B, so that

A=

r’n J-cc 01dE.4(4,

B=

(A2.2)

B ~J%P). sm -m

Then A and B are said to commute if and only if EA(U) E,(P)f = E&)

(A2.3)

EA(4.f

for every f E si and every real 01,/3 [see Von Neumann (13), Ch. II, Section IO]. If this relation is fulfilled, it can be shown that J%(B) Af = AEiG0.L

E,(a) & = md4

g

(A2.4)

for every f E D,(A) and every g E n(B) [Achieser and Glasmann (12), Section 751. Conversely, Eq. (A2.3) follows from Eq. (A2.4). If A and B are bounded and have domain 52, Eq. (A2.4) yields Eq. (A2.1) for every f E R. The relation (A2.3) thus extends the familiar concept of commutativity to self-adjoint operators which are not necessarily bounded. It is also compatible with Eq. (A2.1) for bounded B and unbounded A. Now supposethat exp(isA) and exp(itB) commute for every real s, t. Then J’I, exp(isol) d,[EA(~) exp(itB)] = Jl, exp(isol) d,[exp(itB) EJoi)],

(A2.5)

and so EA(a) exp(itB) = exp(itB) EA(cu),

(A2.6)

by a uniquenessproperty of Fourier-Stieltjes integrals [Bochner (19), Section 18, Theorem 18; Achieser and Glasmann (12), Section 601. Expanding exp(itB) in terms of E&3) now showsthat Eq. (A2.3) is satisfied, hencethat A and B commute.

ORBITAL

ANGULAR

273

MOMENTUM

Now let the spectra of A and B be purely discrete, and let the eigenvalues be a, and A, respectively (m, n = 1, 2,...). If A and B commute, the element (A2.7)

E&@?z+ 0) - &4Jsn)l[cl(%z+ 0) - L(%)lf

is a simultaneous eigenvector for A and B. Every f E A can then be developed in terms of such vectors. Conversely, if A and B have a complete orthonormal set of eigenvectors {cnln}, every f E $3 can be written in the form

This yields

and so A and B commute. Summarizing,

the following

result holds true.

Let A and B be self-adjoint operators on a Hilbert space S, and let their spectra be purely discrete. Then A and B commute if and o&y if they hace a complete set of simultaneous eigemectors. THEOREM

A2.

Note added in proof. Professor E. Merzbacher kindly drew the author’s attention to a paper by Kretzschmar (20) which is devoted to the z-component of the orbital angular momentum. Demanding that the eigenvalues of L, be integral, Kretzschmar pointed out that L, is not necessarily represented by the differential operator -ia/+,. This observation emerged from a study of the Aharonov-Bohm effect (21), (22). In describing this effect, Kretzschmar made use of the functions 0,s defined in Section II. He also considered operators II which are formally related to restrictions of our operators L for nonintegral orbital angular momenta (22). If one requires that the usual commutation relations are satisfied, and further demands that the eigenfunctions of the orbital angular momentum can be expressed in terms of Cartesian coordinates X, I’, z, then it follows that the eigenvalues must be integral. This was shown by Louck (23). RECEIVED:

September 5, 1967 REFERENCES

I. E. MERZBACHER, Am. J. Phys. 30,237 (1962). 2. J. M. BLATT AND V. F. WEISSKOPF, “Theoretical 3. M. L. WHIPPMAN, Am. J. Phys. 34,656 (1966). 4. H. A. BUCHDAHL,

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30, 829 (1962).

Nuclear

Physics.”

Wiley,

New

York,

1952.

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5. H. S. GREEN, “Matrix Mechanics.” Noordhoff, Groningen, 1965. 6. W. PAUI~, Helv. Phys. Acta 12, 147 (1939). 7. D. PANDRES, JR., J. Math. Phys. 6, 1098 (1965). 8. S. S. SANNIKOV, Nucl. Phys. 87, 834 (1967). 9. E. NELSON, Ann. Math. 70, 572 (1959). IO. E. M. STEIN, in “High-Energy Physics and Elementary Particles,” p. 563. International Atomic Energy Agency, Viemra, 1965. II. T. M. MACROBERT, “Spherical Harmonics” (2nd ed.). Dover, New York, 1947. 12. N. I. ACHIESER AND I. M. GLASMANN, “Theorie der Linearen Operatoren im Hilbert-Raum.” Akademie-Verlag, Berlin, 1954. 13. J. VON NEUMANN, “Mathematical Foundations of Quantum Mechanics.” Princeton University Press, Princeton, 1955. 14. F. RIESZ AND B. SZ.-NAGY, “Leccns d’Analyse Fonctionnelle” (3rd ed.). Gauthier-Villars, Paris and Akademiai Kiadb, Budapest, 1955. IS. L. G&WING, Proc. Natl. Acad. Sci. U.S. 33, 331 (1947). 16. HARISH-CHANDRA, Trans. Am. Math. Sot. 75, 185 (1953). 17. S. HEWASON, “Differential Geometry and Symmetric Spaces.” Academic Press, New York, 1962. 18. G. RACAH, Ergeb. Exakt. Naturw. 37, 28 (1965). 19. S. BOCHNER, “Lectures on Fourier Integrals.” Princeton University Press, Princeton, 1959. 20. M. KRETZSCHMAR, Z. Phys. 185, 73 (1965). 21. M. KRETZSCHh4AR, Z. Phys. 185, 84 (1965). 22. M. KRETZSCHMAR, Z. Phys. 185, 97 (1965). 23. J. D. LxXJCK, Am. J. Phys. 31, 378 (1963).