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Unformatted text preview: Integer versus halfinteger angular momentum Ian R. Gatlanda) Georgia Institute of Technology, Atlanta, Georgia 303320430
(Received 3 October 2005; accepted 16 December 2005) The absence of halfinteger orbital angular momentum states is established by considering the
realizations of the angular momentum operators in spherical coordinates, 6 and (p. The angular
momentum raising and lowering operators switch the states between even and odd functions of
6— 1712. Furthermore, the bottom and top states are both even. Hence there must be an odd number
of states. This argument permits integer angular momenta and forbids halfinteger angular momenta. © 2006 American Association of Physics Teachers. [DOI: 10.1119/12166372] I. {NTRODUCTION One problem in presenting the theory of angular momen
tum in the context of quantum mechanics is explaining the absence of orbital angular momentum states that are half— ' integer (in units of it). The angular momentum component
operators are initially deﬁned in terms of position and mo
mentum operators and the commutation relations for the
components are obtained. The remainder of the algebraic
analysis onlyuses the commutation relations for the angular
momentum components so that some of the information con
tained in the positionmomentum commutation relations is
lost. As a result both integer and halfinteger angular mo
mentum states appear. The realization of the angular momentum operators and
states in three—dimensional space recognizes the position—
momentum commutation relations and eliminates the half—
integer angular momentum states. The elimination of the
half—integer states may be established directly by considering
the symmetries of neighboring states and of the bottom and
top states. II. FORMALISM In the usual development, the raising and lowering opera
tors from a state with zcomponent angular momentum num—
ber m to the neighboring m states (expressed in terms of the
polar angle 6 and the azimuthal angle go) are"2 Li = u iaeﬂuita/aa— cot name), (1)
and the z—component angular momentum operator is
LZ = — ind/ago. (2) If Fg’m(6,(P) is an eigenfunction of L2 and Lz with eigen
values €(l?+1)f't2 and and, respectively, then according to Eq (2).
F€,m( 9’ £0) =fr,m(3)€im“p (3) We combine Eqs. (1) and (3) and ﬁnd that the neighboring
states (after normalization) are given by mid!“ ‘ m Cot 9mm = [Jemima » (4)
where
Pe,m=[€(€+1)m(m+ l)]”%. (5) Raising the top state or lowering the bottom state gives zero
for the righthand side of Eq. (4). From the resulting differ
ential equations we ﬁnd 191 Am. J. Phys. 74 (3). March 2006 httpzi/aapmrg/ajp nineteent a. (6) The normalization constants N+ and N_ in Eq. (6) have the
same magnitude but not necessarily the same phase. All of
the states for a given if may be obtained by starting from the
bottom state, Eq. (6) with m=—€, and using Eqs. (4) and (5)
to move up in m. III. SYMMETRY ANALYSIS We now consider the symmetries of the fem functions un
der the transformation 6—>1r—6. Under such a transforma—
tion sin 0 is even (no sign change), cos 0 is odd (changes
sign), and cot 0 is odd. Also, if 3(6) is even (odd), then
rig/dd is odd (even). It follows from Eq. (4) that if fem is an
even function, then famil is an odd function, and vice versa.
But, according to Eq. (6), the bottom state is even, so the
states are alternately even and odd. However, according to Eq. (6), the top state is also even so
there must be an odd number of in states. This requirement in
turn implies that E be an integer. The integer angular momen
tum states are allowed by the symmetry analysis, but the
halfinteger angular momentum states are forbidden. Half
integer'angular momentum states cannot be represented in
terms of threedimensional space functions. IV. COMMENTARY This symmetry analysis is most effective if the spherical
harmonics (integer angular momentum states) are derived by
starting with Eq. (6) and then using Eqs. (4) and (5) to obtain
the other in states, rather than by using Legendre polynomi
als and the associated Legendre functions. If only spin 1/2 is
of interest, then a simple argument is available. The upper
state obtained by applying Eq. (4) to the lower state given by
Eq. (6) is not the same as the upper state obtained directly
from Eq. (6).3'4 This inconsistency is an example of a result
noted by Pauli that the angular momentum operators fail to
be Hermitian for half—integer states generated by the raising
and lowering operators (they lead to wave functions that are
not norrnalizable).5‘7 The analysis based on symmetry considerations may be
compared with other approaches. Many of these depend on
establishing the existence of a state with integer m. Then all
the m are integers and so is 6. The analytic method starts
with the Legendre polynomials, which implicitly assume that
m=0, and hence result in integer 3 . Indeed, any function that
depends only on 2 and r represents m=0 states so such states
seem natural for all 6.8 © 2006 American Association of Physics Teachers 191 A somewhat different approach uses the conditiq ,
wave function be single valued. This requiremen implies
that eim(rp+21r) = eirmp’ _; : which, in turn, However, this conditio‘ condition that the pic; ney"uénsnytestungvalues.The latter condition allowi" elfinteger'vv'alu'es:Fof=m¥and431E OnY the v :'
other hand, Merzbaéhe “has; shownE that étheawave wfun'ction  must be single valued if it is a function of space coordinates.
His analysis involves the transformation properties of the
wave function under coordinate rotations and is more appro—
priate for an advanced courseﬁ‘9 :.:It is not necessary to use spherical coordinates toelirninate
half~integer 6 values. Ballentine uses Cartesian coordinates
to"?(=:_1_t‘151‘essg the difference of"t'w'o' harmenicr'osenlatdr Hamilto a’c'h' with 'ii'nitang’ular seqacne}: The result:
iffs ’e’ise ues‘aiié 'iﬁ'tég‘ér Ii‘tﬁlpl'es' of a3 Balleirtiné’s' mains involves and iCoOrdinates? Whereas thesymmetry analysis is equigglent to the reversal of the z cOor'diira'té; so' there is no obvious connection. 192 Am. 1. Phys, Vol. 74, No. 3, March 2006 the, ""(and: hearse? Eare' int'Ig‘e'rs. is more stringent "than thenatural“ 5 l' ;. . .. ,:_;~ I thank Brian Kennedy for his helpfulcornments and sug ._ gestions. .. “)Eiécaoﬁib mail: new(assassinate :‘LaeonardrI:'Sehiff,'..QLéanrum‘Méchahics, 2nd ed. .(McGraWHill, New : Yolkiﬁl'955”)
zDaviclJG' . Prentice Hal Upper saddened; means)".
3 onetime? Pﬁr‘iﬁﬁlés 76f? ' Meahanics ' (Prentice Hall,
:'.Englewoocl_:€31iffs, NJ;>1999).~:2;.3 ' :": ‘ \ :
fSteﬂren Gasioi't_:>_wi<_:zI Quantum Physics, 2nd .ed. (John. Wiley 8r. Sons,
New York, 1996). 7 . t _ _ _ r ._ U ._
5w. Pauli, “Uber ein kriterium ftir amtdeer ZWciWertig'keit‘dér Eigenfunk
tionen in der Wellenmechanik,” Helv. Phys. Acta 12., 147—168 (1939). ' I. (Pearson l5E. Merzbacher. “Single valuedness of wave ﬁinctionsf' J. 30. 237~Z47 (1962). 4 A t .  t
C. Van Winter, “Orbital angular momentum and group representations,”
Ann. Phys.‘ 4123242740968); = ' ' '3‘ 1"
“David: Perle; Introduction .to :the Quantum, Theory , (McGrawIIili, New 1 ,< 2 ~" i: : _ l' a '5“‘:' '  fl).
_9Kuﬁ Gottfried i'ljung—Mow gYan, Quantum Mechanics; Fundamentals,
25.1917 " V'lsﬁxihgééYeﬂaeNew Ybrk.2004) ' . . ’ 1°15le "E; ’Ballentine. 'Qunntii'm_'Mechﬁniis: A! Moire"; 'Deirelaprhért':
"(World scientiﬁéfsnigaporégtws)._ ' r * "   .l. Ian R. Gatland 192 ...
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