AngularMomentum - Integer versus half-integer angular momentum Ian R Gatlanda Georgia Institute of Technology Atlanta Georgia 30332-0430(Received 3

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Unformatted text preview: Integer versus half-integer angular momentum Ian R. Gatlanda) Georgia Institute of Technology, Atlanta, Georgia 30332-0430 (Received 3 October 2005; accepted 16 December 2005) The absence of half-integer 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 half-integer 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 position-momentum commutation relations is lost. As a result both integer and half-integer 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 z-component 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 right-hand 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 half-integer angular momentum states are forbidden. Half- integer'angular momentum states cannot be represented in terms of three-dimensional 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-"uén-snytestung-values.The latter condition allowi" elf-integer'vv'alu'es:Fof=m¥and431E OnY the v :'- other hand, Merzbaéhe “has; shownE that étheawave w-fun'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 seq-acne}: The result: iffs ’e’ise ues‘aiié '-ifi'tég‘ér Ii‘tfilpl'es' of a3- Balleirtiné’s' mains involves and iCoOrdinates? Whereas the-symmetry analysis is equigglent to the reversal of the z cOor'dii-ra'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 the-natural“ 5 l' ;. .- ..- ,:_;~ I thank Brian Kennedy for his helpfulcornments and sug- ._ gestions. .. “)Eiécaofiib mail: new(assassinate :‘Laeonard-rI:'Sehiff,'..QLéanrum‘Méchahics, 2nd ed. .-(McGraW-Hill, New : Yolkifil'955”) zDaviclJ-G' . 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 amt-deer 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 , (McGraw-I-Iili, New 1 -,< 2- ~" i: :- _ l' a '5“‘:' ' - fl). _9Kufi Gottfried i'ljung—Mow gYan, Quantum Mechanics; Fundamentals, 25.1917 " V'lsfixihgéé-YeflaeNew Ybrk.2004)- ' . . ’ 1°15le "E; ’Ballentine. 'Qunntii'm_'Mechfinii-s: A! Moire"; 'Deirelaprhért': "(World scientifiéfsnigaporégtws)._ ' r * -" - - .l. Ian R. Gatland 192 ...
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AngularMomentum - Integer versus half-integer angular momentum Ian R Gatlanda Georgia Institute of Technology Atlanta Georgia 30332-0430(Received 3

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