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American Journal of Physics

Integer versus half-integer angular momentum

I. R. Gatland

Published Feb 14, 2006• 3 citations
Publisher

Abstract

The absence of half-integer orbital angular momentum states is established by considering the realizations of the angular momentum operators in spherical coordinates, θ and φ. The angular momentum raising and lowering operators switch the states between even and odd functions of θ−π∕2. 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.

DOI
10.1119/1.2166372

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The absence of half-integer orbital angular momentum states is established by considering the realizations of the angular momentum operators in spherical coordinates, θ and φ. The angular momentum raising and lowering operators switch the states between even and odd functions of θ−π∕2. 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.
Abstract Harmonic wave functions for integer and half-integer angular momentum are given in terms of the Euler angles (θ, ϕ, ψ) that define a rotation in SO(3), and the Euclidean norm r in <?CDATA R3{{\mathbb{R}}}^{3}R3?> R 3 , keeping the usual meaning of the spherical coordinates (r, θ, ϕ). They form a Hilbert (super)-space decomposed in the form <?CDATA H=H0⊕H1{ \mathcal H }={{ \mathcal H }}_{0}\oplus {{ \mathcal H }}_{1}H=H0​⊕H1​?>  =  0 ⊕  1 . Following a classical work by Schwinger, 2-dimensional harmonic oscillators are used to produce raising and lowering operators that change the total angular momentum eigenvalue of the wave functions in half units. The nature of the representation space <?CDATA H{ \mathcal H }H?>  is approached from the double covering group homomorphism SU(2) → SO(3) and the topology involved is taken care of by using the Hurwitz-Hopf map <?CDATA H:R4→R3H:{{\mathbb{R}}}^{4}\to {{\mathbb{R}}}^{3}H:R4→R3?> H : R 4 → R 3 . It is shown how to reconsider H as a 2-to-1 group map, <?CDATA G0=R+×SU(2)→R+×SO(3){G}_{0}={{\mathbb{R}}}^{+}\times {SU}(2)\to {{\mathbb{R}}}^{+}\times {SO}(3)G0​=R+×SU(2)→R+×SO(3)?> G 0 = R + × SU ( 2 ) → R + × SO ( 3 ) , translating it into an assignment (z 1, z 2) ↦ (r, θ, ϕ, ψ) whose domain consists of pairs (z 1, z 2) of complex variables, under the appropriate identification of <?CDATA R4{{\mathbb{R}}}^{4}R4?> R 4 with <?CDATA C2{{\mathbb{C}}}^{2}C2?> C 2 . It is shown how the Lie algebra of G 0 is coupled with two Heisenberg Lie algebras of 2-dimensional (Schwinger’s) harmonic oscillators generated by the operators <?CDATA {z1,z2,zˉ1,zˉ2}\{{z}_{1},{z}_{2},{\bar{z}}_{1},{\bar{z}}_{2}\}{z1​,z2​,zˉ1​,zˉ2​}?> { z 1
Chapter 11 is devoted to the quantum theory of angular momentum. As in classical mechanics, working with angular momentum requires leaving the one-dimensional space, with all the complications that this entails. The importance of angular momentum resides in at least three facts: i) central problems are of particular interest; (ii) many particles have an intrinsic angular momentum (the spin); and iii) the physics of angular momentum involves features such as the so-called space quantization. The first part of the chapter deals with the orbital angular momentum and its eigenfunctions, the spherical harmonics. Its algebraic properties, embodied in the matrix representation, allow for an extension of the theory to half-integer angular momentum in terms of Pauli matrices, as well as the addition of angular momenta.
Angular momentum is one of the more abstract topics in quantum mechanics. This chapter begins with a review of classical angular momentum. Quantum-mechanical angular momentum is then introduced by obtaining operators from their classical counterparts. The commutation relations amongst those operators and the corresponding uncertainty relations then follow. Eigenstates and eigenvalues of the angular momentum operators are then introduced (but not derived). From these follow the raising and lowering operators. Physical meaning is imparted to quantum angular momentum by considering predictions for sequential measurements, and by comparing the conceptual pictures of angular momentum in classical and quantum mechanics. Finally, quantum spin and orbital angular momenta, and the deep differences between them, are discussed.
The Fermion Spherical harmonics [Yℓm(θ,ϕ)Y_\ell^{m}(θ,ϕ)Yℓm​(θ,ϕ) for half-odd-integer ℓ\ellℓ and mmm - presented in a previous paper] are shown to have the same eigenfunction properties as the well-known Boson Spherical Harmonics [Yℓm(θ,ϕ)Y_\ell^{m}(θ,ϕ)Yℓm​(θ,ϕ) for integer ℓ\ellℓ and mmm]. The Fermion functions are shown to differ from the Boson functions in so far as the ladder operators M+M_+M+​ (M−M_-M−​) that ascend (descend) the sequence of harmonics over the values of mmm for a given value of ℓ\ellℓ, do not produce the expected result {\em in just one case}: when the value of mmm changes from ±1/2\pm{1/2}±1/2 to ∓1/2\mp{1/2}∓1/2; i.e. when mmm changes sign; in all other cases the ladder operators produce the usually expected result including anihilation when a ladder operator attempts to take mmm outside the range: −ℓ≤m≤+ℓ-\ell\le m\le +\ell−ℓ≤m≤+ℓ. The unexpected result in the one case does not invalidate this scalar coordinate representation of spin angular momentum, because the eigenfunction property is essential for a valid quantum mechanical state, whereas ladder operators relating states with different eigenvalues are not essential, and are in fact known only for a few physical systems; that this coordinate representation of spin angular momentum differs from the abstract theory of angular momentum in this respect, is simply an interesting curiosity. This new representation of spin angular momentum is expected to find application in the theoretical description of physical systems and experiments in which the spin-angular momentum (and associated magnetic moment) of a particle is oriented in space, since the orientation is specifiable by the spherical polar angles, θθθ and ϕϕϕ.
The concept and much of the mathematical treatment of orbital angular momentum was treated in Chapter 3. An unexpected result from that treatment is that using the result of Equation (3.41) along with the result of Equation (3.48), we find that mT −mB = 2` = n , (4.1) so ` is either an integer or a half integer! The integers correspond to orbital angular momentum, while the half-integer values correspond to intrinsic angular momentum, or spin angular momentum. There is no classical model for spin angular momentum, but the most fundamental particles (electrons, quarks, protons, and neutrons) all exhibit this type of intrinsic angular momentum.
This chapter covers various aspects of orbital, total, and spin angular momenta and the Clebsch-Gordon coefficients. An approach that is not found in many books, is used to derive eigenvalues of orbital angular momentum. The Pauli theory of spin half systems, addition of angular momenta, eigenfunctions of orbital angular momentum, computation of Clebsch-Gordon coefficients, solved examples, and unsolved exercises are included in the chapter. The basic purpose of the chapter has been to bring out the quantum mechanical importance of the theory of angular momenta with the use of essential mathematical derivations, which are self-consistent, and to adopt an elegant modern approach. Special mention on why orbital angular momentum cannot have non-integer values is made. The concept of spin angular momentum, which is purely a quantum mechanical quantity, is introduced in a simple and most convincing manner.