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arXiv:quant-ph/0507006

Properties of Fermion Spherical Harmonics

Geoffrey Hunter, Mohsen Emami-Razavi

Published Jul 1, 2005• 4 citations
arXivHTMLPDF

Abstract

The Fermion Spherical harmonics [$Y_\ell^{m}(θ,ϕ)$ for half-odd-integer $\ell$ and $m$ - presented in a previous paper] are shown to have the same eigenfunction properties as the well-known Boson Spherical Harmonics [$Y_\ell^{m}(θ,ϕ)$ for integer $\ell$ and $m$]. The Fermion functions are shown to differ from the Boson functions in so far as the ladder operators $M_+$ ($M_-$) that ascend (descend) the sequence of harmonics over the values of $m$ for a given value of $\ell$, do not produce the expected result {\em in just one case}: when the value of $m$ changes from $\pm{1/2}$ to $\mp{1/2}$; i.e. when $m$ changes sign; in all other cases the ladder operators produce the usually expected result including anihilation when a ladder operator attempts to take $m$ outside the range: $-\ell\le m\le +\ell$. 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, $θ$ and $ϕ$.

DOI
10.48550/arxiv.quant-ph/0507006

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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 recently established existence of spherical harmonic functions, Yℓm(θ,ϕ)Y_\ell^{m}(θ,ϕ)Yℓm​(θ,ϕ) for half-odd-integer values of ℓ\ellℓ and mmm, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles θs,ϕsθ_s, ϕ_sθs​,ϕs​, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators, S2,SzS^2, S_zS2,Sz​, are represented by the usual differential operators in spherical polar coordinates (commonly used for L2,LzL^2, L_zL2,Lz​), and their electron-spin eigenfunctions are sin⁡θsexp⁡(±ϕs/2)\sqrt{\sinθ_s} \exp(\pmϕ_s/2)sinθs​​exp(±ϕs​/2). This eigenfunction representation has the pedagogical advantage over the abstract spin eigenfunctions, α,β,α, β,α,β, that ``integration over spin coordinates'' is a true integration (over the angles θs,ϕsθ_s, ϕ_sθs​,ϕs​). In addition they facilitate construction of many electron wavefunctions in which the electron spins are neither parallel nor antiparallel, but inclined at an intermediate angle. In particular this may have application to the description of EPR correlation experiments.
Angular momentum theory provides a mathematical foundation for the rotation of the eigenvectors of the angular momentum operator, the spherical harmonics functions. A useful property of spherical harmonics is that their rotation in 3D space corresponds to rotation in the virtual space of their azimuthal degrees of freedom. In addition to that, spherical harmonics functions that have different orbital indices do not mix. Given that spherical harmonics functions have known rotation properties, and that any arbitrary function can be expanded in a series of spherical harmonics, this property allows the rotation of an electrostatic potential when it is expanded in the series of spherical harmonics. In this chapter, the exponential form of the rotation operator, the rotation matrix for a 3D rotation of the coordinate system, and the principles of rotation of vectors and functions are described. The chapter provides a detailed self-contained introduction to the theory of angular momentum, derives the commutative properties of the angular momentum operator, and explains shift operators, while minimizing references to quantum mechanics and entirely avoiding group theory.
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
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials
The addition theorem for spherical harmonics is just a special case of the law for the transformation of spherical harmonics under rotation of their arguments. Certain operators are scalars under rotation; this means that they commute with the total angular momentum and are unchanged by the transformation. The types of operators having simple transformation properties under rotation are known generally as tensor operators. The energy eigenstates of an atom, taking into account the spin-orbit interaction, are not usually eigenstates of the z component of the total orbital angular momentum, but are eigenstates of the total angular momentum, spin plus orbital. Sehwinger has looked at the theory of angular momentum from a point of view that enables one to carry out very elegant calculations of the various quantities occurring in the theory of angular momentum. The quanta of some different oscillators are simply the quarks that compose the strongly interacting particles.
The usual explicit expressions for spherical harmonics are derived by a method which does not involve the solution of differential equations or the use of angular momentum operators in spherical polar form. Instead a raising operator constructed from an arbitrary vector is used together with the usual angular momentum raising and lowering operators. The method is first presented as simply as possible in Schrödinger representation using position as the vector. Then it is generalized to an arbitrary vector and arbitrary integral angular momentum without using a particular representation. Finally it is shown how a conserved vector (other than angular momentum) of a central system can be used to reveal properties of the energy eigenstates. As an example, eigenstates of the hydrogen atom are found using the Runge-Lenz vector.