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arXiv:2211.10775 · Physica Scripta

The Hurwitz-Hopf map and harmonic wave functions for integer and half-integer angular momentum

Sergio A. Hojman, Eduardo Nahmad-Achar, O. A. Sánchez-Valenzuela

Published Jun 22, 2023• 0 citations
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Abstract

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 ${{\mathbb{R}}}^{3}$?> R 3 , keeping the usual meaning of the spherical coordinates (r, θ, ϕ). They form a Hilbert (super)-space decomposed in the form <?CDATA ${ \mathcal H }={{ \mathcal H }}_{0}\oplus {{ \mathcal H }}_{1}$?>  =  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 ${ \mathcal H }$?>  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 $H:{{\mathbb{R}}}^{4}\to {{\mathbb{R}}}^{3}$?> H : R 4 → R 3 . It is shown how to reconsider H as a 2-to-1 group map, <?CDATA ${G}_{0}={{\mathbb{R}}}^{+}\times {SU}(2)\to {{\mathbb{R}}}^{+}\times {SO}(3)$?> 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 ${{\mathbb{R}}}^{4}$?> R 4 with <?CDATA ${{\mathbb{C}}}^{2}$?> 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 $\{{z}_{1},{z}_{2},{\bar{z}}_{1},{\bar{z}}_{2}\}$?> { z 1

DOI
10.1088/1402-4896/ace08e

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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 ${{\mathbb{R}}}^{3}$?> R 3 , keeping the usual meaning of the spherical coordinates (r, θ, ϕ). They form a Hilbert (super)-space decomposed in the form <?CDATA ${ \mathcal H }={{ \mathcal H }}_{0}\oplus {{ \mathcal H }}_{1}$?>  =  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 ${ \mathcal H }$?>  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 $H:{{\mathbb{R}}}^{4}\to {{\mathbb{R}}}^{3}$?> H : R 4 → R 3 . It is shown how to reconsider H as a 2-to-1 group map, <?CDATA ${G}_{0}={{\mathbb{R}}}^{+}\times {SU}(2)\to {{\mathbb{R}}}^{+}\times {SO}(3)$?> 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 ${{\mathbb{R}}}^{4}$?> R 4 with <?CDATA ${{\mathbb{C}}}^{2}$?> 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 $\{{z}_{1},{z}_{2},{\bar{z}}_{1},{\bar{z}}_{2}\}$?> { z 1
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.
We study the quantum mechanical harmonic oscillator in two and three dimensions, with particular attention to the solutions as basis states for representing their respective symmetry groups — O(2), O(1,1), O(3), and O(2,1). The goal of this study is to establish a correspondence between Hilbert space descriptions found by solving the Schrodinger equation in polar coordinates, and Fock space descriptions constructed by expressing the symmetry operators in terms of creation/annihilation operators. We obtain wavefunctions characterized by a principal quantum number, the group Casimir eigenvalue, and one group generator whose eigenvalue is m + s , for integer m and real constant parameter s . For the three groups that contain O(2), the solutions split into two inequivalent representations, one associated with s = 0, from which we recover the familiar description of the oscillator as a product of one-dimensional solutions, and the other with s > 0 (in three dimensions, solutions are found for s = 0 and s = 1/2) whose solutions are non-separable in Cartesian coordinates, and are hence overlooked by the standard Fock space approach. The O(1,1) solutions are singlet states, restricted to zero eigenvalue of the symmetry operator, which represents the boost, not angular momentum. For O(2), a single set of creation and annihilation operators forms a ladder representation for the allowed oscillator states for any s, and the degeneracy of energy states is always finite. However, in three dimensions, the integer and half-integer eigenstates are qualitatively different: the former can be expressed as finite dimensional irreducible tensors under O(3) or O(2,1) while the latter exhibit infinite degeneracy. Creation operators that produce the allowed integer states by acting on the non-degenerate ground state are constructed as irreducible tensor products of the fundamental vector representation. However, the half-integer eigenstates are infinite-dimensional, as expected for the non-compact O(2,1), and no Fock space description is found for this case. For all s ≠ 0 solutions, the SU(N) symmetry of the harmonic oscillator Hamiltonian recently discovered by Bars is spontaneously broken by the ground state. The connection of this symmetry breaking to the non-separability into one-dimensional Cartesian solutions is demonstrated. We conclude with a discussion of the O(2,1) solutions, and the problem of timelike ghost excitations.
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
We study the quantum mechanical harmonic oscillator in two and three dimensions, with particular attention to the solutions as represents of their respective symmetry groups: O(2), O(3), and O(2,1). Solving the Schrodinger equation by separating variables in polar coordinates, we obtain wavefunctions characterized by a principal quantum number, the group Casimir eigenvalue, and one observable component of orbital angular momentum, with eigenvalue m+sm+sm+s, for integer mmm and real constant parameter sss. In each symmetry group, sss splits the solutions into two inequivalent representations, one associated with s=0s=0s=0, which recovers the familiar description of the oscillator as a product of one-dimensional solutions, and the other with s>0s>0s>0 (in three dimensions, s=0,1/2s=0, 1/2s=0,1/2) whose solutions are non-separable in Cartesian coordinates, and are hence overlooked by the standard Fock space approach. In two dimensions, a single set of creation and annihilation operators forms a ladder representation for the allowed oscillator states for any sss, and the degeneracy of energy states is always finite. However, in three dimensions, the integer and half-integer eigenstates are qualitatively different: the former can be expressed as finite dimensional irreducible tensors under O(3) or O(2,1), and a ladder representation can be constructed via irreducible tensor products of the vector creation operator multiplet, while the latter exhibit infinite degeneracy and the finite-dimensional ladder representation fails for these states. These results are closely connected to the breaking of a unitary symmetry of the harmonic oscillator Hamiltonian recently discussed by Bars.
The angular wave functions for a hydrogen atom are well known to be spherical harmonics, and are obtained as the solutions of a partial differential equation. However, the differential operator is given by the Casimir operator of the SU(2)SU(2)SU(2) algebra and its eigenvalue l(l+1)ℏ2l(l+1) \hbar^2l(l+1)ℏ2, where lll is non-negative integer, is easily obtained by an algebraic method. Therefore the shape of the wave function may also be obtained by extending the algebraic method. In this paper, we describe the method and show that wave functions with different quantum numbers are connected by a rotational group in the cases of l=0l=0l=0, 1 and 2.