Harmonic Oscillator States with Non-Integer Orbital Angular Momentum

TL;DR

This paper explores harmonic oscillator states with non-integer orbital angular momentum, revealing new solutions and representations in quantum systems that extend beyond traditional integer-based angular momentum models.

Contribution

It introduces solutions with non-integer angular momentum parameter s, uncovering new representations and degeneracies in harmonic oscillators across different symmetry groups.

Findings

Solutions with non-integer s split the oscillator states into two inequivalent representations.

In 2D, a single ladder of states exists for all s, with finite degeneracy.

In 3D, integer and half-integer s states show distinct degeneracy and tensor properties.

Abstract

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 $m+s$, for integer $m$ and real constant parameter $s$. In each symmetry group, $s$ splits the solutions into two inequivalent representations, one associated with $s=0$, which recovers the familiar description of the oscillator as a product of one-dimensional solutions, and the other with $s>0$ (in three dimensions, $s=0, 1/2$) 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 $s$, 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), 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.