Harmonic Oscillator States with Non-Integer Orbital Angular Momentum
| dc.creator | Land, Martin | |
| dc.date | 2009-03-10 | |
| dc.date | 2009-03-27 | |
| dc.date.accessioned | 2026-07-07T12:56:55Z | |
| dc.date.available | 2026-07-07T12:56:55Z | |
| dc.description | 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. | |
| dc.description | 33 pages (based on talk given at IARD 2008) | |
| dc.identifier | https://arxiv.org/abs/0903.1757 | |
| dc.identifier | http://arxiv.org/abs/0903.1757 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/224751 | |
| dc.subject | Mathematical Physics | |
| dc.subject | High Energy Physics - Theory | |
| dc.title | Harmonic Oscillator States with Non-Integer Orbital Angular Momentum | |
| dc.type | text |