A New Look at the Quantum Mechanics of the Harmonic Oscillator
| dc.creator | Kastrup, H. A. | |
| dc.date | 2006-12-05 | |
| dc.date | 2007-07-02 | |
| dc.date.accessioned | 2026-07-07T11:44:35Z | |
| dc.date.available | 2026-07-07T11:44:35Z | |
| dc.description | Classically the Harmonic Oscillator (HO) is the generic example for the use of angle and action variables phi in R mod 2 pi and I > 0. But the symplectic transformation (ϕ,I) to (q,p) is singular for (q,p) = (0,0). Globally {(q,p)} has the structure of the plane R^2, but {(phi,I)} that of the punctured plane R^2 -(0,0). This implies qualitative differences for the QM of the two phase spaces: The quantizing group for the plane R^2 consists of the (centrally extended) translations generated by {q,p,1}, but the corresponding group for {(phi,I)} is SO(1,2) = Sp(2,R)/Z_2, (Sp(2,R): symplectic group of the plane), with Lie algebra basis {h_0 = I, h_1 = I cos phi, h_2 = - I sin phi}. In the QM for the (phi,I)-model the three h_j correspond to self-adjoint generators K_j, j=0,1,2, of irreducible unitary representations (positive discrete series) for SO(1,2) or one of its infinitely many covering groups, the Bargmann index k > 0 of which determines the ground state energy E (k, n=0) = hbar omega k of the (phi,I)-Hamiltonian H(K). For an m-fold covering the lowest possible value is k=1/m, which can be made arbitrarily small! The operators Q and P, now expressed as functions of the K_j, keep their usual properties, but the richer structure of the K_j quantum model of the HO is ``erased'' when passing to the simpler Q,P model! The (phi,I)-variant of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with the (Landau) levels of charged particles in magnetic fields, with the propagation of light in vacuum, passing through electric or magnetic fields. Finally it leads to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant. | |
| dc.description | 100 pages (Latex); v2: abstract simplified, remarks and references added | |
| dc.identifier | https://arxiv.org/abs/quant-ph/0612032 | |
| dc.identifier | http://arxiv.org/abs/quant-ph/0612032 | |
| dc.identifier | AnnalenPhys.16:439-528,2007 | |
| dc.identifier | doi:10.1002/andp.200610245 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/201653 | |
| dc.subject | Quantum Physics | |
| dc.subject | Astrophysics | |
| dc.subject | High Energy Physics - Theory | |
| dc.subject | Mathematical Physics | |
| dc.subject | Atomic Physics | |
| dc.title | A New Look at the Quantum Mechanics of the Harmonic Oscillator | |
| dc.type | text |