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Hadronic J.; (United States)

q-Equivalent particle Hamiltonians. III. The two-dimensional quantum oscillator

Giuseppe Marmo, Eugene J. Saletan

Published Aug 1, 1978• 3 citations
OpenAlex

Abstract

An example is given which shows how a formal ambiguity in classical mechanics can lead to a difference in substance in quantum mechanics.

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Revista Mexicana de Física•Dec 1, 2004•8 citations
We study classical and quantum dynamics of the symmetric harmonic oscillator and the symmetric bouncer defined in 2-D. For these systems we get each of them for two different constants of motion, two Lagrangians and two Hamiltonians which describe the same classical dynamics. However, the quantization of these systems (using Schrödinger equation), using their two equivalents Hamiltonian, describes different quantum dynamics for each of them. This represents an ambiguity on the Hamiltonian formulation of the Quantum Mechanics.
We study the quantum Pais-Uhlenbeck oscillator at the resonant (equal-frequency) point, where the dynamics becomes non-diagonalisable and the conventional Fock-space construction collapses. At the classical level, the degenerate system admits more than one Hamiltonian formulation generating the same equations of motion, leading to a nontrivial quantisation ambiguity. Working first in the ghostly two-dimensional Hamiltonian formulation, we construct differential intertwiners that generate a spectrum-generating algebra acting on the generalised eigenspaces of the Hamiltonian. This algebra organises the generalised eigenvectors into finite Jordan chains and closes into a hidden su(2)su(2)su(2) Lie algebra that exists only at resonance. We then show that quantising a classically equivalent Hamiltonian yields a radically different quantum theory, with a fully diagonalisable spectrum and genuine degeneracies. Our results demonstrate that the resonant Pais-Uhlenbeck oscillator provides a concrete example in which classically equivalent Hamiltonians define inequivalent quantum theories.
Apart from the total energy, the two-dimensional isotropic harmonic oscillator possesses three independent constants of motion which, with the standard symplectic structure, generates a dynamical symmetry group isomorphic to SU(2). We show that, by suitably redefining the symplectic structure, any of these three constants of motion can be used as a Hamiltonian, and that the remaining two, together with the total energy, generate a dynamical symmetry group isomorphic to SU(1,1). We also show that the standard energy levels of the quantum two-dimensional isotropic harmonic oscillator and their degeneracies are obtained making use of the appropriate representations of SU(1,1), provided that the canonical commutation relations are modified according to the new symplectic structure. Whereas in classical mechanics the different symplectic structures lead to equivalent formulations of the equations of motion, in quantum mechanics the modifications of the commutation relations should be accompanied by modifications in the interpretation of the formalism in order to obtain results equivalent to those found with the common relations.