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arXiv:quant-ph/0503187

Existence of Time Operator for a Singular Harmonic Oscillator

Vesna Mikuta-Martinis, Martinis, M.

Published Mar 23, 2005• 2 citations
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Abstract

The time operator for a quantum singular oscillator of the Calogero-Sutherland type is constructed in terms of the generators of the SU(1,1) group. In the space spanned by the eigenstates of the Hamiltonian, the time operator is not self-adjoint. We show, that the time-energy uncertainty relation can be given the meaning within the Barut-Girardello coherent states defined for the singular oscillator.We have also shown the relationship with the time-of-arrival operator of Aharonov and Bohm.

DOI
10.48550/arxiv.quant-ph/0503187

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The time operator for a quantum singular oscillator of the Calogero-Sutherland type is constructed in terms of the generators of the SU(1,1) group. In the space spanned by the eigenstates of the Hamiltonian, the time operator is not self-adjoint. We show, that the time-energy uncertainty relation can be given the meaning within the Barut-Girardello coherent states defined for the singular oscillator.We have also shown the relationship with the time-of-arrival operator of Aharonov and Bohm.
The problem of existence of a self-adjoint time operator conjugate to a Hamiltonian with SU(1,1) dynamical symmetry is investigated. In the space spanned by the eigenstates of the generator K3K_3K3​ of the SU(1,1) group, the time operator for the quantum singular harmonic potential of the form ω2x2+g/x2ω^2x2 + g/x2ω2x2+g/x2 is constructed explicitly, and shown that it is related to the time-of-arrival operator of Aharonov and Bohm. Our construction is fully algebraic, involving only the generators of the SU(1,1) group.
For a time-dependent harmonic oscillator with an inverse squared singular term, we find the generalized invariant using the Lie algebra of SU(2)SU(2)SU(2) and construct the number-type eigenstates and the coherent states using the spectrum-generating Lie algebra of SU(1,1)SU(1,1)SU(1,1). We obtain the evolution operator in both of the Lie algebras. The number-type eigenstates and the coherent states are constructed group-theoretically for both the time-independent and the time-dependent harmonic oscillators with the singular term. It is shown that the squeeze operator transforms unitarily the time-dependent basis of the spectrum-generating Lie algebra of SU(1,1)SU(1,1)SU(1,1) for the generalized invariant, and thereby evolves the initial vacuum into a final coherent vacuum.
The Perelomov and the Barut–Girardello SU(1, 1) coherent states for harmonic oscillator in one-dimensional half space are constructed. Results show that the uncertainty products ΔxΔp for these two coherent states are bound from below [Formula: see text] that is the uncertainty for the ground state, and the mean values for position x and momentum p in classical limit go over to their classical quantities respectively. In classical limit, the uncertainty given by Perelomov coherent does not vanish, and the Barut–Girardello coherent state reveals a node structure when positioning closest to the boundary x = 0 which has not been observed in coherent states for other systems.
We study self-adjoint operators defined by factorizing second order differential operators in first order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum mechanical models like the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero-Sutherland type. 1. Introduction. In this paper we define and study peculiar self-adjoint extensions of simple differential operators like the one defining the quantum mechanical harmonic oscillator. We also find the explicit solution of some of the resulting quantum mechanical models and discuss their physical interpretations. These examples provide amusing illustrations that the
We study self-adjoint operators defined by factorizing second-order differential operators in first-order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum-mechanical models such as the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero–Sutherland type.