arXiv:quant-ph/0211118

Time Operator for a Quantum Singular Oscillator

M. Martinis, Vesna Mikuta-Martinis

Published Nov 19, 2002• 1 citation

Abstract

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 $K_3$ of the SU(1,1) group, the time operator for the quantum singular harmonic potential of the form $ω^2x2 + g/x2$ 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.

DOI: 10.48550/arxiv.quant-ph/0211118

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arXiv:quant-ph/0211118

Time Operator for a Quantum Singular Oscillator

M. Martinis, Vesna Mikuta-Martinis

Published Nov 19, 2002• 1 citation

arXiv HTML PDF

Abstract

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

K_3

of the SU(1,1) group, the time operator for the quantum singular harmonic potential of the form

ω^2x2 + 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.

DOI: 10.48550/arxiv.quant-ph/0211118

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Existence of Time Operator for a Singular Harmonic Oscillator

Vesna Mikuta-Martinis, Martinis, M.

arXiv:quant-ph/0503187Mar 23, 20052 citations

Time operator for a quantum many-body system with SU(1,1) dynamical symmetry

Ivan Andrić, M. Martinis

The time operator canonically conjugated to the Hamiltonian of $N$ interacting particles on the line is constructed using SU(1,1) as a dynamical symmetry. This hidden conformal symmetry enables us to make a group theoretic analysis of the time operator in terms of SU(1,1) generators. At distances very far from the interacting region the time operator is represented as a generalization of the quantum "time-of-arrival" operator.The time operator canonically conjugated to the Hamiltonian of $N$ interacting particles on the line is constructed using SU(1,1) as a dynamical symmetry. This hidden conformal symmetry enables us to make a group theoretic analysis of the time operator in terms of SU(1,1) generators. At distances very far from the interacting region the time operator is represented as a generalization of the quantum "time-of-arrival" operator.

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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

K_3

of the SU(1,1) group, the time operator for the quantum singular harmonic potential of the form

ω^2x2 + 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.

The time operator canonically conjugated to the Hamiltonian of

N

interacting particles on the line is constructed using SU(1,1) as a dynamical symmetry. This hidden conformal symmetry enables us to make a group theoretic analysis of the time operator in terms of SU(1,1) generators. At distances very far from the interacting region the time operator is represented as a generalization of the quantum "time-of-arrival" operator.

For a time-dependent harmonic oscillator with an inverse squared singular term, we find the generalized invariant using the Lie algebra of

SU(2)

and construct the number-type eigenstates and the coherent states using the spectrum-generating Lie algebra of

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)

for the generalized invariant, and thereby evolves the initial vacuum into a final coherent vacuum.

Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of

q

and the eigenstates of

p

. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.