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Theoretical and Mathematical Physics

Relativistic linear oscillator under the action of a constant external force. Transition amplitudes and the Green’s function

Sh. M. Nagiyev, R. M. Mir‐Kasimov

Published Jan 1, 2023• 2 citations
Publisher

Abstract

We discuss an exactly solvable relativistic model of a nonrelativistic linear harmonic oscillator in the presence of a constant external force. We show that as in the nonrelativistic case, the relativistic linear oscillator in an external uniform field is unitarily equivalent to the oscillator without this field. Using two methods, we calculate transition amplitudes between energy states of the discrete spectrum of the relativistic linear oscillator under the action of a suddenly applied uniform field. We find Barut–Girardello coherent states and the Green’s function in the coordinate and momentum representations. We obtain the linear and bilinear generating functions for the Meixner–Pollaczek polynomials. We prove that the relativistic wave functions, the generators of the dynamical symmetry group, and the transition amplitudes have the correct nonrelativistic limit.

DOI
10.1134/s004057792301004x

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The investigation of the generalized coherent states for oscillator-like systems connected with given family of orthogonal polynomials is continued. In this work we consider oscillators connected with Meixner and Meixner-Pollaczek polynomials and define generalized coherent states for these oscillators. The completeness condition for these states is proved by the solution of the related classical moment problem. The results are compared with the other authors ones. In particular, we show that the Hamiltonian of the relativistic model of linear harmonic oscillator can be thought of as the linearization of the quadratic Hamiltonian which naturally arised in our formalism.
We construct approximate analytic solutions of the Logunov–Tavkhelidze equation in the case of a potential that, in the one-dimensional relativistic configuration representation, has the form analogous to the potential of the nonrelativistic harmonic oscillator in the coordinate representation. The wave functions are obtained in both the momentum and relativistic configuration representations. The approximate values of the energy of the relativistic harmonic oscillator are the roots of transcendental equations. The wave functions in the relativistic configuration representation have additional zeros in comparison with the wave functions of the corresponding states of the nonrelativistic harmonic oscillator in the coordinate representation.