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Theoretical and Mathematical Physics

Relativistic linear oscillator under the action of a constant external force. Wave functions and dynamical symmetry group

Sh. M. Nagiyev, R. M. Mir‐Kasimov

Published Sep 1, 2021• 7 citations
Publisher

Abstract

An exactly solvable relativistic model of a linear oscillator is considered in detail in the presence of a constant external force in both the momentum representation and the relativistic configuration representation. It is found that in contrast to the nonrelativistic case, depending on the magnitude of the force, both discrete and continuous energy spectra are possible. It is shown that in the case of a discrete spectrum, the wave functions in the momentum representation are expressed in terms of the Laguerre polynomials, and in the relativistic configuration representation, in terms of the Meixner–Pollaczek polynomials. Integral and differential–difference formulas are found connecting the Laguerre and Meixner–Pollaczek polynomials. A dynamical symmetry group is constructed.

DOI
10.1134/s0040577921090087

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An exactly solvable relativistic model of a linear oscillator is considered in detail in the presence of a constant external force in both the momentum representation and the relativistic configuration representation. It is found that in contrast to the nonrelativistic case, depending on the magnitude of the force, both discrete and continuous energy spectra are possible. It is shown that in the case of a discrete spectrum, the wave functions in the momentum representation are expressed in terms of the Laguerre polynomials, and in the relativistic configuration representation, in terms of the Meixner–Pollaczek polynomials. Integral and differential–difference formulas are found connecting the Laguerre and Meixner–Pollaczek polynomials. A dynamical symmetry group is constructed.
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.
Meixner oscillators have a ground state and an `energy' spectrum that is equally spaced; they are a two-parameter family of models that satisfy a Hamiltonian equation with a {\it difference} operator. Meixner oscillators include as limits and particular cases the Charlier, Kravchuk and Hermite (common quantum-mechanical) harmonic oscillators. By the Sommerfeld-Watson transformation they are also related with a relativistic model of the linear harmonic oscillator, built in terms of the Meixner-Pollaczek polynomials, and their continuous weight function. We construct explicitly the corresponding coherent states with the dynamical symmetry group Sp(2,ℜ\Reℜ). The reproducing kernel for the wavefunctions of these models is also found.
A relativistically invariant representation of the generalized momentum of a particle in an external field is proposed. In this representation, the dependence of the potentials of the interaction of the particle with the field on the particle velocity is taken into account. The exact correspondence of the expressions of energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions to the problems of mechanics with relativistic and nonrelativistic approaches. The invariance of the proposed representation of the generalized momentum makes it possible to equivalently describe a physical system in geometrically conjugate spaces of kinematic and dynamic variables. Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the constant interaction of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus is Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion, and the expression for the force acting on the charge are derived when moving in an external electromagnetic field. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), and also the problems of a hydrogen atom are given.
We investigate the quantum dynamics of a particle subjected to a linear potential using the Lewis--Riesenfeld invariant operator method. Starting from the time-dependent Schrödinger equation associated with a constant external force, we construct the most general Hermitian quadratic invariant and derive the corresponding coupled differential equations for its time-dependent coefficients. By means of an appropriate sequence of unitary transformations, the invariant operator is reduced to the form of a harmonic oscillator Hamiltonian. This reduction enables a clear classification of the system according to the sign of the conserved quantity ω2. Particular attention is devoted to the physically relevant case ω2 >0, which yields a discrete eigenspectrum. Explicit analytical expressions for the invariant coefficients, the displacement parameters, and the transformed wave functions are obtained. The resulting formalism provides an exact quantum description of a particle under a constant force and establishes a direct connection between invariant theory and harmonic oscillator quantization.
We constructed a new exactly solvable model of the non-relativistic quantum linear harmonic oscillator with the position-dependent mass Mx=2m0/eαx+1. We solved the Schrödinger equation for this model in the absence and presence of an external homogeneous field. We showed that in both cases the wave functions of the bound states are expressed through Jacobi polynomials, and the corresponding energy spectra are non-equidistant. We also showed that in the limit α→0 the wave functions and energy spectra coincide with the corresponding expressions for a linear harmonic oscillator with the constant mass m0 in the absence and presence of an external homogeneous field. We present also two new limit relations that reduce the Jacobi polynomials directly to the Hermite polynomials with shifted and non-shifted arguments. The proofs of these limit relations are based on the method of mathematical induction.