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Theoretical and Mathematical Physics

On two direct limits relating pseudo-Jacobi polynomials to Hermite polynomials and the pseudo-Jacobi oscillator in a homogeneous gravitational field

Sh. M. Nagiyev

Published Jan 1, 2022• 6 citations
Publisher

Abstract

We present two new limit relations that reduce the orthogonal pseudo-Jacobi polynomials directly to the Hermite polynomials with shifted and nonshifted arguments. The proofs of these limit relations are based on the method of mathematical induction. These limits open up the prospects for studying new exactly solvable harmonic oscillator models in homogeneous external fields in quantum mechanics in terms of pseudo-Jacobi polynomials. As an application of these limit relations, a model of a linear harmonic oscillator with a position-dependent mass in an external homogeneous gravitational field (a pseudo-Jacobi oscillator in an external field) is considered. The form of the generalized Hamiltonian for describing quantum mechanical systems with a position-dependent mass is presented.

DOI
10.1134/s0040577922010093

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We present two new limit relations that reduce the orthogonal pseudo-Jacobi polynomials directly to the Hermite polynomials with shifted and nonshifted arguments. The proofs of these limit relations are based on the method of mathematical induction. These limits open up the prospects for studying new exactly solvable harmonic oscillator models in homogeneous external fields in quantum mechanics in terms of pseudo-Jacobi polynomials. As an application of these limit relations, a model of a linear harmonic oscillator with a position-dependent mass in an external homogeneous gravitational field (a pseudo-Jacobi oscillator in an external field) is considered. The form of the generalized Hamiltonian for describing quantum mechanical systems with a position-dependent mass is presented.
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.
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Abstract In the present work we use a point transformation defined in the extended configuration space to show that, in both classical and quantum formalisms, the evolution of a particle under the influence of a constant gravitational field can be related to the evolution of the harmonic oscillator. That is, in the classical mechanics regime, we show that this transformation allows us to establish the connection in the Newtonian, Lagrangian, Hamilton-Jacobi and Hamiltonian formalisms. In the quantum mechanics regime, the transformation determines a relationship between the wave functions, solutions to the corresponding Schrödinger equations, the propagators and the Wigner functions. Furthermore, by using the quantum potential approach to quantum mechanics, we show that the classical point transformation, in a natural manner, induces a quantum point transformation in the corresponding extended quantum configuration space, that determines the connection between the corresponding Madelung-Bohm trajectories. We find that the quantum transformation has the same functional form as the classical one even though the quantum or Madelung-Bohm trajectories are in general totally different from the classical ones. Then, we show that the general results can also be applied to the corresponding paraxial wave equations, and by using the stationary solutions of the quantum Harmonic oscillator and the Airy solution obtained by Berry and Balazs, as two applications, we determine the evolution of the Hermite-Gaussian and Airy beams in free space, in a linear and in a quadratic optical media. Finally, by using the point transformation, we construct two metrics whose geodesics correspond precisely to the classical trajectories for the harmonic oscillator and a particle in a uniform gravitational field.
We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of uncoupled oscillators. This motivates us to propose a generalized Pais-Uhlenbeck Hamiltonian in terms of the usual harmonic oscillator creation and annihilation operators by including terms quadratic and linear in them. Such a Hamiltonian turns out to be essentially non-Hermitian but has an equivalent Hermitian representation which is reducible to a typically position-dependent reduced mass form.