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arXiv:2001.06764 · Journal of Physics Conference Series

Nonstationary Deformed Singular Oscillator: Quantum Invariants and the Factorization Method

Kevin Zelaya

Published Apr 1, 2020• 5 citations
arXivHTMLPDF

Abstract

Abstract New families of time-dependent potentials related with the stationary singular oscillator are introduced. This is achieved after noticing that a nonstationary quantum invariant can be constructed for the singular oscillator. Such a quantum invariant depends on coefficients related to solutions of the Ermakov equation, where the latter guarantees the regularity of the solutions at each time. In this form, after applying the factorization method to the quantum invariant rather than to the Hamiltonian, one manages to introduce the time parameter into the transformation, leading to factorized operators that become the constants of motion for the new time-dependent Hamiltonians. At the appropriate limit, the initial quantum invariant reproduces the stationary singular oscillator Hamiltonian. Some families of stationary potentials already reported by other authors are also recovered as particular cases. A striking feature of the method is that the singular barrier of the potential can be managed to vanish, which leads to non-singular time-dependent potentials.

DOI
10.1088/1742-6596/1540/1/012017

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Abstract New families of time-dependent potentials related with the stationary singular oscillator are introduced. This is achieved after noticing that a nonstationary quantum invariant can be constructed for the singular oscillator. Such a quantum invariant depends on coefficients related to solutions of the Ermakov equation, where the latter guarantees the regularity of the solutions at each time. In this form, after applying the factorization method to the quantum invariant rather than to the Hamiltonian, one manages to introduce the time parameter into the transformation, leading to factorized operators that become the constants of motion for the new time-dependent Hamiltonians. At the appropriate limit, the initial quantum invariant reproduces the stationary singular oscillator Hamiltonian. Some families of stationary potentials already reported by other authors are also recovered as particular cases. A striking feature of the method is that the singular barrier of the potential can be managed to vanish, which leads to non-singular time-dependent potentials.
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Quantum features of a dynamical system subjected to time-dependent non-central potentials are investigated. The entire potential of the system is composed of the inverse quadratic potential and the Coulomb potential. An invariant operator that enables us to treat the time-dependent Hamiltonian system in view of quantum mechanics is introduced in order to derive Schrödinger solutions (wave functions) of the system. To simplify the problem, the invariant operator is transformed to a simple form by unitary transformation. Quantum solutions in the transformed system are easily obtained because the transformed invariant operator is a time-independent simple one. The Nikiforov-Uvarov method is used for solving eigenvalue equation of the transformed invariant operator. The double ring-shaped generalized non-central time-dependent potential is considered as a particular case for further study. From inverse transformation of quantum solutions obtained in the transformed system, the complete quantum solutions in the original system are identified. The quantum properties of the system are addressed on the basis of the wave functions.
Quantum solutions of a time-dependent Hamiltonian for the motion of a time-varying mass subjected to time-dependent singular potentials in three dimensions are investigated. A time-dependent inverse quadratic potential and a Coulomb-like potential are considered as the components of the singular potential of the system. Because the Hamiltonian is a function of time, special techniques for deriving quantum solutions of the system are necessary. A quadratic invariant operator is introduced, and its eigenstates are derived using the Nikiforov-Uvarov method together with a unitary transformation method. The Nikiforov-Uvarov method enables us to solve the eigenvalue equations of the invariant operator, which are second-order linear diffierential equations, by reducing the original equation to a hypergeometric type. According to the invariant operator theory, the wave functions of the system are represented in terms of the eigenstates obtained in such a way. The difference of the wave functions from the eigenstates of the invariant operator is that the wave functions have time-dependent phases while the eigenstates do not. By determining the phases of the wave functions via the help of the Schr¨odinger equation, we identify the full wave functions of the system and address their physical implications.
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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.
The harmonic oscillator with a time-dependent frequency has a family of linear quantum invariants for the time-dependent Schrödinger equation, which are determined by any two independent solutions to the classical equation of motion. Ermakov and Pinney have shown that a general solution to the time-dependent oscillator with an inverse cubic term can be expressed in terms of two independent solutions to the time-dependent oscillator. We explore the connection between linear quantum invariants and the Ermakov-Pinney solution for the time-dependent harmonic oscillator. We advance a novel method to construct Ermakov-Pinney solutions to a class of time-dependent oscillators and the wave functions for the time-dependent Schrödinger equation. We further show that the first and the second Pöschl-Teller potentials belong to a special class of exact time-dependent oscillators. A perturbation method is proposed for any slowly-varying time-dependent frequency.