Exact solutions of a damped harmonic oscillator in a time dependent noncommutative space

TL;DR

This paper derives exact eigenstates and energy expectation values for a damped harmonic oscillator in a time-dependent noncommutative space, revealing how specific parameters influence quantum states and energies over time.

Contribution

It provides the first exact solutions for a damped harmonic oscillator in a noncommutative space with time-dependent parameters, linking solutions to the Ermakov-Pinney equation.

Findings

Exact eigenstates obtained for specific damping and frequency choices

Energy expectation values vary with time based on solutions to the Ermakov-Pinney equation

Analytic expressions for phase and matrix elements of coordinate operators

Abstract

In this paper we have obtained the exact eigenstates of a two dimensional damped harmonic oscillator in time dependent noncommutative space. It has been observed that for some specific choices of the damping factor and the time dependent frequency of the oscillator, there exists interesting solutions of the time dependent noncommutative parameters following from the solutions of the Ermakov-Pinney equation. Further, these solutions enable us to get exact analytic forms for the phase which relates the eigenstates of the Hamiltonian with the eigenstates of the Lewis invariant. We then obtain expressions for the matrix elements of the coordinate operators raised to a finite arbitrary power. From these general results we then compute the expectation value of the Hamiltonian. The expectation values of the energy are found to vary with time for different solutions of the Ermakov-Pinney equation corresponding to different choices of the damping factor and the time dependent frequency of the oscillator.