Exact Solutions of a Damped Harmonic Oscillator in a Time Dependent Noncommutative Space

Abstract

In this paper we have obtained the exact eigenstates of a two dimensional damped harmonic oscillator in the presence of an external magnetic field varying with respect to time in time dependent noncommutative space. It has been observed that for some specific choices of the damping factor, the time dependent frequency of the oscillator and the time dependent external magnetic field, 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. Then we 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, the time dependent frequency of the oscillator and the time dependent applied magnetic field. We also compare our results with those in the absence of the magnetic field obtained earlier.

A study of an Exotic Quantum Damped

“…We now summarize the results obtained from this study. In this paper we have used the model considered in 12 to study a two-dimensional damped harmonic oscillator in time dependent noncommutative space. We map the Hamiltonian to commutative space by using a shift of variables connecting the noncommutative and commutative space, known in literature as Bopp-shift.…”

“…The solution of the EP equation under various physically significant conditions shall be discussed later. Using this form of the invariant the form of the corresponding solution of the Hamiltonian (as seen from our previous study 12 ) is given by,…”

“…A simple method of obtaining a solution of the EP equation is provided in 12 . We first choose ρ(t) to be any arbitrary time dependent function.…”

“…The eigenfunction of the invariant operator I(t), φ n,m−n (r, θ), for this generic solution set (as deciphered from our earlier publication 12 ) is given by,…”

“…Researchers have done quantum mechanical analysis of damped harmonic oscillators extensively. In an earlier communication we had studied the quantum damped harmonic oscillator in noncommutative [NC] space 12 . By specifying NC space we indicate a space in which we have additional NC commutation relations between the position and momentum canonical coordinates of the form, [X 1 , X 2 ] = iθ and [P 1 , P 2 ] = iΩ, where θ and Ω are positive real constants.…”