Exact Solutions of a Damped Harmonic Oscillator in a Time Dependent Noncommutative Space
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Exact Solutions of a Damped Harmonic Oscillator in a Time Dependent Noncommutative Space Manjari Dutta1 · Shreemoyee Ganguly2 · Sunandan Gangopadhyay1 Received: 8 August 2020 / Accepted: 19 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Noncommutative space · Ermakov-Pinney equation
1 Introduction The study of time dependent classical as well as quantum harmonic oscillators has appealed to theoretical physicists since time immemorial. In the literature the work by Lewis et al. [1] has lead to an upsurge of analysis of the Hamiltonian for the time dependent quantum Sunandan Gangopadhyay
[email protected]; [email protected] Manjari Dutta [email protected] Shreemoyee Ganguly [email protected] 1
Department of Theoretical Sciences, S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700106, India
2
Department of Basic Science and Humanities, University of Engineering and Management (UEM), B/5, Plot No.III, Action Area-III, Newtown, Kolkata 700156, India
International Journal of Theoretical Physics
harmonic oscillator using a class of exact invariants designed for such systems [2, 3]. The problem becomes even more fascinating when one has a system of two such oscillators in two-dimensional space. Now, in order to address practical situations one needs to include damping in the system. Although there are several studies on the one-dimensional damped quantum harmonic oscillator in the past [4–8], it’s two-dimensional equivalent is a less explored system [9]. The work by Lawson et al. [9] is one of the very few which analyses a two-dimensional damped quantum harmonic oscillator system. The solutions obtained by them for the mentioned system provides a platform to explore the construction of various coherent states with intriguing properties. In the present work we extend the study by Lawson et al. [9] and consider the twodimensional damped quantum harmonic oscillator in noncommuta
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