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

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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.

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Data sharing is not applicable to this article as no new data were created or analysed in this study.

Notes

  1. We shall be considering NC phase space in our work. However, we shall generically refer this as NC space.

  2. The polar angle 𝜃 should not be confused with the time dependent NC parameter 𝜃(t).

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Acknowledgements

MD would like to thank Ms. Riddhi Chatterjee and Ms.Rituparna Mandal for their helpful assistance to operate the software Mathematica.

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Authors and Affiliations

  1. Department of Theoretical Sciences, S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata, 700106, India

    Manjari Dutta & Sunandan Gangopadhyay

  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

    Shreemoyee Ganguly

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  1. Manjari Dutta
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  2. Shreemoyee Ganguly
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  3. Sunandan Gangopadhyay
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Correspondence to Sunandan Gangopadhyay.

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Dutta, M., Ganguly, S. & Gangopadhyay, S. Exact Solutions of a Damped Harmonic Oscillator in a Time Dependent Noncommutative Space. Int J Theor Phys 59, 3852–3875 (2020). https://doi.org/10.1007/s10773-020-04637-4

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