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In this paper, we formulate and study the quantum counterpart of the energyequipartition theorem for a charged quantum particle moving in a harmonicpotential in the presence of a uniform external magnetic field and linearlycoupled to a passive quantum heat bath through coordinate variables. The bathis modelled as a collection of independent quantum harmonic oscillators. Wederive the closed form expressions for the mean kinetic and potential energiesof the charged-dissipative-magneto-oscillator in the form $E_k = \langle\mathcal{E}_k \rangle$ and $E_p = \langle \mathcal{E}_p \rangle$ respectively,where $\mathcal{E}_k$ and $\mathcal{E}_p$ denote the average kinetic andpotential energies of individual thermostat oscillators. The net averaging istwo-fold, the first one being over the Gibbs canonical state for thethermostat, giving $\mathcal{E}_k$ and $\mathcal{E}_p$ and the second onedenoted by $\langle . \rangle$ being over the frequencies $\omega$ of the bathoscillators which contribute to $E_k$ and $E_p$ according to probabilitydistributions $\mathcal{P}_k(\omega)$ and $\mathcal{P}_p(\omega)$ respectively.The relationship of the present quantum version of the equipartition theoremwith that of the fluctuation-dissipation theorem (within the linear-responsetheory framework) is also explored. Further, we investigate the influence ofthe external magnetic field and the effect of different dissipation processesthrough Gaussian decay, Drude and radiation bath spectral density functions, onthe typical properties of $\mathcal{P}_k(\omega)$ and $\mathcal{P}_p(\omega)$.Finally, the role of system-bath coupling strength and the memory effect isanalyzed in the context of average kinetic and potential energies of thedissipative charged magneto-oscillator.

Quantum counterpart of energy equipartition theorem for a dissipative charged magneto-oscillator: Effect of dissipation, memory, and magnetic field