  In this paper, we formulate and study the quantum counterpart of the energy
equipartition theorem for a charged quantum particle moving in a harmonic
potential in the presence of a uniform external magnetic field and linearly
coupled to a passive quantum heat bath through coordinate variables. The bath
is modelled as a collection of independent quantum harmonic oscillators. We
derive the closed form expressions for the mean kinetic and potential energies
of 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 and
potential energies of individual thermostat oscillators. The net averaging is
two-fold, the first one being over the Gibbs canonical state for the
thermostat, giving $\mathcal{E}_k$ and $\mathcal{E}_p$ and the second one
denoted by $\langle . \rangle$ being over the frequencies $\omega$ of the bath
oscillators which contribute to $E_k$ and $E_p$ according to probability
distributions $\mathcal{P}_k(\omega)$ and $\mathcal{P}_p(\omega)$ respectively.
The relationship of the present quantum version of the equipartition theorem
with that of the fluctuation-dissipation theorem (within the linear-response
theory framework) is also explored. Further, we investigate the influence of
the external magnetic field and the effect of different dissipation processes
through Gaussian decay, Drude and radiation bath spectral density functions, on
the typical properties of $\mathcal{P}_k(\omega)$ and $\mathcal{P}_p(\omega)$.
Finally, the role of system-bath coupling strength and the memory effect is
analyzed in the context of average kinetic and potential energies of the
dissipative charged magneto-oscillator.
