  We study the quantum counterpart of the theorem on energy equipartition for
classical systems. We consider a free quantum Brownian particle modelled in
terms of the Caldeira-Leggett framework: a system plus thermostat consisting of
an infinite number of harmonic oscillators. By virtue of the theorem on the
averaged kinetic energy $E_k$ of the quantum particle, it is expressed as $E_k
= \langle \mathcal E_k \rangle$, where $\mathcal E_k$ is thermal kinetic energy
of the thermostat per one degree of freedom and $\langle ...\rangle$ denotes
averaging over frequencies $\omega$ of thermostat oscillators which contribute
to $E_k$ according to the probability distribution $\mathbb P(\omega)$. We
explore the impact of various dissipation mechanisms, via the Drude, Gaussian,
algebraic and Debye spectral density functions, on the characteristic features
of $\mathbb{P}(\omega)$. The role of the system-thermostat coupling strength
and the memory time on the most probable thermostat oscillator frequency as
well as the kinetic energy $E_k$ of the Brownian particle is analysed.
