# Quantum analogue of energy equipartition theorem

One of the fundamental laws of classical statistical physics is the energy equipartition theorem

which states that for each degree of freedom the mean kinetic energy E k equals ##IMG##

[http://ej.iop.org/images/1751-8121/52/15/15LT01/aab03f2ieqn001.gif] , where ##IMG##

[http://ej.iop.org/images/1751-8121/52/15/15LT01/aab03f2ieqn002.gif] is the Boltzmann constant and T

is the temperature of the system. Despite the fact that quantum mechanics has already been developed

for more than 100 years, still there is no quantum counterpart of this theorem. We attempt to fill

this far-reaching gap and consider the simplest system, i.e. the Caldeiraâ€“Leggett model for a free

quantum Brownian particle in contact with a thermostat consisting of an infinite number of harmonic

oscillators. We prove that the mean kinetic energy E k of the Brownian particle equals the mean

kinetic energy