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 Caldeira–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

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