What is the energy of a quantum Brownian particle?

In this talk, I will describe the notion of ‘thermally-averaged
energy’ which can be ascribed to a quantum particle in contact with a heat
bath. Two scenarios will be considered: (a) the particle is free (no
potential), (b) the particle moves in a harmonic potential. In both these
scenarios, the notion of thermally-averaged energy will be assessed in the
strong and weak-coupling regimes by taking the heat bath to be composed of
an infinite number of independent quantum oscillators with certain
spectral distributions. The notion of energy in the strong-coupling regime
will be provided an interpretation in the language of superstatistics
which also leads us consistently to the weak-coupling results. For the
harmonic potential case, the validity of the virial theorem will be
assessed.