A use of the central limit theorem to obtain a classical limit for the center-of-mass tomogram

Abstract

We investigate the dependence of the center-of-mass tomogram of a system with many degrees of freedom N on the Planck constant . It is shown that to use the central limit theorem under taking the limit N → +∞ one should keep the energy of the system to be constant. In the case, the resulting distribution is Gaussian if the initial distribution is a product of independent excited states of a quantum oscillator or even and odd coherent states either. Then, if one turns the Planck constant → 0 we get δ-function associated with the distribution concentrated in zero with the probability equal to one.