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

TL;DR

This paper explores how the center-of-mass tomogram of a large quantum system approaches a classical distribution using the central limit theorem, highlighting the roles of system size, energy, and Planck's constant.

Contribution

It demonstrates the conditions under which the center-of-mass tomogram converges to a Gaussian distribution and connects quantum to classical limits via the central limit theorem.

Findings

The distribution becomes Gaussian for large N with constant energy.

As Planck's constant approaches zero, the distribution concentrates at zero.

The approach links quantum states to classical probability distributions.

Abstract

We investigate the dependence of the center-of-mass tomogram of a system with many degrees of freedom $N$ on the Planck constant $\hbar $. It is shown that to use the central limit theorem under taking the limit $N\to +\infty $ 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 $\hbar \to 0$ we get $\delta $-function associated with the distribution concentrated in zero with the probability equal to one.