Exact time-dependent solution of the Schr\"odinger equation, its generalization to the phase space and relation to the Gibbs distribution

Abstract

Using the simplest but fundamental example, the problem of the infinite potential well, this paper makes an ideological attempt (supported by rigorous mathematical proofs) to approach the issue of {\guillemotleft}understanding{\guillemotright} the mechanism of quantum mechanics processes, despite the well-known examples of the EPR paradox type. The new exact solution of the Schr\"odinger equation is analyzed from the perspective of quantum mechanics in the phase space. It is the phase space, which has been extensively used recently in quantum computing, quantum informatics and communications, that is the bridge towards classical physics, where understanding of physical reality is still possible. In this paper, an interpretation of time-dependent processes of energy redistribution in a quantum system, probability waves, the temperature and entropy of a quantum system, and the transition to a time-independent {\guillemotleft}frozen state{\guillemotright} is obtained, which is understandable from the point of view of classical physics. The material of the paper clearly illustrates the solution of the problem from the standpoint of continuum mechanics, statistical physics and, of course, quantum mechanics in the phase space.