Ensemble in phase space: Statistical formalism of quantum mechanics

We present an alternative formalism of quantum mechanics tailored to statistical ensemble in phase space. The purpose of our work is to show that it is possible to establish an alternative autonomous formalism of quantum mechanics in phase space using statistical methodology. The adopted perspective leads to obtaining within the framework of its theory the fundamental quantum-mechanical equation without recourse to the other formulations of quantum mechanics, and gives the idea for operators pertaining to dynamical quantities. The derivation of this equation starts with the ensemble in phase space and, as a result, reproduces Liouville’s theorem and virial theorem for quantum mechanics. We have explained with the help of this equation the structure of quantum mechanics in phase space and the approximation to the Schrodinger equation. Furthermore, we have shown that this formalism provides reasonable results of quantisation such as the quantisation of harmonic oscillation, the two-slit interference and the uncertainty relation. In particular, we have demonstrated that this formalism can easily give the relativistic wave equation without using the linearisation of the Hamiltonian operator. The ultimate outcome this formalism produces is that primary and general matters of quantum mechanics can be studied reasonably within the framework of statistical mechanics.