On the conection between the Liouville equation and the Schrodinger equation

According to the widely accepted notion, the Schr{\"o}dinger equation (SE) is not derivable in principle. Contrary to this belief, we present here a straightforward derivation of SE. It is based on only two fundamentals of mechanics: the classical Hamilton-Jacobi equation(HJE) and the Planck postulate about the discrete transfer of energy at micro-scales. Our approach is drastically different from the other published derivations of SE which either employ an ad hoc underlying ...

In recent years, a surprisingly direct and simple rigorous understanding of quantum Liouville theory has developed. We aim here to make this material more accessible to physicists working on quantum field theory.

It is a classical derivation that the Wigner equation, derived from the Schr\"odinger equation that contains the quantum information, converges to the Liouville equation when the rescaled Planck constant $\epsilon\to0$. Since the latter presents the Newton's second law, the process is typically termed the (semi-)classical limit. In this paper, we study the classical limit of an inverse problem for the Schr\"odinger equation. More specifically, we show that using the initial c...

We discuss how the language of wave functions (state vectors) and associated non-commuting Hermitian operators naturally emerges from classical mechanics by applying the inverse Wigner-Weyl transform to the phase space probability distribution and observables. In this language, the Schr\"odinger equation follows from the Liouville equation, with $\hbar$ now a free parameter. Classical stationary distributions can be represented as sums over stationary states with discrete (qu...

We consider how to describe Hamiltonian mechanics in generalised probabilistic theories with the states represented as quasi-probability distributions. We give general operational definitions of energy-related concepts. We define generalised energy eigenstates as the purest stationary states. Planck's constant plays two different roles in the framework: the phase space volume taken up by a pure state and a dynamical factor. The Hamiltonian is a linear combination of generalis...

Using a simple geometrical construction based upon the linear action of the Heisenberg--Weyl group we deduce a new nonlinear Schr\"{o}dinger equation that provides an exact dynamic and energetic model of any classical system whatsoever, be it integrable, nonintegrable or chaotic. Within our model classical phase space points are represented by equivalence classes of wavefunctions that have identical position and momentum expectation values. Transport of these equivalence clas...

We discuss a family of quasi-distributions (s-ordered Wigner functions of Agarwal and Wolf) and its connection to the so called phase space representation of the Schroedinger equation. It turns out that although Wigner functions satisfy the Schroedinger equation in phase space they have completely different interpretation.

We show that the quantum wavefunction, interpreted as the probability density of finding a single non-localized quantum particle, which evolves according to classical laws of motion, is an intermediate description of a material quantum particle between the quantum and classical realms. Accordingly, classical and quantum mechanics should not be treated separately, a unified description in terms of the Wigner distribution function being possible. Although defined on classical p...

Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function $S^{(K)}(q,p,t)$ whose physical meaning is unknown. We show that a different $S(q,p,t)$, with well-defined physical meaning, may be introduced without destroying the attractive "quantum-like" mathematical features of the KvN theory. This new $S(q,p,t)$ is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which pres...

A classical theorem of Stone and von Neumann says that the Schr\"{o}dinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl ...