A Phase Space Formulation of Quantum State Functions

Abstract

Allowing for virtual paths in phase space permits an extension of Hamilton’s principle of least action, of lagrangians and of hamiltonians to phase space. A subsequent canonical quantization, then, provides a framework for quantum statistical mechanics. The classical statistical mechanics and the conventional quantum mechanics emerge as special case of this formalism. Von Neumann’s density matrix may be inferred from it. Wigner’s functions and their evolution equation may also be obtained by a unitary transformation.

Symmetry Transformation in Extended Phase Space: the Harmonic Oscillator in the Husimi Representation

“…It can be easily shown that the Wigner representation can be obtained by a canonical transformation or by the corresponding unitary transformation in the EPS [11]. The same technique could be applied to obtain the Wigner equation from the Schrödinger equation in the phase space [16] showing the relation between the EPS technique and the Later one.…”

“…The canonical transformations that leaves the extended Hamilton equations form invariant are the extended canonical transformations [11]. Let us consider the following linear transformation…”

“…(See the paragraph before equation (10) and the Section 5.5 in [11].) The above Hamiltonian for f = mω changes to the Q-function Hamiltonian, H Q .…”

“…Carroll [10] has shown that there are generalized quantum theories for which the quantum potential depends on the wave function. Using the extended phase space formulation of quantum mechanics [11,12,13], Nasiri [9] has shown that in the Wigner representation of phase space quantum mechanics [14] the quantum potential is removed from the dynamical equation of a particle in linear and harmonic potentials keeping the Hamilton-Jacobi equation form invariant. It seems that the Husimi representation could be another candidate to release the quantum potential.…”

Quantum Potential and Symmetries in Extended Phase Space

“…Although, the arguments invoked on the basis of the Heisenberg uncertainty principle made the physical meaning of the "phase space points" problematic, things have changed and phase space techniques mainly formulated by the theory of deformation quantization [23] and a family of Schrödinger equations in phase space [24,25,26,27,28] are now widely accepted and used. As an approach of the latter type, Sobouti and Nasiri [29] have proposed a formulation of quantum statistical mechanics by generalizing the principle of least action to the trajectories in phase space, and a canonical quantization procedure in this space. In the extended phase space (EPS) formalism, positions and momenta are assumed to be independent c-number variables.…”