Hydrodynamic interpretation of generic squeezed coherent states: A kinetic theory

TL;DR

This paper explores the hydrodynamic interpretation of quantum mechanics for squeezed coherent states, deriving phase space distributions, entropy, and quantum thermodynamic quantities, and linking quantum and classical dynamics through kinetic theory.

Contribution

It introduces a kinetic theory framework for squeezed coherent states, connecting quantum statistical concepts with classical phase space dynamics and deriving new quantum thermodynamic relations.

Findings

Exact Fokker-Planck equations for phase space distributions

Decomposition of entropy into position and momentum contributions

Identification of quantum pressure, temperature, and internal energy

Abstract

The hydrodynamic interpretation of quantum mechanics treats a system of particles in an effective manner. In this work, we investigate squeezed coherent states within the hydrodynamic interpretation. The Hamiltonian operator in question is time dependent, n-dimensional and in quadratic order. We start by deriving a phase space Wigner probability distribution and an associated equilibrium entropy for the squeezed coherent states. Then, we decompose the joint phase space distribution into two portions: a marginal position distribution and a momentum distribution that is conditioned on the post-selection of positions. Our conditionally averaged momenta are shown to be equal to the Bohm's momenta whose connection to the weak measurements is already known. We also keep track of the corresponding classical system evolution by identifying shear, magnification and rotation components of the symplectic phase space dynamics. This allows us to pinpoint which portion of the underlying classical motion appears in which quantum statistical concept. We show that our probability distributions satisfy the Fokker-Planck equations exactly. They can be used to decompose the equilibrium entropy into the missing information in positions and in momenta as in the Sackur-Tetrode entropy of the classical kinetic theory. Eventually, we define a quantum pressure, a quantum temperature and a quantum internal energy which are related to each other in the same fashion as in the classical kinetic theory. We show that the quantum potential incorporates the kinetic part of the internal energy and the fluctuations around it. This allows us to suggest a quantum conditional virial relation. In the end, we show that the kinetic internal energy is linked to the fractional Fourier transformer part of the underlying classical dynamics similar to the case where the energy of a quantum oscillator is linked to its Maslov index.