Method of Hydrodynamic Images and Quantum Calculus in Fock-Bargmann Representation of Quantum States

TL;DR

This paper introduces a novel classical hydrodynamics approach to quantum states in Fock space, using conformal mapping and complex potentials to interpret wave function zeros as point vortices, linking quantum states with fluid dynamics and q-calculus.

Contribution

It presents a new hydrodynamic framework for quantum states in Fock-Bargmann space, incorporating conformal mapping, complex potentials, and q-calculus to describe quantum vortices and coherent states.

Findings

Wave function zeros as point vortices in fluid flow

Representation of quantum states using Jackson q-exponential functions

Description of q-deformed quantum oscillator states in fluid dynamics

Abstract

We propose a new approach to quantum states in Fock space in terms of classical hydrodynamics. By conformal mapping of complex analytic function, representing the wave function of quantum states in Fock-Bargmann representation, we define the complex potential, describing these quantum states by incompressible and irrotational classical hydrodynamic flow. In our approach, zeros of the wave function appear as a set of point vortices (sources) in plane with the same strength, allowing interpretation of them as images in a bounded domain. For the cat states we find fluid representation as descriptive of a point source in the oblique strip domain, with infinite number of periodically distributed images. For the annular domain, the infinite set of images is described by Jackson $q$-exponential functions. We show that these functions represent the wave functions of quantum coherent states of the $q$-deformed quantum oscillator in q-Fock-Bargmann representation and describe the infinite set of point vortices, distributed in geometric progression.