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The 2D free particle in the phase space quantum mechanics

Hubert Jóźwiak, Jaromir Tosiek

TL;DR

This work develops a concrete phase-space treatment of a 2D free quantum particle by applying a Fedosov-adapted Moyal product in carefully chosen Darboux coordinates $(T,\chi,H,L)$. It derives and solves the $\ ext{star}$-eigenvalue equations for two commuting observables sets—Cartesian momentum components and the energy–angular-momentum pair—yielding explicit Wigner eigenfunctions $W_{Em}$ with a positivity constraint that enforces integer angular quantum numbers $m$. It also constructs cross-Wigner functions $W_{Emm'}$ and shows how a fixed Cartesian-momentum Wigner function can be expanded in this energy–angular-momentum basis, $W_{p_{x0}p_{y0}} = \frac{1}{2\pi}\sum_{m,m'} e^{i(-m+m')\chi_0} W_{Emm'}$, providing a phase-space analogue of the Jacobi–Anger expansion. Overall, the paper clarifies how two natural sets of commuting observables for a free particle in the plane can be represented and interrelated within phase-space quantum mechanics, with implications for state decompositions and semiclassical analyses.

Abstract

The Wigner eigenfunctions of a free quantum particle propagating on a plane are derived. Two possibilities are analysed. First, the particle with fixed components of the Cartesian momentum is considered. Then the particle of given energy and angular momentum is discussed. In that second case, a special choice of coordinates on the symplectic space $(\mathbb{R}^{4},\,ω)$ suitable for the representation of eigenstates of the discussed particle is presented. Further, the Moyal $\star_{(\text{M})}$-product on the phase space is derived with the use of the Fedosov algorithm adapted to these coordinates on a flat phase space. Next, the eigenvalue equations for the Wigner eigenfunction are solved and the physically acceptable solutions are identified. Finally, a relationship between the Wigner eigenfunction of the particle with the fixed components of the Cartesian momentum and the cross-Wigner functions of the particle with the given energy and angular momentum is found.

The 2D free particle in the phase space quantum mechanics

TL;DR

This work develops a concrete phase-space treatment of a 2D free quantum particle by applying a Fedosov-adapted Moyal product in carefully chosen Darboux coordinates . It derives and solves the -eigenvalue equations for two commuting observables sets—Cartesian momentum components and the energy–angular-momentum pair—yielding explicit Wigner eigenfunctions with a positivity constraint that enforces integer angular quantum numbers . It also constructs cross-Wigner functions and shows how a fixed Cartesian-momentum Wigner function can be expanded in this energy–angular-momentum basis, , providing a phase-space analogue of the Jacobi–Anger expansion. Overall, the paper clarifies how two natural sets of commuting observables for a free particle in the plane can be represented and interrelated within phase-space quantum mechanics, with implications for state decompositions and semiclassical analyses.

Abstract

The Wigner eigenfunctions of a free quantum particle propagating on a plane are derived. Two possibilities are analysed. First, the particle with fixed components of the Cartesian momentum is considered. Then the particle of given energy and angular momentum is discussed. In that second case, a special choice of coordinates on the symplectic space suitable for the representation of eigenstates of the discussed particle is presented. Further, the Moyal -product on the phase space is derived with the use of the Fedosov algorithm adapted to these coordinates on a flat phase space. Next, the eigenvalue equations for the Wigner eigenfunction are solved and the physically acceptable solutions are identified. Finally, a relationship between the Wigner eigenfunction of the particle with the fixed components of the Cartesian momentum and the cross-Wigner functions of the particle with the given energy and angular momentum is found.
Paper Structure (6 sections, 107 equations, 2 figures)

This paper contains 6 sections, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Wigner eigenfunctions of Cartesian momentum and energy–angular momentum
  3. The Wigner eigenfunction of Cartesian momentum expanded in the cross-Wigner functions of energy-angular momentum
  4. An adaptation of the construction of the Fedosov star product to symplectic flat manifolds
  5. Calculation of the Moyal product of functions $(W_{Emm'}\star_{(\text{M})}W_{\widetilde{E}\widetilde{m}\widetilde{m}'})(T,\,\chi,\,H,\,L)$
  6. Calculation of the Moyal product of Wigner functions $(W_{Emm'}\star_{(\text{M})}W_{\widetilde{p}_{x0}\widetilde{p}_{y0}})(T,\,\chi,\,H,\,L)$

Figures (2)

  • Figure 1: The numerical integral $P_{Em}(r,\,\phi)$, given by the formula (\ref{['Pmk01']}), as a function of radius $r$ for energy $E = 100\,\mathrm{eV}$, mass $M = 9\cdot 10^{-31}\,\mathrm{kg}$ and three non-integer quantum magnetic numbers $m$. The normalization constant $N_{Em}$ is chosen as (\ref{['NEm']}).
  • Figure 2: The Wigner eigenfunction $W_{Em}(T,\,\chi,\,H,\,L)$, given by the formula (\ref{['WmE-final']}) for energy $E = 100\,\mathrm{eV}$, mass $M = 9\cdot 10^{-31}\,\mathrm{kg}$ and magnetic quantum number (a) $m=0$, (b) $m=5$. The normalization constant $N_{Em}$ equals (\ref{['NEm']}).