# Quantum mechanics on phase space and the Coulomb potential

Research paper by P. Campos, M.G.R. Martins, J.D.M. Vianna

Indexed on: 09 Mar '17Published on: 06 Feb '17Published in: Physics Letters A

#### Abstract

Symplectic quantum mechanics (SMQ) makes possible to derive the Wigner function without the use of the Liouville–von Neumann equation. In this formulation of the quantum theory the Galilei Lie algebra is constructed using the Weyl (or star) product with <img height="20" border="0" style="vertical-align:bottom" width="227" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0375960117301226-si1.gif">Qˆ=q⋆=q+iħ2∂p,Pˆ=p⋆=p−iħ2∂q, and the Schrödinger equation is rewritten in phase space; in consequence physical applications involving the Coulomb potential present some specific difficulties. Within this context, in order to treat the Schrödinger equation in phase space, a procedure based on the Levi-Civita (or Bohlin) transformation is presented and applied to two-dimensional (2D) hydrogen atom. Amplitudes of probability in phase space and the correspondent Wigner quasi-distribution functions are derived and discussed.