LAUSR

Abstract Symplectic quantum mechanics (SQM) considers a non-commutative algebra of functions on a phase space Γ and an associated Hilbert space H Γ , to construct a unitary representation for the Galilei group. From this unitary representation the Schrodinger equation is rewritten in phase space variables and the Wigner function can be derived without the use of the Liouville–von Neumann equation. In this article the Coulomb potential in three dimensions (3D) is resolved completely by using the phase space Schrodinger equation. The Kustaanheimo–Stiefel(KS) transformation is applied and the Coulomb and harmonic oscillator potentials are connected. In this context we determine the energy levels, the amplitude of probability in phase space and correspondent Wigner quasi-distribution functions of the 3D-hydrogen atom described by Schrodinger equation in phase space.