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Resenhas do Instituto de Matemática e Estatística da Universidade de São Paulo

Schrõdinger Equation in Phase Space, Irreducible Representations of the Heisenberg Group, and Deformation Quantization

Maurice A. de Gosson

Published Mar 18, 2005• 1 citation
Publisher
DOI
10.11606/resimeusp.v6i4.75425

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A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation
The hydrogenic Schrödinger equation may be transformed into an integral equation on the group space of SO(3). The solutions of this integral equation are the Wigner functions Djmn, the matrix elements of the irreducible representations of SO(3). In this article we examine a q-deformation of the Coulomb problem for which the new wave functions are the matrix elements of the irreducible representations of the quantum group SUq(2). The new integral equation is formulated in terms of the Woronowicz integral. The eigenvalues are given by a modified Balmer formula which lifts the Coulomb degeneracy and replaces n by [n].