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Springer eBooks

Formal Solutions

Martin Janßen

Published Jan 1, 2016• 0 citations
Publisher
DOI
10.1007/978-3-662-49696-1_3

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The two most important equations of Wigner quantum mechanics are the Moyal and the stargenvalue equations. The latter equation is the Weyl-Wigner transform of the eigenvalue equation of standard operator quantum mechanics and, in the context of the Wigner formulation, is of identical importance. The aim of this letter is to present the formal solution of a general stargenvalue equation in arbitrary dimension, both for continuous and discrete spectra. The properties of the formal solution will be studied and a geometrical interpretation given in terms of star-hypersurfaces in quantum phase space. These results provide further insight into the mathematical structure of phase space quantum mechanics and are especially relevant for the construction of a complete formal solution of Wigner quantum mechanics in the Heisenberg picture. 1

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Abstract As the first stage in a project aimed at finding exact bound-state solutions for the few-particle Schrödinger equation, some restricted two-and three- body systems are treated by Rayleigh–Schrödinger perturbation theory. Formal solutions for the perturbation equations are obtained by using standard techniques for solving differential equations. Particular attention is given to justifying the various boundary conditions, since the same conditions apply to formal solutions of the eigenvalue equations. It is shown that boundary conditions designed to preserve the Hermitian character of the Hamiltonian are sufficient to determine all parameters in the formal solutions. The procedure is demonstrated by application to a hydrogen atom perturbed by stationary point charges. Results are given for the ground and some excited states.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Nov 8, 1982•6 citations
The two most important equations of Wigner quantum mechanics are the Moyal and the stargenvalue equations. The latter equation is the Weyl-Wigner transform of the eigenvalue equation of standard operator quantum mechanics and, in the context of the Wigner formulation, is of identical importance. The aim of this letter is to present the formal solution of a general stargenvalue equation in arbitrary dimension, both for continuous and discrete spectra. The properties of the formal solution will be studied and a geometrical interpretation given in terms of star-hypersurfaces in quantum phase space. These results provide further insight into the mathematical structure of phase space quantum mechanics and are especially relevant for the construction of a complete formal solution of Wigner quantum mechanics in the Heisenberg picture. 1
As the first stage in a project aimed at finding exact bound-state solutions for the few-particle Schrodinger equation, some restricted two- and threebody systems are treated by Rayleigh-Schrddinger perturbation theory. Formal solutions for the perturbation equations are obtained by using standard techniques for solving differential equations. Particular attention is given to justifying the various boundary conditions, since the same conditions apply to formal solutions of the eigenvalue equations. It is shown that boundary conditions designed to preserve the Hermitian character of the Hamiltonian are sufficient to determine all parameters in the formal solutions. The procedure is demonstrated by application to a hydrogen atom perturbed by stationary point charges. Results are given for the ground and some excited states.
Abstract As the first stage in a project aimed at finding exact bound-state solutions for the few-particle Schrödinger equation, some restricted two-and three- body systems are treated by Rayleigh–Schrödinger perturbation theory. Formal solutions for the perturbation equations are obtained by using standard techniques for solving differential equations. Particular attention is given to justifying the various boundary conditions, since the same conditions apply to formal solutions of the eigenvalue equations. It is shown that boundary conditions designed to preserve the Hermitian character of the Hamiltonian are sufficient to determine all parameters in the formal solutions. The procedure is demonstrated by application to a hydrogen atom perturbed by stationary point charges. Results are given for the ground and some excited states.