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The Journal of Chemical Physics

On coherent-state representations of quantum mechanics: Wave mechanics in phase space

Klaus B. Mo ller, Thomas Godsk Jørgensen, Gabino Torres-Vega

Published May 1, 1997• 43 citations
Publisher

Abstract

In this article we argue that the state-vector phase-space representation recently proposed by Torres-Vega and co-workers [introduced in J. Chem. Phys. 98, 3103 (1993)] coincides with the totality of coherent-state representations for the Heisenberg-Weyl group. This fact leads to ambiguities when one wants to solve the stationary Schrödinger equation in phase space and we devise two schemes for the removal of these ambiguities. The physical interpretation of the phase-space wave functions is discussed and a procedure for computing expectation values as integrals over phase space is presented. Our formal points are illustrated by two examples.

DOI
10.1063/1.473684

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In this article we argue that the state-vector phase-space representation recently proposed by Torres-Vega and co-workers [introduced in J. Chem. Phys. 98, 3103 (1993)] coincides with the totality of coherent-state representations for the Heisenberg-Weyl group. This fact leads to ambiguities when one wants to solve the stationary Schrödinger equation in phase space and we devise two schemes for the removal of these ambiguities. The physical interpretation of the phase-space wave functions is discussed and a procedure for computing expectation values as integrals over phase space is presented. Our formal points are illustrated by two examples.
Theories of Torres-Vega and Fredrick (1993 J. Chem. Phys. 98 3103), Harriman (1994 J. Chem. Phys. 100 3651) and of Ban (1998 J. Math. Phys. 39 1744), in which phase space points ( p , q ) are used as configurational variables to formulate quantum mechanics are considered from the standpoint of a class of quantization schemes associating phase space functions with operators. The connection between these schemes and the theories given in Torres-Vega and Fredrick (1993 J. Chem. Phys. 98 3103), Harriman (1994 J. Chem. Phys. 100 3651), Dirac (1930 The Principles of Quantum Mechanics (Oxford: Oxford University Press)), Møller, Jørgensen and Torres-Vega (1997 J. Chem. Phys. 106 7228), Klauder and Skagerstam (1985 Coherent States: Applications in Physics and Mathematical Physics (Singapore: World Scientific)), Li, Wei and Lü (2004 Phys. Rev. A 70 022105), Ban (1998 J. Math. Phys. 39 1744) is made by means of augmented wavefunctions ψ (λ) σ ( p , q ; t ), where λ = 0 corresponds to the ordering of Wigner and Weyl. For that case we use these functions to define a family of positive operator-valued measures for the phase angle of an harmonic oscillator.
The rigorous solutions of the stationary Schrodinger equation for hydrogen atom are solved with the wave-mechanics method within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The Fourier-like projection transformations of wave function from the phase space to position and momentum spaces are extended to three-dimensional systems. The eigenfunctions in general position and momentum spaces could be obtained through the transformations from eigenfunction in the phase space.
The theory of the quantum mechanical representation in phase space recently proposed by Torres-Vega and Frederick is rigorously verified to be extended to the bound systems whose potential energy functions may contain terms of negative powers of coordinate and the quantum Liouville equation in the theory is also generalized to the same systems. It is suggested that the kernels of projection transformation take the forms e i pq/ 2 ħ and e -(i pq/ 2 ħ ) instead of {1/√(4π ħ )}e i pq/ 2 ħ and {1/√(4π ħ )}e -(i pq/ 2 ħ ) in order to keep the consistency for projections of the wave functions in all quantum problems. The validity of quantum average values and the virial theorem in the phase space representation are discussed in detail.
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