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Physica Scripta

A new formulation of the potential scattering in quantum phase space

Xu‐Guang Hu, Qian‐Shu Li, Au‐Chin Tang

Published Aug 1, 1996• 4 citations
Publisher

Abstract

In the present paper the quantum mechanical representation in phase space introduced by Torres-Vega and Frederick (1993 J. Chem. Phys. 98 3103) has been successfully applied to the phase space description of the quantum potential scattering. 3D potential scattering is formulated from the view-points of the wave function and density operator in this new quantum phase-space representation. It is found that in the region where the potential acts predominantly, the scattered part of the wave function or density in quantum phase space depends closely not only on the potential, but also on a non-locality factor arising from the use of the eigenfunctions of the coordinate and momentum operators in the phase-space representation. Transition from the present formulation to the classical scattering is made.

DOI
10.1088/0031-8949/54/2/001

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In the present paper the quantum mechanical representation in phase space introduced by Torres-Vega and Frederick (1993 J. Chem. Phys. 98 3103) has been successfully applied to the phase space description of the quantum potential scattering. 3D potential scattering is formulated from the view-points of the wave function and density operator in this new quantum phase-space representation. It is found that in the region where the potential acts predominantly, the scattered part of the wave function or density in quantum phase space depends closely not only on the potential, but also on a non-locality factor arising from the use of the eigenfunctions of the coordinate and momentum operators in the phase-space representation. Transition from the present formulation to the classical scattering is made.
We have formulated the quantum rearrangement scattering of atom-diatomic molecules in phase space from the viewpoint of the density operators within the framework of Torres-Vega's and Frederick's phase-space representation of quantum mechanics [J. Chem. Phys. 98, 3103 (1993); 93, 8862 (1990)]. This formalism has a remarkable feature in that it naturally includes the on-the-energy-shell coherences of the initial system that are important for depicting actual scattering experiments. We have found that the full density operator describing the rearrangement scattering is closely related not only to the usual reactive transition operator T${\mathrm{^}}_{\mathrm{\ensuremath{\beta}}\mathrm{\ensuremath{\alpha}}}$, but also to the function F defined in the paper which is nonlocal in position vector and could reflect some fine structures of the reactive scattering in strong interaction region. The time-dependent reactive scattering in the quantum phase-space representation is also discussed.
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.
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.