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Journal of Mathematical Physics

Quantum two-particle scattering in fuzzy phase space

Eduard Prugovečki

Published Sep 1, 1976• 33 citations
Publisher

Abstract

The concepts of configuration and momentum representation space for state vectors are generalized to that of fuzzy-phase-space representation spaces L2(Γs), 0<s<∞, which are interpolated in between these two standard representations. It is shown that the wavepacket in L2(Γs) displays the familiar evanescence property from any region Ks × Ms in the fuzzy phase space Γs if that region is bounded in its configuration part Ks; also, that the probability of detecting the system in Ks × Ms has a finite asymptotic time limit if Ks is a (fuzzy) cone. For scattering states the existence of free states that are asymptotic in Γs is established, and a formula for differential cross section in Γs is derived.

DOI
10.1063/1.523115

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The concepts of configuration and momentum representation space for state vectors are generalized to that of fuzzy-phase-space representation spaces L2(Γs), 0<s<∞, which are interpolated in between these two standard representations. It is shown that the wavepacket in L2(Γs) displays the familiar evanescence property from any region Ks × Ms in the fuzzy phase space Γs if that region is bounded in its configuration part Ks; also, that the probability of detecting the system in Ks × Ms has a finite asymptotic time limit if Ks is a (fuzzy) cone. For scattering states the existence of free states that are asymptotic in Γs is established, and a formula for differential cross section in Γs is derived.
The problem of expressing quantum mechanical expectation values as averages with respect to nonnegative density functions on phase space, by analogy with classical mechanics, is reexamined in the light of some earlier work on fuzzy phase spaces. It is shown that such phase space representations are possible if ordinary phase space is replaced by a so-called fuzzy phase space, on which the usual marginal distribution functions are redefined to conform to the fact that arbitrarily precise simultaneous measurements on position and momentum are not compatible with quantum mechanics. In the process a generalization of Wigner’s theorem on the nonexistence of phase space representations of quantum mechanics, which also satisfy the standard (classical) marginality conditions in position and momentum, is obtained. It is shown that a (continuous) representation of quantum mechanics exists on a given fuzzy phase space if an only if the corresponding confidence functions for position and momentum measurements satisfy the Heisenberg uncertainty relations.
The phase-space formulation of quantum scattering is discussed and the asymptotic quantities describing potential scattering in one and three dimensions (transmittances, cross sections) are analyzed. It is found that under certain conditions the transmittance in time-dependent scattering only depends on the momentum distribution of the incident packet, being independent of its particular shape in coordinate space. For three-dimensional scattering, generalized cross sections are defined for momentum coherences and a physical interpretation of the generalized optical theorem is proposed.
We derive and explain the key ideas behind a time-dependent formulation of quantum scattering theory, applicable generally to systems with a finite-range scattering potential. The scattering is initiated and probed by plane wave packets, which are localized just outside the range of the potential. The asymptotic limits of conventional scattering theory (initiation in the remote past; detection in the remote future) are not taken. Instead, the differential cross section (DCS) is obtained by projecting the scattered wave packet onto the probe plane wave packets. The projection also yields a time-dependent version of the DCS. Cuts through the wave packet, just as it exits the scattering potential, yield time-dependent and time-independent angular distributions that give a close-up picture of the scattering which complements the DCS. We have previously applied the theory to interpret experimental cross sections of chemical reactions [e.g., S. C. Althorpe, F. Fern\'andez-Alonso, B. D. Bean, J. D. Ayers, A. E. Pomerantz, R. N. Zare, and E. Wrede, Nature (London) 416, 67 (2002)]. This paper gives the derivation of the theory, and explains its relation to conventional scattering theory. For clarity, the derivation is restricted to spherical-particle scattering, though it may readily be extended to general multichannel systems. We illustrate the theory using a simple application to hard-sphere scattering.
A quantum mechanical representation suitable for studying the time evolution of quantum densities in phase space is proposed and examined in detail. This representation on ℒ2 (2) phase space is based on definitions of the operators P̂ and Q̂ in phase space that satisfy various correspondences for the Liouville equation in classical and quantum phase space, as well as quantum position and momentum ℒ2 (1) spaces. The definitions presented here, P̂=p/2−iℏ∂/∂q and Q̂=q/2+iℏ∂/∂p, are related to definitions that have been recently proposed [J. Chem. Phys. 93, 8862 (1990)]. The resulting quantum phase space representation shares many of the mathematical properties of usual representations in coordinate and momentum spaces. Within this representation, time evolution equations for complex-valued functions (wave functions) and their square magnitudes (distribution functions) are derived, and it is shown that the coordinate and momentum space time evolution equations can be recovered by a simple Fourier projection. The phase space quantum probability conservation equation obtained is a good illustration of the quantization rule that requires one to replace the classical Poisson bracket between the Hamiltonian and the probability density with the quantum commutator between the corresponding operators. The possible classical analogs to quantum probabilities densities are also considered and some of the present results are illustrated for the dynamics of the coherent state.