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NATO ASI series. Series B : Physics

Quantum Kinetic Equations

Herbert E. Spohn

Published Jan 1, 1994• 22 citations
Publisher
DOI
10.1007/978-1-4615-2460-1_1

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The article outlines key topics related to deriving kinetic equations and, in general, to the relationship between the evolution equations governing quantum many-particle systems. The main issues being considered are the mathematical origin of quantum kinetic equations, novel methods for deriving kinetic equations from hierarchies of evolution equations for quantum systems, specifically focusing on kinetics within the framework of the evolution of observables, and the generalization of the quantum kinetic equation specified by non-separable initial states.
A quantum kinetic master equation (QKME) for bosonic atoms is formulated. It is a quantum stochastic equation for the kinetics of a dilute quantum Bose gas, and describes the behavior and formation of Bose condensation. The key assumption in deriving the QKME is a Markov approximation for the atomic collision terms. In the present paper the basic structure of the theory is developed, and approximations are stated and justified to delineate the region of validity of the theory. Limiting cases of the QKME include the quantum Boltzmann master equation and the Uehling-Uhlenbeck equation, as well as an equation analogous to the Gross-Pitaevskii equation.
The Schrödinger equation, Klein‐Gordon equation (KGE), and Dirac equation are believed to be the fundamental equations of quantum mechanics. Schrödinger’s equation has a defect in that there are no negative kinetic energy (NKE) solutions. Dirac’s equation has positive kinetic energy (PKE) and NKE branches. Both branches should have low-momentum, or nonrelativistic, approximations: One is the Schrödinger equation, and the other is the NKE Schrödinger equation. The KGE has two problems: It is an equation of the second time derivative so that the calculated density is not definitely positive, and it is not a Hamiltonian form. To overcome these problems, the equation should be revised as PKE- and NKE-decoupled KGEs. The fundamental equations of quantum mechanics after the modification have at least two merits. They are unitary in that all contain the first time derivative and are symmetric with respect to PKE and NKE. This reflects the symmetry of the PKE and NKE matters, as well as, in the author’s opinion, the matter and dark matter of our universe. The problems of one-dimensional step potentials are resolved by utilizing the modified fundamental equations for a nonrelativistic particle.