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Asymptotic analysis of the quantum Liouville equation
Author(s) -
Steinrück Herbert
Publication year - 1990
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670130205
Subject(s) - mathematics , action (physics) , quantum , constant (computer programming) , planck constant , fokker–planck equation , liouville equation , mathematical analysis , mathematical physics , statistical physics , quantum mechanics , physics , differential equation , computer science , programming language
We present an asymptotic analysis of the quantum Liouville equation with respect to the Planck's constant, which models the temporal evolution of the (quasi)distribution of an ensemble of electrons under the action of a potential. We consider two cases: firstly a smooth potential, and secondly a potential modelled by a δ‐distribution. In both cases the zeroth‐order term behaves classically. In the smooth case the classical Liouville equation is satisfied and in the case for the δ‐potential an interface condition is derived, so that everything is reflected at the potential barrier.