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Long time asymptotics for quantum particles in a periodic potential. (English) Zbl 0944.81012

Summary: We study a quantum particle in a periodic potential and subject to slowly varying electromagnetic potentials. It is proved that, in the Heisenberg picture, the scaled position operator \(\epsilon \mathbf x(\epsilon^{-1}t)\) has a limit as \(\epsilon\to 0\). The limit operator is determined by the semiclassical equations of motion, which implies that for long times the wave packet is well approximated by the semiclassical evolution. From our result we infer the hydrodynamic limit, \(\mathbf q\to 0\), \(t\to\infty\), \(\mathbf qt=\text{const}\), of the structure function \(S(\mathbf q,t)\) of a fluid of noninteracting fermions in a crystal potential.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
82D25 Statistical mechanics of crystals
47N50 Applications of operator theory in the physical sciences
47N55 Applications of operator theory in statistical physics (MSC2000)
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