Abstract
We derive and analyse the three-dimensional quantum Liouville equation for an electron with spin in an external electromagnetic field. By using methods of semigroup theory, we prove existence and uniqueness of the initial value problem. Expanding the solution into a series of pure states, we derive the existence of a generalized particle density and anL ∞-estimate on the solution. The last section is devoted to an analysis of the classical and electrostatic limits.
Zusammenfassung
Die drei-dimensionale Quanten-Liouville Gleichung für ein Elektron mit Spin in einem äuβeren elektromagnetischen Feld wird hergeleitet und analysiert. Mit Hilfe der Halbgruppen-Theorie werden Existenz und Eindeutigkeit des Anfangswertproblems bewiesen. Aus der Entwicklung der Lösung in eine Reihe von reinen Zuständen schließen wir die Existenz einer verallgemeinerten Teilchendichte und erhalten eineL ∞-Abschätzung für die Lösung. Im letzten Kapitel werden der klassische und elektrostatische Limes untersucht.
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Arnold, A., Steinrück, H. The ‘electromagnetic’ Wigner equation for an electron with spin. Z. angew. Math. Phys. 40, 793–815 (1989). https://doi.org/10.1007/BF00945803
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DOI: https://doi.org/10.1007/BF00945803