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arXiv:1003.3372 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

A sharp version of Ehrenfest’s theorem for general self-adjoint operators

Gero Friesecke, Bernd Schmidt

Published Feb 17, 2010• 9 citations
arXivHTMLPDF

Abstract

Abstract We prove the Ehrenfest theorem of quantum mechanics under sharp assumptions on the operators involved.
DOI
10.1098/rspa.2009.0351

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We point out that two of the most important theorems of Quantum Mechanics, the Ehrenfest theorem and the Hellmann-Feynman theorem, lack in their standard form important information: there are cases where non-Hermitian boundary contributions emerge. These contributions actually appear naturally, in order for the above theorems to be valid and applicable (i.e. in multiply-connected spaces), and this occurs for physical quantities that are not represented by well-defined self-adjoint operators (such as the position operator in a periodic potential, or in general Aharonov-Bohm configurations, either in real or in an arbitrary parameter-space, in the sense of Berry's adiabatic and cyclic procedures). In this short note, we report modifications of these two theorems when such non-Hermiticities appear, and we demonstrate how they resolve certain Quantum Mechanical paradoxes (most of them having been noticed in the past as violations of the so-called Hypervirial theorem in Quantum Chemistry). This resolution of paradoxes (essentially the re-establishment of applicability of the Ehrenfest theorem even in multiply-connected spaces) always proceeds through the appearance of certain generalized currents, in a theoretical picture with interesting structure (where a generalized continuity equation with a sink term shows up naturally).
For a particle in a box, the operator −i∂x is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg–Robertson–Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it.