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Quantum Mechanics with Trajectories

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Book cover A Trajectory Description of Quantum Processes. I. Fundamentals

Part of the book series: Lecture Notes in Physics ((LNP,volume 850))

Abstract

Bohmian mechanics allows to understand the quantum world in a classical-like fashion, by means of trajectories evolving throughout configuration space. This appealing feature has stimulated its application nowadays to many different problems from atomic and molecular physics, condensed matter physics, chemical physics or quantum chemistry, for example. This is a remarkable growth if one takes into account that this causal theory of quantum motion started as a simple hidden-variable model to disproof von Neumann’s theorem on the impossibility of hidden variables in quantum mechanics, and its applications initially covered fundamental problems. In this Chapter, the main elements of this theory are briefly revisited and they will be further developed in Volume 2. Furthermore, a contextualization of Bohmian trajectories with respect to alternative trajectory-based approaches to quantum mechanics, such as Feynman’s path integral, the semiclassical approximation, mixed/hybrid (quantum-classical) formulations or quantum (causal) stochastic trajectories, is also presented.

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Notes

  1. 1.

    For example, when trying to solve the time-dependent Schrödinger equation by means of standard grid methods, \({{\mathcal{K}}}\) has to receive a special consideration. In order to avoid truncations of its nonlocal nature, the action of the kinetic operator is assumed in the momentum space (by means of the fast Fourier transform technique, for example), where this operator is local. Then, after acting on \(\varPsi\), the result (which is already affected by the value of \(\varPsi\) in all points of the grid) is put back in the configuration space.

  2. 2.

    Note that, unlike (6.6), the expression for \(\bar{v}\) is not symmetric with respect to \(\bar{\varPsi}^{\ast}.\) This is because, as previously mentioned, in this case the transformation is one to one, and therefore the complex conjugate wave field is not needed.

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    Ángel S. Sanz & Salvador Miret-Artés

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Sanz, Á.S., Miret-Artés, S. (2012). Quantum Mechanics with Trajectories. In: A Trajectory Description of Quantum Processes. I. Fundamentals. Lecture Notes in Physics, vol 850. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18092-7_6

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