AMS Fall Western & Central Sampler: Semiclassical Quantum Dynamics and Bohmian Trajectories

Christof Sparber
2016 Notices of the American Mathematical Society  
T he early days of the twentieth century saw the establishment of quantum mechanics as a novel description of our physical world. Ever since its invention, a basic problem concerns the connection between quantum mechanics and the older, well-understood theory of classical mechanics. It was accepted early on that classical mechanics should be understood as an emergent phenomenon of quantum mechanics, i.e., it should be recovered from the underlying quantum mechanical description when considered
more » ... on when considered over special values of its physical parameters. When trying to follow this basic idea, one immediately faces an obstacle: quantum mechanics and classical mechanics are usually treated within two entirely different mathematical formalisms. While the former is based on the time-evolution of vectors in (infinite-dimensional) Hilbert spaces, the latter is concerned with the dynamics of point particles on a (finite-dimensional) phase space. Focussing on one of the simplest quantum mechanical systems, we consider the dynamics of a single quantum particle, say, an electron, under the influence of a static external force field mediated by a given real-valued potential ( ), where ∈ ℝ 3 denotes the spatial coordinate. The state of the electron at time ∈ ℝ is described by a wave function ( , ⋅) ∈ 2 (ℝ 3 ; ℂ) whose time evolution is governed by Erwin Schrödinger's fundamental equation from 1926: Here we have rescaled the original equation including all physical parameters (mass, charge, the Planck's constant ℏ, etc.) into dimensionless units such that only one (small) parameter > 0 remains. The latter plays the role of a
doi:10.1090/noti1429 fatcat:q4rwc2bv5rh3hlrx3tm7pbgcy4