Some Selected Applications of Bohmian Mechanics

Abstract

In this chapter, our purpose is to simply show that Bohmian mechanics is a powerful route to bring about new solutions to problems discussed by conventional quantum mechanical approaches, apart from allowing some striking correspondence between both frameworks. This goal is carried out by choosing some key quantum mechanical problems in the framework of Bohmian mechanics such as, for exam-ple, the so-called Ermakov–Bohm invariants, boundary conditions and uncertainty principle in tunneling, the quantum traversal time, Airy wave packets and Airy slits, the detection of inertial and gravitational masses with Airy wave packets, the geomet-ric phase analyzing the Aharonov–Bohm effect and quantum vortices, the reformu-lation of the Gross–Pitaevskii equation within the hydrodynamical framework and, finally, the study of simple dissipative dynamics by using the well-known Caldirola-Kanai Hamiltonian. In this dissipative scenario, the motion of a free particle, the quantum interference of two wave packets and the dynamics in a linear potential as well as the corresponding of a damped harmonic oscillator (within the underdamped, critically damped and overdamped regimes) are finally analyzed for ulterior refer-ences. 2.1 Introduction Recently, Bernstein has presented a pedagogically clear and historically brilliant review of the Bohmian theory [1] showing that it is sharp where the usual one is fuzzy and general where the usual one is special. Bell and Bernstein argued convincingly that the de Broglie-Bohm interpretation of quantum mechanics should be known from the very beginning and part of any college curriculum on the subject. In this theory, the wave function provides only a partial description of the system. The description is completed by the specification of the actual positions of the parti-cles which evolve according to the so-called guidance condition or guiding equation, Eq. (1.20). This equation provides the velocity of the particles in terms of the wave function. However, the trajectory of a particle is not at all classical; it is instead determined by the structure of the associated quantum wave which guides or pilots the particle along its path.