The quantum particle in a box: what we can learn from classical electrodynamics

• Published in: The European physical journal. ST, Special topics Vol. 227; no. 15-16; pp. 2155 - 2169
• Main Authors: de la Peña, L., Cetto, A. M., Valdés-Hernández, A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2019; Springer Nature B.V
• Subjects: Atomic; Charged particles; Classical and Continuum Physics; Condensed Matter Physics; Echoes; Electrodynamics; Lorentz force; Materials Science; Measurement Science and Instrumentation; Molecular; Non-equilibrium Dynamics: Quantum Systems and Foundations of Quantum Mechanics; Optical and Plasma Physics; Physics; Physics and Astronomy; Regular Article
• ISSN: 1951-6355; 1951-6401
• DOI: 10.1140/epjst/e2018-800048-x

The problem of a charged particle enclosed in an infinite square potential well is analysed from the point of view of classical theory with the addition of the electromagnetic zero-point radiation field, with the aim to explore the extent to which such an analysis can contribute to enhance our understanding of the quantum behavior. First a proper treatment is made of the freely moving particle subject to the action of the radiation field, involving a frequency cutoff ω c . The jittering motion and the effective structure of the particle are sustained by the permanent action of the zero-point field. As a result, the particle interacts resonantly with the traveling field modes of frequency ω c in its proper frame of reference, which superpose to give rise to a modulated wave accompanying the particle. This is identified with the de Broglie wave, validating the choice of Compton’s frequency for ω c . For the stationary states of particles confined in the potential well, the Lorentz force produced by the accompanying field is shown to lead to discrete values for the mean speed and to an uneven probability distribution that echoes the corresponding quantum distribution. The relevance of the results obtained and the limitations of the classical approach used, are discussed in the context of present-day stochastic electrodynamics.