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American Journal of Physics

Correspondence principle demonstrated

Robert L. W. Chen, Michael Barry Rhodes

Published Nov 1, 1984• 3 citations
Publisher

Abstract

A method of producing computer graphics for quantum mechanics has been adapted to portray processes with much shorter wavelengths in comparison to those in an earlier work by Chen. A set of pictures are produced which demonstrates the correspondence principle—that quantum mechanics reduces to classical mechanics in appropriate limits.
DOI
10.1119/1.13774

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The correspondence principle addresses the connection between classical and quantum physics. The simple statement that quantum mechanics reduces to classical mechanics in the limit where the principal quantum number n approaches infinity, while found in many textbooks, is not true in general. In this article we will give special attention to the notion that the quantum frequency spectrum of a periodic system reduces to the classical spectrum in this limit. Two simple counter-examples—a particle in a cubical box, and a rigid rotator—will show us that the classical result is not always recovered in the limit of large quantum numbers.
A presentation of basic quantum mechanics for non-physical science majors is described which, by relying on computer-generated graphic display, circumvents the usual mathematical limitations and allows a detailed treatment of free particle motion in a wave picture. The development is based on a computer-simulated, single-slit, particle diffraction experiment and culminates in a complete description and graphic display of the progress of the diffraction pattern from a square pulse at the slit to the Fraunhofer pattern at infinity.
Quantum mechanics and classical mechanics are distinctly different theories, but the correspondence principle states that quantum particles behave classically in the limit of high quantum number. In recent years much research has been done on extending both quantum and classical mechanics into the complex domain. These complex extensions continue to exhibit a correspondence, and this correspondence becomes more pronounced in the complex domain. The association between complex quantum mechanics and complex classical mechanics is subtle and demonstrating this relationship requires the use of asymptotics beyond all orders.
Because atoms and molecules are small, their behavior can only be understood using quantum mechanics, rather than classical mechanics. Quantum mechanics amounts to studying the behavior of electrons and nuclei and their interactions. We will limit ourselves to slowly moving electrons and nuclei, and thus will not need to discuss relativistic quantum mechanics. Large particles also obey quantum mechanics, but the relevant wavelengths are so small that we don't notice them. Small particles absolutely require the use of quantum mechanics for any meaningful understanding. This treatment is not merely an inconvenience: it leads to useful and interesting phenomena such as the Pauli exclusion principle, the Aufbau principle, Hund's rules, the Heisenberg uncertainty principle, wave-particle duality, the photoelectric effect, Ehrenfest's theorem, and the correspondence principle.