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Journal of Chemical Education

The particle in the one-dimensional champagne bottle

Gerald Ray Miller

Published Nov 1, 1979• 3 citations
Publisher

Abstract

Introducing a half-sine wave bottom to the usual one-dimensional particle-in-the-box problem.
DOI
10.1021/ed056p709

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If the valley encountered by the particle is infinitely deep, then the particle must be trapped inside. The best analogy is an infinitely deep well. Therefore, this potential is called the infinite square well , or the particle in a box (PIB). The TISE for the PIB is very easy to solve, again using the physical principles from Chapter 6 . The solutions are sine waves with an integral or half-integral number of wavelengths fitting the width of the well. This fact that only certain wavelengths are allowed is a perfect example of quantization of a trapped particle. From the allowed wavelengths, we can also derive the quantized momentum and kinetic energy of the particles. Since momentum is a vector quantity, care must be taken to consider the direction in which the particle might be moving. With this caveat, these simple sine wave solutions are ideal for illustrating principles and procedures such as eigenfunctions, normalization, orthogonality, orthonormality, and expectation values.