Abstract
The problem of a particle moving freely inside a box with rigid, perfectly reflecting walls is a standard exercise in basic quantum mechanics (QM), the box being taken to be a line interval, a square, and a cube, respectively, in the 1, 2, and 3 dimensional cases. The problem illustrates several aspects of QM, such as stationary states, uncertainty relations, eigenvalues, eigenfunctions, orthonormality, completeness, and so on. The purpose of this two-part article is to provide some insights into how the particle-in-a-box problem acts as a paradigm for the understanding of these and other aspects, such as the roles played by dimensionality, discrete and continuous symmetry, degeneracy, etc. Further, we shall also see (in Part 2) how the same problem leads directly to some non-trivial matters such as quantum chaos and deep results in mathematical physics. Finally, it must be emphasized that the particle-in-a-box problem is no longer just a theoretical exercise. With modern technology, it is physically realized in different kinds of nanostructures such as nanowires, atomic corrals, quantum dots, conjugated polymers, among others. This makes it all the more important to analyze various problems of this class and their solutions.
Similar content being viewed by others
Suggested Reading
E. Merzbacher, Quantum Mechanics, 3rd edition, Wiley, 1998.
L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, 3rd edition, Pergamon, 1977.
V. Balakrishnan, Mathematical Physics, Ane Books, 2018; Springer, 2020.
See, for instance, K. R. McLean, Math. Gaz., Vol.85, 470, 2001.
Wikipedia Contributors, Sum of squares function, Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Sum_of_squares_function&oldid=980277364, [Online] (2021).
Acknowledgement
I thank Sunethra Ramanan for discussions and technical assistance. I also thank Suresh Govindarajan, S. Lakshmibala, and Arul Lakshminarayan for a discussion of the additional degeneracy of the energy levels of a particle in a square box. I am deeply grateful to the reviewer for a very thorough reading of the manuscript and for numerous helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
V. Balakrishnan is with the Department of Physics, IIT Madras, Chennai. His current research interests are quantum dynamics, stochastic processes and dynamical systems.
Rights and permissions
About this article
Cite this article
Balakrishnan, V. Particle in a Box: A Basic Paradigm in Quantum Mechanics — Part 1. Reson 27, 1135–1153 (2022). https://doi.org/10.1007/s12045-022-1411-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12045-022-1411-5