Particle in a Box: A Basic Paradigm in Quantum Mechanics - Part 1

Resonance

article

V Balakrishnan¹ 

[1] Department of Physics Indian Institute of Technology

关键词: Quantum mechanics;    stationary states;    degeneracy;    numerical degeneracy;    uncertainty relations;    eigenvalues;    eigenfunctions;    orthonormality;    completeness.;   

DOI  :  10.1007/s12045-022-1411-5

学科分类：社会科学、人文和艺术（综合）

来源: Springer

【 摘 要 】

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, eigen-values, 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.

【 授权许可】

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