Abstract
One dimensional quantum mechanics problems, namely the infinite potential well, the harmonic oscillator, the free particle, the Dirac delta potential, the finite well and the finite barrier are generalized for finite arbitrary dimension in a radially symmetric, or angular invariant, manner. This generalization enables the Schrödinger equation solutions to be visualized for Bessel functions and Whittaker functions, and it also enables connections to multi-dimensional physics theories, like string theory.
Highlights
An introductory quantum mechanics course deals with solutions to one-dimensional problems, as can be seen in the commonly used textbooks on the subject
The importance of multi-dimensional problems has increased in physics, since string theory has given rise to the possibility that there could be more than three dimensions of space
The multidimensional approach provides a deeper understanding of the physics of the one-dimensional problems, as well as the capabilities and limitations of the mathematical apparatus
Summary
An introductory quantum mechanics course deals with solutions to one-dimensional problems, as can be seen in the commonly used textbooks on the subject. The more dimensions a problem has, the more possibilities of motion, and the more symmetries it can have to restrict these possibilities This means that a generalization can be a choice, depending of the symmetries of the n−dimensional solution of the problem. The multidimensional approach provides a deeper understanding of the physics of the one-dimensional problems, as well as the capabilities and limitations of the mathematical apparatus. As some of these results may be scattered throughout literature, it is useful, for both students and researchers, to have them presented in a single place. This article is organized as follows: in section two, the infinite n−dimensional cylindrical quantum well is solved.
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