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Bound states in a semi-infinite square potential well

Nivaldo A. Lemos

Abstract

The finite square potential well is a staple problem in introductory quantum mechanics. There is an extensive literature on the determination of the allowed energies, which requires the solution of a transcendental equation by numerical, graphical or approximate analytic methods. Here we investigate the less explored problem of a particle in a semi-infinite potential well. The energy eigenvalues, which are also determined by a transcendental equation, are found by a standard graphical method, and a simple rule that yields the number of stationary states is provided. Next a simplification of the aforementioned transcendental equation is attempted. During the process pitfalls are encountered and a purportedly simpler graphical treatment of the problem given in the solutions manual to a fine textbook is shown to be flawed. A more careful analysis brings forth the correct simplification, which is shown to be particularly suitable for finding highly accurate approximations to the energy levels. Finally, a class of exact solutions is produced, the associated normalized eigenfunctions are constructed and the probability of finding the particle inside the well is computed.

Bound states in a semi-infinite square potential well

Abstract

The finite square potential well is a staple problem in introductory quantum mechanics. There is an extensive literature on the determination of the allowed energies, which requires the solution of a transcendental equation by numerical, graphical or approximate analytic methods. Here we investigate the less explored problem of a particle in a semi-infinite potential well. The energy eigenvalues, which are also determined by a transcendental equation, are found by a standard graphical method, and a simple rule that yields the number of stationary states is provided. Next a simplification of the aforementioned transcendental equation is attempted. During the process pitfalls are encountered and a purportedly simpler graphical treatment of the problem given in the solutions manual to a fine textbook is shown to be flawed. A more careful analysis brings forth the correct simplification, which is shown to be particularly suitable for finding highly accurate approximations to the energy levels. Finally, a class of exact solutions is produced, the associated normalized eigenfunctions are constructed and the probability of finding the particle inside the well is computed.
Paper Structure (10 sections, 53 equations, 5 figures)

This paper contains 10 sections, 53 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Semi-infinite square well in quantum mechanics
  3. Graphical determination of the energy levels
  4. Simplifying the search for the energy eigenvalues
  5. Another simplification attempt
  6. Yet another simplification attempt
  7. The correct simplification
  8. Approximate energy spectrum
  9. A class of exact solutions
  10. Conclusion

Figures (5)

  • Figure 1: Semi-infinite potential energy square well. The particle is excluded from the region $x < 0$ by an impenetrable wall of infinitely high potential energy.
  • Figure 2: Graphs of the function ${\tilde{z}} = -z \cot z$ and of the first quadrant of the circle $z^2 + {\tilde{z}}^2 = z_0^2$ for $z_0=15$ and $z_0=25$. The circular arcs are slightly distorted because the horizontal and vertical scales are not quite the same. Values of $z$ are on the horizontal axis while those of $\tilde{z}$ are on the vertical axis. The value of $z$ for each intersection of the two graphs is a solution to equation \ref{['transcendental-equation-exact']}. For $z_0=15$ there are 5 solutions, whereas for $z_0=25$ there are 8 solutions. The number of solutions is equal to the positive integer $N$ that satisfies inequalities \ref{['number-of-solutions']}.
  • Figure 3: Graphical solutions of equation \ref{['ztilde-function-z-no-square']} for $z_0=15$ (steeper straight line) and $z_0=25$ (less slanted straight line).
  • Figure 4: Left: solutions of equation \ref{['ztilde-function-z-square-root']} for $z_0=15$ and $z_0=25$. Right: solutions of equation \ref{['ztilde-function-z-negative-square-root']} for $z_0=15$ and $z_0=25$. In both cases, the steeper straight line corresponds to $z_0=15$.
  • Figure 5: Graphical solutions of equation \ref{['transcendental-sinz-modulus-cosz']} for $z_0=15$ (steeper straight line) and $z_0=25$ (less slanted straight line).