Eigenfunctions and Energies for Sloped-Bottom Square-Well Potential

Eigenenergies and eigenfunctions of the potential for and for are obtained numerically. This is treated as a perturbation of the infinite square-well potential, with , for and for . The eigenfunction of the unperturbed problem is denoted by . Note that, for increasing quantum number , the effect of the perturbation diminishes.

Energies are in units with , and the mass of the particle is 1/2. The eigenenergies of the unperturbed potential (the infinite square well of width 1) are . The case with and is the "V-bottom" potential, which is studied by perturbation methods in Appendix J of [1]. The case with leads to the so-called infinite tilted well. Its exact solution is given in [2].

References

[1] R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed., New York: Wiley, 1985.

[2] J. N. Churchill and F. O. Arntz, "The Infinite Tilted-Well: An Example of Elementary Quantum Mechanics with Applications toward Current Research," American Journal of Physics, 37(7), 1969 pp. 693–697. doi:10.1119/1.1975775.