Eigenfunctions and Energies for Sloped-Bottom Square-Well Potential
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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.
Contributed by: Santos Bravo Yuste (February 2018)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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.
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