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Physics 567

Solving the Schrodinger Equation for an Arbitrary Potential

Introductory Example

Graphical solution for eigenvalues of a finite square well.

Codes

Solutions for Various Potentials

Three Straightforward Examples

First, look at the Plot of the potential functions.

Potential Terminal output err(E) Eigenstates Comments
v1square.m v1square__failure_output.txt v1square__failure_err.pdf v1square__failure_eigenstates.pdf Do you see the problem?
v1square.m v1square_output.txt v1square_err.pdf v1square_eigenstates.pdf Higher precision helped.
v1rounded.m v1rounded_output.txt v1rounded_err.pdf v1rounded_eigenstates.pdf Exponential instability of some eigenfunctions.
v1v1skewed.m v1skewed_output.txt v1skewed_err.pdf v1skewed_eigenstates.pdf A skewed well.

Two Peculiar Potentials

View a plot of the potential functions.

Potential Terminal output err(E) Eigenstates Comments
v1dktail.m v1dktail_output.txt v1dktail_err.pdf v1dktail_eigenstates.pdf Required a tight tolerance, ftol = 1e-14.
v1dkhead.m v1dkhead_output.txt v1dkhead_err.pdf v1dkhead_eigenstates.pdf Ground state is nearly degenerate.

The W Potentials

View a plot of the potential functions.

Potential Terminal output err(E) Eigenstates Comments
v1doubledip.m v1doubledip__failure_output.txt v1doubledip__failure_err.pdf v1doubledip__failure_eigenstates.pdf Oversampling = 100.
v1doubledip.m v1doubledip_output.txt v1doubledip_err.pdf v1doubledip_eigenstates.pdf Oversampling = 1000.
v1localization.m v1localization_output.txt v1localization_err.pdf v1localization_eigenstates.pdf Can this be right?

Questions regarding the W potentials:

  1. Regarding v1doubledip,
    1. Why are n = 1, 2 nearly degenerate? The following questions may or may help you to think it through:
      1. What would the single particle eigenstates look like for a potential consisting of two widely separated square wells of equal depth?
      2. How would this change if the two wells are joined into one well twice as wide?
      3. Discuss the transition between these two extreme cases.
    2. What is the probability density, and how would it evolve in time, if the state is
      1. n = 0 ?
      2. n = 1 ?
      3. an equal superposition of n = 0, 1 ?
    3. Regarding v1localization,
      1. How do the lowest two states of this potential differ from the lowest two states of the symmetric W potential?
      2. Assume that the state functions are correct. What is the probability density, and how would it evolve in time if the state is
        1. n = 0 ?
        2. n = 1 ?
        3. an equal superposition of n = 0, 1 ?
      3. Two colleagues have expressed differing reactions (below) to results of v1doubledip and v1localization. You may also have a third option in mind.
        1. Can the dispute be settledfrom the information at hand?
        2. Would more numerical experimentation help?

      Interpretation I. This looks like interesting physics. Due to the slight asymmetry in v1localization, each of the two wells has acquired a distinct level structure, so that an electron can literally be placed in one well or the other. This is analogous to electron localization in a semiconductor.

      Interpretation II. It is suspicious that such a slight change in the potential function would lead to qualitatively different modes in the well. The large difference in the eigenfunctions found for these two potentials is probably a numerical artifact. Note the instability of the solution in evanescent regions such as the hump between the two wells. The ODE solver is probably not capable of moving accurately between the wells.

      Page maintained by Charles Kankelborg