Index of /kankel/ph567/examples/Octave/ODE_shooting/schrodinger
Physics 567
Solving the Schrodinger Equation for an Arbitrary Potential
Introductory Example
Graphical solution
for eigenvalues of a finite square well.
Codes
- oneshot.m
takes a guess at an eigenvalue E as its argument
and returns an error. The roots of this function are the
eigenvalues.
- oneshotTest.m
demonstrates that oneshot.m works correctly.
Plot generated by
oneshotTest.m.
- schrodinger.m
implements the Schrodinger equation in 1D for an arbitrary
potential.
- eigenstates.m
makes a systematic search for all the eigenvalues and eigenfunctions
of a potential. The results are refined using
bisect.m.
- A subfolder contains code for various
potential functions.
(discussed in more detail below)
Solutions for Various Potentials
Three Straightforward Examples
First, look at the
Plot of the potential functions.
Two Peculiar Potentials
View a
plot of the potential functions.
The W Potentials
View a
plot of the potential functions.
Questions regarding the W potentials:
- Regarding v1doubledip,
- Why are n = 1, 2 nearly degenerate? The following questions
may or may help you to think it through:
- What would the single particle eigenstates look like for a potential
consisting of two widely separated square wells of equal depth?
- How would this change if the two wells are joined into one well twice
as wide?
- Discuss the transition between these two extreme cases.
- What is the probability density, and how would it evolve in time, if
the state is
- n = 0 ?
- n = 1 ?
- an equal superposition of n = 0, 1 ?
- Regarding v1localization,
- How do the lowest two states of this potential differ from the
lowest two states of the symmetric W potential?
- Assume that the state functions are correct. What is the
probability density, and how would it evolve in time if the state is
- n = 0 ?
- n = 1 ?
- an equal superposition of n = 0, 1 ?
- Two colleagues have expressed differing reactions (below) to
results of v1doubledip and v1localization.
You may also have a third option in mind.
- 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 may be 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.
- Can the dispute be settled from the information at hand?
- If not, what would you do next?
The Big Picture
- Codes are black boxes that give answers.
- Be suspicious.
- How might the code be fooled?
- Get a feel for the physics. Test the results against your physical intuition. What are some examples?
- Hand calculations usefully support numerical work. What are the limitations of analytic work?
Page maintained by
Charles Kankelborg