Modern 2:More on Square Wells and Review for Monday's Exam
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Before starting the review, I would like to go back and redo the infinite square well potential but with a slight variation.
Download a trivial Mathematica notebook with some plots of the results |
review
There is a review of important topics at the end of chapter 3, pages 58-59. I will go over this in class today. But briefly, it includes the time dependent and time independent Schrodinger equations. Momentum space representation. The Heisenberg uncertainty relation (for x, p_x and E, t). The idea that to each physical quantity A there is an associated observable given as a self-adjoint operator . And that the expected value of a measurement associated with this quantity is given by .
In addition to this summary, you should understand
- the infinite square well sufficiently to be able to derive the energy eigenstates, the energy levels and normalization.
- the eigenstates are orthogonal and complete. This lets us represent any wavefunction as a superposition of eigenstates.
- if the wavefunction IS an eigenfunction of some observable with eigenvalue a, the the measurement of A is equal to a with probability 1.
- it would be good to refresh your memory on Fourier series, in case I ask you to represent a non-eigenstate in terms of a superposition of eigenstates. E.g., a Gaussian.
There are two potential problems that will be fair game for Monday's exam. First is the finite step potential:
and zero otherwise
and infinite otherwise
A first order expectation would be that you can set up the solutions in the different regions (knowing under what circumstances you would get a propagating wave and under what you would get an exponentially damped (tunneling) state. Then you should be able to apply any boundary conditions.
I don't expect you to be able to derive the probability current (at least on an exam) but you should know how to compute it and you should understand it's physical significance.