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The Square Well in the Phase Plane
Students are now encountering quantum mechanics very much earlier in the course of their education than was common a decade or so ago, and consequently one frequently finds that ones students are not as comfortable with eigenvalue problems as one might hope. The purpose of this paper is to describe a film made to offer a different perspective on a standard eigenvalue problem, ie the bound states of a square well.
There are several films now available that plot the wave function as a function of distance for successive values at a trial eigenvalue. If the boundary condition at the origin is satisfied, then one sees the exponential growth of the solution to the Schrodinger equation at large distances when the trial eigenvalue does not correspond to one of the allowed energies of the system. In the linear superposition of exponentials that is the solution the coefficient of the growing exponential term goes through zero as the eigenvalue is passed. This presentation gives a vivid feeling for the behaviour of the solutions to the differential equation.
Because this sort of problem is in all likelihood new to the student, it is worth considering alternative presentations that may tend to emphasize other aspects of the problem. One such presentation and the one we have chosen to depict in a film, is the presentation of the phase plane in which the solution is plotted against its spatial derivative. A description of the film follows.
The Film: The square Well in the phase plane
The reduced radial wave equation with zero angular momentum (or the one, dimensional square well with odd parity wave functions) is:
-d2y / dx2 + V(x) u = Eu
V(x) = -V0 when x ≤ a
V(x) = 0 when x > a
and h = 2m = 1
A straightforward calculation indicates that for x < a the trajectory in the u, du/dx plane is elliptical and that for x > a the solution u(x) is such that u(x) approaches esqrt(Ex) as x approaches infinity when E is not one of the eigenvalues of the system.
Since in this region the derivative is proportional to the function the trajectory in the phase plane is a straight line whose slope is governed by the trial eigenvalue. If the trial eigenvalue is one of the allowed energies of the system the trajectory asymptotically approaches the origin.
For this film we chose a=π and V0=30.
In the first scene, the differential equation is integrated numerically with a trial eigenvalue and the trajectory in the phase plane is traced out. This process is repeated several times as the trial eigenvalue is changed in an attempt to converge on one of the eigenvalues of the well. In watching this sequence a student should be able to tell on which side of the real eigenvalue the trial eigenvalue is and also to be able to account for the varying rate at which the trajectory is traced out. Further the student should be able to describe how to deduce the trial eigenvalue from the asymptotic slope of the phase plane trajectory in the units of the plot and be able to describe qualitatively how this scene would differ if even functions were being generated rather than odd ones.
In the second scene, the trial eigenvalue is changed smoothly from -V0 ≤ E ≤ 0 and the resulting trajectories shown. An energy scale keeps track of the value of E and a mark is made on the scale each time an eigenvalue is passed. Since the initial value of the slope is the same for each value of E, the elliptical portion of the trajectory gets smaller as the trial energy increases. The student should be able to explain this fact and be able to discuss the problem of normalizing the eigenstates. If these trajectories are interpreted as being appropriate to the odd solutions to the Schrodinger equation in one dimension, the student should be able to discuss the possible number of bound states of even as well as odd parity.
In the final scene the value of E is set to zero and the well depth is varied so that -30 ≤ V0 ≤ 0 and the resulting trajectories shown as V0 varies smoothly. The student should be able to account for all the features of any trajectory in a qualitative way as well as be able to make inferences about the number of bound states such a well could support.
This problem is a simple one. It is analytically soluble. Perhaps for these very reasons it is useful to look at it from a quite different point of view. The author will consider himself well rewarded if some students do so with relish and are led to re-examine other things they already know.