Teaching Time Development
in Quantum Mechanics

This page written by Dan Styer, Oberlin College Physics Department;
last updated 24 October 1997.

A paper contributed to the meeting of the Ohio Section of the American Physical Society
at Miami University, Oxford, Ohio, 11 October 1997.

Abstract

Attention devoted to time development in a quantum mechanics course makes the subject less formal and more tangible. This can be done effectively through computer simulation or through analytic problems. Six specific problems are suggested, including an Ehrenfest treatment of the simple harmonic oscillator, displaced energy states of the simple harmonic oscillator, and quantal recurrence in the Coulomb problem.

Why do students find quantum mechanics so hard?

It's weird.
Or you could say difficult, or unfamiliar, or counterintuitive. The only thing to do about this is to acknowledge it.
It's mathematically intricate.
Hermitian operators, spherical harmonics, eigenfunction expansions. You name it, quantum mechanics has got it. It's hard for students to see whether they're having problems with the new physics or with the new math.
Students lack physical intuition.
Everyone is grossly familiar with pushes, with pulls, and with balls in flight. Even those who think that force is proportional to velocity instead of to acceleration have a reasonable idea of what a particle is. But students of quantum mechanics have little or no physical intuition to back up their intricate mathematics.
Courses emphasize the energy eigenproblem.
Most of a classical mechanics course is devoted to solving the initial value problem, but most of a quantum mechanics course is devoted to solving the more formal energy eigenproblem. It is the quantal time development problem that is analogous to the classical initial value problem, and it is in pursuing its solution that students develop a picture for quantum mechanics and its classical limit.

Time development through computer simulation

I shamelessly recommend my own award-winning simulation program Quantum mechanical time development (QMTime), published as part of J.R. Hiller, I.D. Johnston, and D.F. Styer, Quantum Mechanics Simulations (Wiley, New York, 1995).

Time development through analytic problems

Any of the six topics below can be worked into an assignment for students at the junior level. (I plan to provide downloadable TeX source for such assignments . . . but that's still under construction.)

Full treatment of the free particle.
The time evolution of a Gaussian wave packet can be worked out analytically: See, for example, Richard L. Liboff, Introductory Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1980), pages 151-155. This gives a complete solution, but it involves a lot of Fourier transforms and it's easy to bog down in the details.
Ehrenfest treatment of the free particle.
The time development of the mean and uncertainties of both position and momentum can be worked out through a few simple commutators. See Daniel F. Styer, "The motion of wavepackets through their expectation values and uncertainties", American Journal of Physics 58 (1990) 742-744, parts II and III.
Ehrenfest treatment of harmonic oscillator.
Same as above, except parts II and IV. The mean position bobs back and forth with the classical period, and the mean uncertainty is periodic with half that period.
Displaced energy states of the harmonic oscillator.
Displace any energy eigenstate and let it fly. The probability density will bob back and forth rigidly. A very surprising result! C.C. Yan, "Soliton-like solution of the Schrodinger equation for simple harmonic oscillator", American Journal of Physics 62 (1994) 147-151.

Quantal recurrence in the infinite square well.
Any wavefunction in an infinite square well of width L is periodic in time with a period

                           4 m L^2
                           -------.
                           hbar pi

Quantal recurrence in the Coulomb problem.
Any quantal state consisting of a superposition of two or more bound energy eigenstates with principal quantum numbers n₁, n₂, . . ., n_r evolves in time with a period of
```
                           h  N^2
                           ------,
                             Ry
```
where the integer N is the least common multiple of n₁, n₂, . . ., n_r.

A fundamental misconception

Even the best of us make errors. Linus Pauling and Roger Hayward, The Architecture of Molecules (Freeman, San Francisco, 1964), plates 2 and 3, suggest that in quantum mechanics particles really are tiny classical entities with definite values for both position and momentum, but that they move in a blur, too quickly to be pinned down.

Similarly, the video Eureka! The Conduction of Heat "illustrates how electrons whiz so quickly around the nucleus that they appear to form layers". (See the 1997-98 physics catalog from "Films for the Humanities and Sciences", 1-800-257-5126, page 3.) [This reference was brought to my attention through a comment at my Miami University talk by Professor Joseph West of Wabash College.]

In both cases powerful teaching tools (beautiful visualization and dynamic video, respectively) are used to effectively propagate a misconception. It is hard to see how such a picture could give rise to quantal interference. And it leaves completely open the question of how much time development is due to this postulated blur and how much is encompassed by regular quantal time evolution.