## Quantum States in a One Dimensional Box

By Neil Fleishon, California Polytechnic State University, San Luis Obispo, CA 93407

On the right is a little program which shows the time evolution of a quantum state in a potential well corresponding to an (essentially) impenetrable box. You will make some observations, take some data, make a some graphs, and calculate a few things. This will, hopefully, give you some idea of what a stationary state is, what it is good for, and how it differs from a non-stationary state.

NOTE: This applet seems to work on all browsers I have tried (Netscape 3.xx, 4.xx and IE4.xx). If you have trouble, see me.

### (Almost) Infinite Square Well

Command-click (Mac) or Right-click (PC) on the display to see applet help in a new window.

1. Stationary States. When the simulation comes up, it should be showing the time development of a stationary state wave function. Click the Stop button, then the Reset button to see the wave function at t=0.
(a) Sketch the wave function (at t=0) you see on the graph.
Click the run button and watch the state evolve.
(b) Stop the program at t=5 and t=10. Sketch the wave function at these times on the same graph as the t=0 wave function above.
(c) Answer these questions: What aspect(s) of the display change as time progresses? What aspect(s) don't?
(d) You can click and hold down the mouse over the display to read the value of the wave function. Doing this, estimate the probability that the particle is in the region between 6.995 and 7.005. Show your work (no integrals!).

2. Stationary State Energy Measurement. A stationary state has the form
PSI(x)=|PSI(x)|exp(-i2*pi*E*t/h). This means that the energy of the state can be determined from the rate of change of the phase. The phase is something we cannot determine in a simple experiment but, through the magic of this program, we can read the phase as a function of time by holding down the mouse (The phase is indicated by a "<" and measured in degrees.) and using the Stop/Run button.
(a) Make a table of at least 5 data points of phase vs time. The times should be as close together as possible. Why? From this data, find a good estimate for the energy E. (The units will be a problem. See the accompanying "Help" notes.) Show your work.
(b) Make a book calculation of the energy. What is the wavelength of the particle? What is its momentum? Find its energy? Show your work.

3. Make your own stationary state. Click on the tab marked QM above the display. This will take you to a place to input wave functions. Your job is to change the existing formula in the box marked "Real" to make a different stationary state.
(a) Write down the formula you entered in the box.
Click on the tab marked "Wavefunction" to go back to the display
(b) Sketch the graph of the wave function. Now run the program. How do you know that your state is truly stationary?

4. Make a non-stationary state. Click on the tab marked QM above the display. This will take you to a place to input wave functions. Enter "(sin(3*pi*x/10)+sin(4*pi*x/10)*(step(x)-step(x-10))" into the box marked "Real" part. Click on the tab marked "Wavefunction" to go back to the display .
(a) Sketch the wavefunction at t=0.
(b) Run the program and watch the wave function develop. On the sketch of part (a) overlay a sketch of the wave function at t=5 and t=10. Label your sketches. Is this a non-stationary state? How is its behavior different from the stationary states above? Explain.

Original file name: QTime help - witten by Neil Fleishon, nfleisho@calpoly.edu