At this time launch the QM1D Javascript Applet (below, or here: http://www.falstad.com/qm1d-phys214/qm1d/). Java is required

Infinite Well

Open the program, which will by default give you the simulation of an infinite well potential. Drag the lower right corner to make the window as large as possible

The top portion of the display will now show the potential plus energy levels; the middle portion shows both the probability density (once you've selected a solution -- just click on one of the energy levels -- or made a superposition), i.e., the absolute square of the wave function, as a filled solid white area, as well as the eigenstate (as a yellow curve) corresponding to the energy level your cursor is nearest to; and the bottom portion with the circles shows the phasors corresponding to each of the eigenstate solutions for the problem.

Sliders on the right side of the display allow you to adjust various parameters. For now leave the Well Width setting at 2 nm. The program opens with the default mass set to mc² = 511 keV, the electron mass. Do not change this setting unless directed otherwise.

Force the system to be in its lowest energy state by clicking on Ground State. Note that the probability density is constant in time; however, you can see the rotation of the associated phasorremember, the rotation rate of the phasor is proportional to the energy of the state. (If you don't see the rotation, make sure the "Stopped" box is unchecked.)

Now use the cursor to make an equal superposition of the ground and first excited state: Do this by clicking the mouse inside the second phasor disk (at a point near the rim). You may need to reduce the Simulation Speed (slider to the left). Observe what happens.

Q1. What is the ratio of the energy of the n = 2 state to the n = 1 state for an infinite square well potential? You can use the program to find out: use the mouse to click on the ground state energy level, and read the energy value above the phasors; then use the mouse to click on the first excited state energy level, and read its energy. What is the ratio of these two energies?

Q2. In class we learned that the period of oscillation of a superposition of two energy eigenstates (with energies E_m and E_n) is just 1/f, where f = f_m - f_n, and f_m = E_m/h, f_n = E_n/h. Calculate this period using the energies you determined for the ground and first excited states.

Q3. Click the "Stopped" box and then "Clear" (to rest the clock to 0), and again create an equal superposition of the ground and first excited states by first clicking on the "Ground State" button, and then using the mouse to draw the second phasor to its maximum length; again point them in the same direction, so that the particle is entirely on one side of the well. Note the time should say "t = 0 sec". Move the "Simulation Speed" slider all the way to the left, and then unclick the "Stopped" box to start the simulation; stop it when the wave function has returned to its initial location. Enter the time on the clock; NOTE: the units are femtoseconds (10^-15 s).

Q4. Assuming the ground state and first excited state phasors are both maximal, youve created an EQUAL superposition of these two energy eigenstates. From the top "Measure" pulldown menu select "Measure energy". Note what energy you observe. Now start again with the EQUAL superposition of the ground and first excited state wave function, and repeat the "Measure energy" option; do this at least 10 times, noting the energy that is measured each time you collapse the wave function. What energies did you observe?

Q5. Given that youve created an EQUAL superposition, half the time you should measure one energy, half the time the other. What is the average energy for particles in your superposition?

1. E₁

2. E₂

3. E₁ + E₂

4. (E₁+ E₂)/2

Q6. The listed energy E above the phasors is the AVERAGE energy of a large ensemble of particles in this superposition state. What is that average energy?

Q7. Finally, put the system solely in the ground state by clicking the "Ground State" button. Note the value of the energy (in meV). Next slightly increase the width of the well using the slider.

Which of the following statements correctly describes the effect of increasing the well width?

1. The energy of the ground state decreases, as does the wavelength of the particle.

2. The energy of the ground state increases, as does the wavelength of the particle.

3. The energy of the ground state decreases, but the wavelength stays the same.

4. The energy of the ground state increases, but the wavelength stays the same.

5. None of the above