S1: The Infinite Square Well

For the first portion of the lab, we will solve for the states of fixed energy for a particularly simple potential: an infinite square well. The potential V(x) is zero for 0 < X < Xshelf = 1nm, and infinite everywhere else. (We call it Xshelf because it will be the start of the shelf for the next section.) Since the electron must be inside the box, and the wave-function is continuous, we have the boundary conditions

at both X=0 and X=Xshelf.

Since V(x)=0 inside the well, we are solving a familiar equation: the second derivative of with respect to X is a constant times . We need to find a solutions which vanish at the two ends. This is exactly the same equation we solved for the simple harmonic oscillator, except we have X instead of time t. It's also exactly the same problem as solving for the standing waves on a string with fixed boundary conditions.

(S1a) Analytic Solutions

In your writeup, draw a picture of the ground state (the lowest energy state, with the fewest antinodes). Either by following the derivation in class, or by analogy to the pendulum or the standing waves on a string, or by consulting your text, calculate the energy of this lowest energy state, as a formula involving m and

. Draw pictures of the first three excited states, and calculate their energies too. Include these in your writeup.

To evaluate your formulas for these energies, you'll need to figure out . You can do this in two different ways. You need the answer in energy units of eV and length units in nm (10^{-9} m). You can work everything out in eV: the rest mass of the electron

m c^2 = 0.511 MeV,

h = 4.1357x10^{-15} eV s, and

c = 2.997925x10^8 m/s.
Alternatively, you can work in MKS units:
m = 9.1095x10^{-31} kg,

h = 6.6262x10^{-34} J s, and

1 eV = 1.60219x10^{-19} J.
Remember that

is h divided by two pi.

(S1b) Numerical Solutions

Using the program schrdgr, vary the energy to find values which make

at X=1nm. Show that they agree with the values you calculated, to at least three significant figures. (You'll need to zoom in to your final point to make sure the last few digits are correct.) Include the comparison in your writeup: either show that the best energies agree with your theoretical estimates, or that using your theory numbers make the wavefunction vanish to the required accuracy.

(S1c) Too low and too high energies.

Make a freehand sketch in your writeup of what happens to the wave function when the ground state energy guess is a little too low. Explain why this is true, using the Schrödinger equation and discussing the curvature of the wave function and its dependence on the energy E.

(S1d) Energy level diagram.

Draw in your writeup an energy level diagram. This is pretty trivial: draw a vertical axis, label it E, and mark with short (1 or 2cm) horizontal lines the values of the four energies you derived. When the electron shifts from one energy state En to another Em, a photon (a quantum particle of light) is emitted, whose energy is En-Em, and whose frequency (in cycles per second) is its energy divided by Planck's constant h=4.1357 x 10^{-15} ev sec, and whose wavelength is thus the frequency divided into the speed of light (3x10^{8}m/sec). Will you be able to see light which is emitted when the electron drops from the first excited state into the ground state? What color will the light be when the electron drops from the second excited state into the first excited state (i.e., level 3 into level 2 on your diagram)?

Links Back

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).