Calculate the energies of the lowest three non-degenerate

Question

With the help of a computer and a commercial software package such as Excel, Mathematica, Mathcad, Maple, or Matlab to plot solutions to Schrodinger's Wave Equation (SWE) for the following problems:

1. Consider an electron in a plane crystal of a conducting material. Such an electron is free to move throughout the plane of the crystal, but cannot escape to the outside. It is trapped in a two-dimensional infinite well. The electron can move in two dimensions. The potential function is given by V (x, y) for 0 < x < L/2, 0 < z < L/2, and V(x,y) = ∞ elsewhere. Where, L = 0.5 x 10^-10 m

a. Start with Schrodinger's wave equation, use the separation of variables technique and find the wave solution (Ψ (x, y) ). (This is a hand derivation. Scan or provide a picture of your derivation steps for full credit.)

b. Prove that the total energy of this electron is given by

En = (h² / 8L²m) (n_x² +n_y²),      Equation 1

Is this Energy quantized or continuous? Explain why.

c. Calculate the energies of the lowest three non-degenerate states for an electron moving in a 2-D crystal of edge length L = 0.5 x 10^-10 m.

d. Using Matlab, graph the normalized wave solutions (ie. |Ψ (x, y)|²) for L = 0.5 x 10^-10 m associated with the energies of the lowest three nondegenerate states. (The lowest energy states must not have the same energy.

2. Consider an electron with energy E is incident on a step potential barrier as shown in Figure 1 below. Assume that the electron is traveling in the +x direction and that it originated at x = -∞. Assume that the total energy of the particle is less than the barrier height, or E < V_o. Also note that V(x) for x < 0; V(x) =V_o for x > 0.

a. Start with Schrodinger's time-independent wave equation, find the wave solution (Ψ (x)).

b. Find the ratio of the reflected wave coefficient to the incident wave coefficient.

c. Find the ratio of the transmitted wave coefficient to the incident wave coefficient.

(Note: Remember to label all axes and DO NOT FORGET UNITS!! Make sure to SHOW ALL WORK!!

Figure 1: Step Barrier Potential Well