Assignment: Solid State Devices

1. The wavefunction of a 1-D particle is: Ψ = Be^-2x for x ≥ 0 and Ψ = Ce^3x for x < 0, where B and C are real constants. Find the values of B and C to make y a valid wavefunction. Also, where in space is the particle most likely to be?

2. A particle is described by a plane-wave wavefunction Ψ(x,t) = Ae^{j(10x + 3y - 4t)}. You need to calculate the expectation value of a physical quantity U, which is given by the function U = 4p_x²+ 2p_z²+ 7mE. Here p_x and p_z are the x and z components of the momentum, m is the mass and E is the energy of the particle and your result should be be given in terms of the Plank's constant.

3. Calculate the first 3 energy levels for an electron in a 1-D quantum well with width of 2nm and infinitely high and steep walls.

4. Schematically show the number of electrons in the various subshells of an atom with the electronic shell structure 1s²2s²2p⁴ and an atomic weight of 21. How many protons and neutrons are in the nucleus of this atom? Is this atom chemically reactive or chemically inert and why?

5. Calculate the values for the Fermi-Dirac distribution function f(E) at 300K and plot (preferably by using Matlab) these values vs. the energy in eV. Choose the Fermi level to be E_F = 1eV and make the calculated points closer together in the vicinity of E_F to obtain a smooth curve. You will notice that f(E) varies quite rapidly within a few kT of E_F. Since f(E) is related to the probability that an energy level at E is occupied, show that the probability that a state located ΔE above E_F is occupied is the same as the probability that a state located ΔE bellow E_F is empty.