Question

Problem 16.1

P16.1 In this problem, you will calculate the transmission probability through the barrier illustrated in Figure 16.10. We first go through the mathematics leading to the solution. You will then carry out further calculations. The domain in which the calculation is carried out is divided into three regions for which the potentials are Aetikx + Be-ikx Region I ψ(x)-cexpFPWh-x] - 1 V(x) =0 for x 0 V(x) = Vo for 0 < x < a V(x) =0 for x za 2m(Vo -E) Region I Region II Region III + D expl + Ce-KX De+KX Region II + = 2mE 2mE だ The spatial part of the wave functions must have the follow- ing form in the three regions if E Vo: = Fe+ikx + Ge-ikx Region III

Answer 1

In this problem, you will solve for the total energy eigenfunctions and eigenvalues for an electron in a finite depth box. We first go through the calculation for the box parameters used in Figure 16.1. You will then carry out the calculation for a different set of parameters We describe the potential in this way: V(x) = V_0 for x lessthanorequalto -a/2 Region I V(x) = 0 for -a/2 < x < a/2 Region II V(x) = V_0 for x greaterthanorequalto a/2 Region III The eigenfunctions must have the following form in these three regions: Psi(x) = B exp[+ squareroot 2m(V_0 - E)/h^2 x] + B' exp[squareroot 2m(V_0 - E)/h^2 x] = Be^+kx + B' e^-kx Region I Psi(x) = C sin squareroot 2 m E/h^2 x + D cos squareroot 2 m E/h^2 x = sin kx + D cos kx Region II Psi(x) = A exp[-squareroot 2m(V_0 - E/h^2 x] + A' exp[+squareroot 2m(V_0 - E)/h^2 x] = A e^-kx + A' e^+kx Region III So that the wave functions remain finite at large positive and negative values of x,