# Quantum Well & Barrier w/ Phase Shift

1. Jan 25, 2009

### darkfall13

1. The problem statement, all variables and given/known data

V(x) = $$\inf$$ if x $$\leq$$ 0
= -V if 0 $$<$$ x $$<$$ a
= 0 if x $$>$$ a

NOTE: E>V

Find the wave functions in each region and the stated phase shift.

2. Relevant equations

Schrodinger Eq.

Instructions note that wave functions are:
$$\Psi_I$$ = $$A\sin{kx}$$
$$\Psi_{II}$$ = $$B\sin{k'x+\phi}$$

where,

k = $$\sqrt{2m(V+E)}/\hbar$$
k' = $$\sqrt{2mE}/\hbar$$

Show $$2\phi = 2\left[\cot^{-1}\left(\frac{k}{k'}\cot\left({ka}\right)\right)-k'a\right]$$

3. The attempt at a solution

I can get Region I just fine (0:a well)

$$\frac{-\hbar^2}{2m} \frac{d^2 \Psi}{dx^2} - V \Psi = E \Psi$$
$$-\frac{d^2 \Psi}{dx^2} = \left(E+V\right)\Psi\left(\frac{2m}{\hbar^2}$$
let $$k = \frac{\sqrt{2m\left(V+E\right)}}{\hbar}$$
$$\Psi_x\left(x\right) = A\sin{kx}$$

Region II

$$\frac{-\hbar^2}{2m} \frac{d^2 \Psi}{dx^2} = E \Psi$$
$$-\frac{d^2\Psi}{dx^2} = \frac{2mE}{\hbar^2}\Psi$$
$$\Psi_{II}\left(x\right) = B\sin{k'x}$$
where $$k' = sqrt{2mE}{\hbar}$$

But my problem lies in the phase shift. How is it related to the wave functions and how can it be calculated?