Physics 401

More Rectangular Barrier Problems

In class we found the general solution to the square barrier problem is the following set of equations

$$\begin{eqnarray*}\phi_1 &=& Ae^{ik_1 x} + Be^{-ik_1 x} \\ \phi_2 &=& Ce^{ik_2 x} + De^{-ik_2 x} \\ \phi_3 &=& Fe^{ik_1 x} + Ge^{-ik_1 x} \end{eqnarray*}$$

where

$$k_1 = \sqrt{ 2mE \over \hbar^2 } \qquad k_2 = \sqrt{ 2m(E-V_0) \over \hbar^2} \qquad .$$

We were able to express the boundary conditions in the form of a transfer matrix so that

$$\begin{eqnarray*}\zeta_1 &=& {\rm\bf t} \zeta_3 \\ &=& {\rm\bf d}_{12} {\rm\bf p}_2 {\rm\bf d}_{21} \zeta_3 \end{eqnarray*}$$

where $\zeta_1$ and $\zeta_3$ are vectors representing the amplitudes of the incoming and outgoing waves, t is the transfer matrix, and ${\tt d}_{12}$, ${\tt p}_2,$ and ${\tt d}_{21}$ are the propagation and discontinuity matrices that compose t. We then found an expression for the transmission coefficient in terms of the components of t, namely

$$T = {1 \over \vert t_{11}\vert^2 }$$

where t₁₁ is a component of the transfer matrix.

1.

Obtain an analytical expression for the amplitude that is necessary to calculate the reflection coefficient. In other words, calculate the reflection coefficient without using the fact that R+T=1. How would you be able to check your result? Outline at least one method in detail.

2.

Obtain expressions for the remaining amplitudes that form the wave function.