Reflection and Transmission

While the derivations for reflection and transmission coefficients are a bit arduous, eventually we can find the following for the finite well:

Finite Well ($E > 0$)

\[T = \dfrac{\vert F \vert^2}{\vert A \vert^2} = \dfrac{1}{1+\dfrac{(k_1^2-k_2^2)^2}{4k_1^2k_2^2}\sin^2(2k_2a)}\] \[R = \dfrac{\vert B \vert^2}{\vert A \vert^2} = \dfrac{1}{1+\dfrac{4k_1^2k_2^2}{(k_1^2-k_2^2)^2\sin^2(2k_2a)}}\]

where $k_1 = \sqrt{\dfrac{2mE}{\hbar^2}}$ and $k_2 = \sqrt{\dfrac{2m(E+V_0)}{\hbar^2}}$.

  1. Question: Consider a low energy beam. Show what happens to $T$ and $R$ under these conditions.
    • Discussion: Why do these results make physical sense?
  2. Question: Consider a high energy beam. Show what happens to $T$ and $R$ under these conditions.
    • Discussion: Why do these results make physical sense?
  3. Question: Set $2k_2a=n\pi$. Show what happens to $T$ and $R$ under these conditions.
    • Discussion: Why do these results make physical sense?