Solution to 2008 Problem 89

In order for the wavefunction and its first derivative to be continuous at x = 0, we must have that
\begin{align*}A + B &= C \\i k_1 A - i k_1 B &= ik_2 C\end{align*}
Solving for B in terms of A gives
\begin{align*}B = A\frac{\frac{k_1}{k_2} - 1 }{\frac{k_1}{k_2} + 1 }\end{align*}
So, the reflection coefficient is
\begin{align*}R = \left|\frac{B}{A}\right|^2 = \boxed{\left(\frac{k_1 - k_2}{k_1 +k_2} \right)^2}\end{align*}
Therefore, answer (D) is correct.

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