In 1921, German physicist Karl Ramsauer observed that at certain energies, electrons scattered abnormally little off argon atoms. This effect cannot be explained in terms of classical mechanics, so it played an important role in popularizing quantum mechanics in its early stages of development (Schrödinger's equation was first published five years after the discovery of the Ramsauer effect).

Consider a well of finite depth $U_0$ and width $a$:
\[U(x) = \begin{cases}
0, \quad & x < 0 \text{ or } x > a \\
-U_0, \quad & 0 < x < a
\end{cases}
\]

Let an electron with mass $m$ and momentum $\hbar k$ fall on it, i.e., its wave function is a de Broglie wave:
\[\Psi = e^{-ikx}\]This wave is partially reflectes from the well and partially transmits trough it. In other words, when $x<0$:
\[ \Psi_- = e^{-ikx} + r e^{ikx},\]where $r$ is the reflection coefficient. When $x>0$:
\[ \Psi_+ = t e^{-ikx},\]where $t$ is the transmission coefficient.

A1 Show that in the well the wavefunction $\Psi_\text{in}$ is
\[ \Psi_\text{in} = A e^{i q x} + B e^{-iqx}. \]Express $q$ in terms of $k$, $m$ and $U_0$.

The wave function and its derivative (if there are no infinitely large jumps in potential energy) must be continuous.

A2 Suggest the problem about wave optics, which is completely analogous to the one currently being considered.

A3 Find $|t|^2$ — the probability that the particle transmits through the well.

A4 Sketch the graph of $|t|^2$ vs $k$.