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C.3 The Rectangular Potential Barrier

    We calculate the tunnel transmission coefficient for an electron wave packet which hits the rectangular potential barrier with height E0 and width d (see Fig. C.3).
Figure C.3: Tunneling through a rectangular barrier with width d and height E0.

Starting with an Ansatz for the three regions
\psi_1 &= A_1e^{\text{i}k_1x}+B_1e^{-\text{i}k_1x} &\qquad k_...{1}{\hbar}
\psi_3 &= B_3e^{-\text{i}k_1x}
and the boundary conditions
\begin{gather}\psi_1(0) = \psi_2(0) \qquad \frac{\partial\psi_1(0)}{\partial x} ...
...rtial\psi_2(d)}{\partial x} =
\frac{\partial\psi_3(d)}{\partial x}
one obtains
A_1 &= \frac{1}{2}\left[A_2\left(1-\frac{\text{i}k_2}{k_1}\ri...
The ratio of incoming and transmitted probability amplitude is
\begin{gather}\frac{\vert B_1\vert^2}{\vert B_3\vert^2}=\frac{1}{2}-\frac{1}{8}\...
The transmission probability depends exponentially on thickness and height of the barrier.

Christoph Wasshuber