Lets say we have a potential step with regions 1 with zero potential ##W_p\!=\!0## (this is a free particle) and region 2 with potential ##W_p##. Wave functions in this case are:(adsbygoogle = window.adsbygoogle || []).push({});

\begin{align}

\psi_1&=Ae^{i\mathcal L x} + B e^{-i\mathcal L x} & \mathcal L &\equiv \sqrt{\frac{2mW}{\hbar^2}}\\

\psi_2&=De^{-i\mathcal K x} & \mathcal K &\equiv \sqrt{\frac{2m(W_p-W)}{\hbar^2}}

\end{align}

Where ##A## is an amplitude of an incomming wave, ##B## is an amplitude of an reflected wave and ##D## is an amplitude of an transmitted wave. I have sucessfuly derived a relations between amplitudes in potential step:

\begin{align}

\dfrac{A}{D} &= \dfrac{i\mathcal L-\mathcal K}{2i\mathcal L} & \dfrac{A}{B}&=-\dfrac{i \mathcal L - \mathcal K}{i \mathcal L + \mathcal K}

\end{align}

I know that if i want to calculate transmittivity coefficient ##T## or reflexifity coefficient ##R## i will have to use these two relations that i know from wave physics.

\begin{align}

T &= \frac{j_{trans.}}{j_{incom.}} & R &= \frac{j_{ref.}}{j_{incom.}}

\end{align}

In above equations ##j## is a probability current which i also know how to derive (nice way is described here):

\begin{align}

j = -\frac{\hbar i }{2m}\left(\frac{d \psi}{dx}\psi^* - \frac{d\psi^*}{dx}\psi\right)

\end{align}

OK so far so good, but in our lectures professor somehow derived below equations which i can't derive. Could anyone please tell me how i can use all my knowledge described above to derive it?

\begin{align}

\boxed{R = \frac{(\mathcal{L - K})^2}{(\mathcal{L + K})^2}} && \boxed{T=\frac{4\mathcal{LK}}{\mathcal{(L+K)^2}}}

\end{align}

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# Transmission and Reflection on a potential step

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