- #1
71GA
- 208
- 0
Homework Statement
Beam of electrons with energy ##10eV## hits the potential step (##8eV## high and ##0.5nm## wide). How much of the current is transmitted?
Homework Equations
I know that energy mentioned in the statement is kinetic energy so keep in mind when reading that ##E\equiv E_k##.
In our case the kinetic energy is larger than the potential energy ##\boxed{E>E_p}## and this is why stationary states for the regions 1, 2, 3 (picture) are as folows:
\begin{align}
\psi(x)_1 &=Ae^{iLx}+B^{-iLx} && L=\sqrt{\tfrac{2mE}{\hbar^2}}=1.619\times 10^{10}\tfrac{1}{m}\\
\psi(x)_2 &=Ce^{iKx} +De^{-iKx} && K=\sqrt{\tfrac{2m(E-E_p)}{\hbar^2}}=0.724\times 10^{10}\tfrac{1}{m}\\
\psi(x)_3 &=Ee^{iLx}
\end{align}
The Attempt at a Solution
I first used the boundary conditions for the border 1-2 and got a system of equations:
\begin{align}
A+B &= C+D\\
iLA-iLB&=iKC-iKD\\
&\downarrow\\
A+B &= C+D\\
A-B &= \tfrac{K}{L}(C-D)\\
&\left\downarrow \substack{\text{I used the assumption that}\\\text{$E \gg E_p$ and got the simplified}\\ \text{relation between L and K:}\\ \tfrac{L}{K}=\sqrt{E/E_p}\gg 1 \Longrightarrow L\gg 1\\ \text{and hence}\\ \tfrac{K}{L}=\sqrt{E_p/E}\ll 1 \Longrightarrow K\ll 1} \right.\\
A + B &= C+ D\\
A-B &= 0\\
&\downarrow~\Sigma\\
2A &=C+D\\
&\downarrow\\
C&=2A-D
\end{align}
Now i used the boundary conditions for the border 2-3 and got a system of equations in which i inserted the above result:
\begin{align}
Ce^{iKd} + De^{-iKd} &= Ee^{iLd}\\
iKCe^{iKd} -iK De^{-iKd} &= iLEe^{iLd}\\
&\downarrow\\
Ce^{iKd} + De^{-iKd} &= Ee^{iLd}\\
Ce^{iKd} -De^{-iKd} &= \tfrac{L}{K}Ee^{iLd}\longleftarrow\substack{\text{Things get weird}\\\text{when i insert }C=2A-D}\\
&\downarrow\\
2Ae^{iKd} -De^{iKd} + De^{-iKd} &= Ee^{iLd}\\
2Ae^{iKd}-De^{iKd} -De^{-iKd} &= \tfrac{L}{K}Ee^{iLd}\\
&\downarrow \substack{ \text{$L$ is very big while $K$ is very small}}\\
2A &= Ee^{iLd}\\
2A -2D &= \tfrac{L}{K}Ee^{iLd}
\end{align}
Now this result is what i don't understand. First equation gives me a ratio ##E/A=2\exp[-iLd]## which i think i need to calculate the transmissivity, but it is a complex exponential. Is this possible? How do i continue? Is this the way to solve this kind of case or did i go completely wrong?
This is my first finite potential well problem and i have been searching the web for a loong time to even find 2 videos (video 1 and video 2) for a similar problem but for the case when ##E<E_p##. The professor in the video allso uses similar approximation - but in his case ##E \ll E_p##.
In the appendix there is allso a scan of the problem solved on the paper just in case anyone likes it this way. It is in Slovenian language but i added some headings to help you reading.
Attachments
Last edited by a moderator: