Question

Quantum Mechanics

A steady stream of 5eV electrons impinges on a square-hill barrier produced by a 10V potential. If half the electrons penetrate the barrier, how thick is it? How many would penetrate if the barrier were (a) twice as thick, (b) twice as high and © a 10V square well of the original thickness rather than a square hill?

Answer 1

question a)
the situation of your problem can be described as below:

Answer 2

into the zone #1 the wave function can be approximated by this function:
\[\begin{gathered}
\varphi \left( x \right) = \exp \left( {ikx} \right) + A\exp \left( { - ikx} \right) \hfill \\
\hfill \\
\exp \left( {ikx} \right) - - - > incident\;wave \hfill \\
A\exp \left( { - ikx} \right) - - - > reflected\;wave \hfill \\
\end{gathered} \]

Answer 3

where:\[\large k = \sqrt {\frac{{2mE}}{{{\hbar ^2}}}} \]
into the zone #2, the wave function can be approximated by this function:
\[\large \varphi \left( x \right) \approx B\exp \left( { - qx} \right),\quad q = \sqrt {\frac{{2m\left( {{V_0} - E} \right)}}{{{\hbar ^2}}}} \]

Answer 4

now, the transmission coefficient T of the transmitted wave, namely of the wave into the zone#3, is given by the subsequent quantity:
\[\Large T = {\left| {\varphi \left( a \right)} \right|^2} \approx \exp \left( { - 2qa} \right)\]
where a is the thickness of the barrier.
So we have:
\[\large T \approx \exp \left\{ { - 2a\sqrt {\frac{{2m\left( {{V_0} - E} \right)}}{{{\hbar ^2}}}} } \right\}\]
Applying the condition of your problem, we have to solve this equation, for a:
\[\Large \exp \left\{ { - 2a\sqrt {\frac{{2m\left( {{V_0} - E} \right)}}{{{\hbar ^2}}}} } \right\} = \frac{1}{2}\]