Infinitely deep and thin potential well: bound states
Consider the following one-dimensional potential:
$U(x) = u_0 \delta(x)$,
where $\delta$ denotes the Dirac delta. In this question we take $u_0 < 0$, which describes an
infinitely deep and thin potential well.
(a) Show that any normalisable energy eigenfunction must be given by the following
expressions in region 1 ($x < 0$) and in region 2 ($x > 0$), respectively:
$\psi_1(x) = A_1 \exp(\lambda x)$,
$\psi_2(x) = A_2 \exp(-\lambda x)$,
where in your answer you should find an explicit formula for $\lambda > 0$ in terms of
the energy $E$, the mass $m$ and the reduced Planck's constant $\hbar$.
Hint: First show that normalisability implies $E < 0$.
[8 marks]
(b) Using a matching condition at $x = 0$ for the wavefunction, find the constant $A_2$
in terms of the constant $A_1$.
[4 marks]
(c) Using the discontinuity in the derivative of the wavefunction at $x = 0$, implied
by the presence of the Dirac delta, find a formula for the allowed energy levels in
terms of $u_0$, $m$ and $\hbar$. How many energy levels are allowed?
[6 marks]
(d) Finally, normalise the wavefunction in order to find the constant $A_1$ in terms of
$u_0$, $m$ and $\hbar$.
[2 marks]
