5. Consider a particle of mass $m$ moving in 1-D for the following \"semi-infinite\" rectangular
well potential energy:
$U(x) = \begin{cases} +\infty, & x < 0 \\ -U_0, & 0 \le x \le a \\ 0, & x > a \end{cases}$
where $U_0 > 0$.
a) Write down the general form of the wavefunction in the three regions of the $x$-axis for
$E < 0$ in terms of
$k_1 = \left(\frac{2m(U_0 + E)}{\hbar^2}\right)^{\frac{1}{2}}$
$k_2 = \left(\frac{-2mE}{\hbar^2}\right)^{\frac{1}{2}}$,
and in terms of two arbitrary constants A, B.
b) Using the boundary conditions, derive the following relationship, which determines the
bound state energy eigenvalues
$E: k_1a \cot(k_1a) = -k_2a$.
Show that the roots $E$ for this transcendental equation can be obtained from the inter-
section of
and the circle
$k_2a = -k_1a \cot(k_1a)$,
$k_2a = \left(\frac{2mU_0a^2}{\hbar^2} - (k_1a)^2\right)^{\frac{1}{2}}$,
on a \"$k_2a$ v. $k_1a$\" graph.
