Consider a particle of mass $m$ subject to the one-dimensional potential $V(x)$ that is given by
$$
\begin{array}{ll}
V=0 & \text { if }-a \leqslant x \leqslant a \quad \text { or } \quad \text { if }|x|>b \\
V=V_{0} & \text { if } a<|x| \leqslant b
\end{array}
$$
where $b>a$. Write down the form of an even-parity solution to the Schrödinger equation in each region in the case where $E<V_{0}$. Note that the particle is not bound in this potential as there is always a probability of quantum-mechanical tunnelling, so solutions exist for all values of $E$. Show, however, that if $\kappa(b-a) \gg 1$ (where $\kappa$ is defined as in problem 2.3) the probability of finding the particle inside the region $|x|<a$ is very small unless its energy is close to that of one of the bound states of a well of side $2 a$ bounded by potential steps of height $V_{0}$. In the case where this condition is fulfilled exactly, obtain an expression for the ratio of the amplitudes of the wavefunction in the regions $|x| \leqslant a$ and $|x|>b$.