= Solution
Before the change, the normalized <ground state> of the infinite square well is
$$
\psi_0(x)=\sqrt{\frac2a}\sin\frac{\pi x}{a}.
$$
For an allowed post-quench energy $E\in(0,U_0)$ satisfying part (i), define the unnormalized eigenfunction
$$
\chi_E(x)=
\begin{cases}
\sin(kx),&0\leq x\leq a/2,\\[2pt]
\dfrac{\sin(ka/2)}{\sinh(la/2)}
\sinh(l(a-x)),&a/2\leq x\leq a.
\end{cases}
$$
Its normalization factor is
$$
N_E^{-2}=
\int_0^{a/2}\sin^2(kx)\,dx
+\frac{\sin^2(ka/2)}{\sinh^2(la/2)}
\int_{a/2}^{a}\sinh^2(l(a-x))\,dx.
$$
The <Born rule> therefore gives
$$
\boxed{
\operatorname{prob}(E)
=\frac{2N_E^2}{a}
\left|
\int_0^{a/2}\sin\frac{\pi x}{a}\sin(kx)\,dx
+\frac{\sin(ka/2)}{\sinh(la/2)}
\int_{a/2}^{a}\sin\frac{\pi x}{a}\sinh(l(a-x))\,dx
\right|^2 }.
$$
For a value of $E$ that is not an eigenvalue, this probability is zero. The sudden change leaves the wavefunction fixed, while the energy eigenbasis changes.
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