= Solution
The specified initial state is normalized. The requested energy is $E_2$, and its <probability amplitude> is
$$
c_2=\int_0^{a/2}
\sqrt{\frac2a}\sin\frac{2\pi x}{a}
\frac2{\sqrt a}\sin\frac{2\pi x}{a},dx
=\frac1{\sqrt2}.
$$
Thus the <Born rule> gives
$$
\boxed{\mathbb P(E=E_2)=|c_2|^2=\frac12}.
$$
Unitary evolution changes each energy coefficient only by a phase, so this probability remains $1/2$ at both $T$ and $2T$. If an <energy measurement> returns $E_2$, the state collapses to the nondegenerate eigenstate $\chi_2$; a subsequent energy measurement therefore returns $E_2$ with probability
$$
\boxed{1}.
$$
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