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Use the normalized eigenfunctions of the infinite square well,The sine identity expands the initial wavefunction asThe normalization factor $4/\sqrt{5a}$ yields precisely these coefficients. By the Born rule, the energy measurement probabilities are $9/10$ for $E_1$ and $1/10$ for $E_3=9E_1$. Thus the energy expectation isThe two probabilities sum to one, also checking the wavefunction normalization.

A stationary state has the formFor a time-independent quantum observable $Q$, the global phases cancel in the expectation value:It is therefore independent of time.

For the infinite square well,The possible energy measurement results arewith respective Born rule probabilities $|c_1{|}^2$ and $|c_2{|}^2$. Normalization gives $|c_1{|}^2+|c_2{|}^2=1$.

At time $t$,The required matrix elements areConsequentlywhere the angular frequency isFinally, $2|c_1{c}_2|\le |c_1{|}^2+|c_2{|}^2=1$, so

The specified initial state is normalized. The requested energy is $E_2$, and its probability amplitude isThus the Born rule givesUnitary 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