Past exam of the options I course of the University of Oxford / 2004 / J / J1

The Time-dependent Schrödinger equation isA normalized quantum state has ${\int }_{{\mathbb{R}}^3}|\psi {|}^2{d}^{3}x=1$; the Hamiltonian operator acts on wave functions satisfying the appropriate domain and boundary conditions.

For hard-wall confinement, the stationary spatial modes and energies areTheir stationary time dependence is $e^{-i{E}_{n}t/\hslash }$. The correct triple-angle identity is $\sin 3\theta =3\sin \theta -4{\sin }^3\theta$; the printed hint has its right side reversed in sign. Thus the specified initial state expands asThe coefficients have squared moduli summing to one. The first requested energy is $E_1$, so its probability is $\boxed{9/10}$. The other requested energy is $2E_1$, which is not $n^2{E}_1$ for any positive integer $n$, so its probability is $\boxed{0}$. The occupied excited energy is instead $E_3=9E_1$, with probability $1/10$.