Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S3 / 3

The one-dimensional time-independent Schrödinger equation isThe differential term is the kinetic energy operator acting on the wave function, $V(x)$ supplies the multiplicative potential energy, and their sum is the Hamiltonian operator. The eigenvalue $E$ is the stationary state's total energy. This form assumes a nonrelativistic particle, a time-independent scalar potential, and no additional internal degrees of freedom in the stated model.

The wave function is a probability amplitude: by the Born rule, $|\psi (x){|}^{2}dx$ is the probability of a position measurement in an interval of width $dx$. For a normalized bound state $\int |\psi {|}^{2}dx=1$, the expected values areThe integrals run over the allowed domain. If the given state is not normalized, divide both by $\int |\psi {|}^{2}dx$. Its variance is $⟨x^2⟩-⟨x{⟩}^2$; neither $\psi$ itself nor its overall phase is a directly measured position probability.

Each energy eigenstate acquires its own phase under the time-independent Hamiltonian operator:Orthonormality removes all off-diagonal terms from the expected value of the energy, givingReturn up to an overall phase requires the relative phase to return to one. For $E_1\ne E_2$, the shortest positive recurrence time isIf the two energies coincide, the whole superposition is stationary up to a common phase for every time; there is then no smallest strictly positive recurrence time. A half-period changes the relative sign and produces the orthogonal antisymmetric state rather than the original symmetric state.

Projection onto the antisymmetric quantum superposition gives amplitude $\frac{1}{2}(e^{-i{E}_{1}t/\hslash }-e^{-i{E}_{2}t/\hslash })$. Its squared modulus isIt vanishes initially, reaches one at a half-recurrence time, and returns to zero after $\tau$. This interference depends on a relative phase even though each energy-eigenstate population and the mean energy remain constant.

An ideal projective measurement yielding the antisymmetric state prepares that state at $t_1$. Starting from it, the probability of reaching the symmetric state after an additional interval $s$ is again ${\sin }^2((E_2-E_1)s/(2\hslash ))$. At the full recurrence interval found above,The full-period propagator is proportional to the identity on the two-dimensional subspace, so it returns the antisymmetric state to itself, not to its orthogonal partner. The result assumes the stated time-independent evolution continues after the measurement.

Choose the infinite well as $0<x<a$ with real eigenfunctions ${\psi }_n=\sqrt{2/a}\sin (n\pi x/a)$. This fixes the relative phase convention for the specified initial plus state. Each of the first two states has probability $1/2$ in the left half. Their cross integral isThe evolving superposition therefore givesSince $E_n=n^2{\pi }^2{\hslash }^2/(2m{a}^2)$, the angular frequency and ordinary frequency areThe probability varies between $1/2\pm 4/(3\pi )$ and remains in $[0,1]$. Changing the sign convention of one eigenfunction while keeping the same physical initial state requires changing its superposition coefficient too; otherwise one has prepared the other initial state and reversed the initial spatial bias.