Past exam of the quantum mechanics quantum theory and quantum computers course of the University of Oxford / 2008 / A / 1 / Solution

Use periodic plane waves on the ring. For $n\in \mathbb{Z}$,They are orthonormal on one circumference and complete by Fourier series. Thus $\boxed{E_k={\hslash }^2{k}^2/(2m{a}^2)}$, with degeneracy $\boxed{1\text{ for }k=0,2\text{ for }k\ge 1}$: the two signs of nonzero angular momentum have the same energy. Equivalently one may use normalized sine and cosine eigenfunctions for each positive $k$.

For the position measurement put $\theta =x/a$. Direct integration givesThe energy measurement requires projection onto whole degenerate eigenspaces, rather than just onto one traveling wave. The expansion coefficients areFor instance the imaginary integral vanishes by reflection about $\theta =\pi$, and the product-to-sum identity evaluates the real integral as $4/(1-4n^2)$. ConsequentlyEvery numerator is positive and no integer $k$ makes a denominator zero, so every energy has nonzero probability. Parseval's identity guarantees that these probabilities sum to one. The supplied state is a valid normalized periodic $L^2$ state; its derivative need not obey the operator-domain periodic condition for the spectral probabilities to be defined.