= Solution
Use periodic <plane waves> on the ring. For $n\in\mathbb Z$,
$$
\phi_n(x)=\frac{e^{inx/a}}{\sqrt{2\pi a}},\qquad H\phi_n=\frac{\hbar^2n^2}{2ma^2}\phi_n.
$$
They are <orthonormal> on one circumference and <complete> by <Fourier series>. Thus $\boxed{E_k=\hbar^2k^2/(2ma^2)}$, with <quantum degeneracy>[degeneracy] $\boxed{1\text{ for }k=0,\quad2\text{ for }k\geq1}$: the two signs of nonzero <angular momentum> have the same energy. Equivalently one may use <normalization>[normalized] sine and cosine <eigenfunctions> for each positive $k$.
For the position measurement put $\theta=x/a$. Direct integration gives
$$
\boxed{\mathbb P(\tfrac12\pi a\leq x\leq\tfrac32\pi a)=\frac1\pi\int_{\pi/2}^{3\pi/2}\sin^2(\theta/2)\,d\theta=\frac12+\frac1\pi.}
$$
The energy measurement requires projection onto whole degenerate <eigenspaces>, rather than just onto one traveling wave. The expansion coefficients are
$$
c_n=\langle\phi_n,\psi\rangle=\frac1{\pi\sqrt2}\int_0^{2\pi}\sin(\theta/2)e^{-in\theta}\,d\theta=\frac{2\sqrt2}{\pi(1-4n^2)}.
$$
For instance the imaginary integral vanishes by reflection about $\theta=\pi$, and the product-to-sum identity evaluates the real integral as $4/(1-4n^2)$. Consequently
$$
\boxed{p_0=\frac8{\pi^2},\qquad p_k=|c_k|^2+|c_{-k}|^2=\frac{16}{\pi^2(4k^2-1)^2}\quad(k\geq1).}
$$
Every 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 <normalization>[normalized] periodic $L^2$ state; its derivative need not obey the operator-domain periodic condition for the spectral probabilities to be defined.
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