Particle in a ring

A quantum particle constrained to a circle has integer angular-momentum eigenvalues and periodic position eigenfunctions. Its propagator can be written either as a sum over angular momenta or as a sum over winding number sectors.

For a free particle on the line or a circle, $H_0=P^2/(2m)$ commutes with the momentum operator $P$. The commuting operator exponentials in the title therefore factorize. Since $P=-i\hslash {\partial }_x$, the factor $e^{isPt/\hslash }=e^{st{\partial }_x}$ translates a wavefunction to $f(x+st)$, as is verified on each momentum eigenstate. Thus the probability density translates with velocity $-s$ relative to the $H_0$ evolution. A real constant $C$ changes only the global phase. In particular a superposition in one degenerate energy eigenspace translates rigidly under the added term $-sP$.

For a particle on a circle with arc length $x\sim x+2\pi R$ and periodic boundary conditions, the Hamiltonian operator $P^2/(2m)$ has orthonormal basis ${\varphi }_n=e^{inx/R}/\sqrt{2\pi R}$, $n\in \mathbb{Z}$, and energy eigenvalues $E_n={\hslash }^2{n}^2/(2m{R}^2)$. To see this, periodicity forces the wave number to be $n/R$, and applying the momentum operator $-i\hslash {\partial }_x$ gives $\hslash n/R$. The integral of $|{\varphi }_n{|}^2$ over one turn is one. The zero energy eigenvalue is simple; each positive energy eigenvalue has quantum degeneracy two, from $n$ and $-n$.