Rigid translation from a momentum term in the Hamiltonian ($e^{-i(H_0-sP+C)t/\hslash }f(x)=e^{-iCt/\hslash }(e^{-i{H}_{0}t/\hslash }f)(x+st)$)

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 $-sP$.