= Rigid translation from a momentum term in the Hamiltonian
{title2=$e^{-i(H_0-sP+C)t/\hbar}f(x)=e^{-iCt/\hbar}(e^{-iH_0t/\hbar}f)(x+st)$}
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 $P=-i\hbar\partial_x$, the factor $e^{isPt/\hbar}=e^{st\partial_x}$ 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 $-sP$.
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