= Current of a translated Gaussian coherent state
{title2=$\psi=\left(\frac\alpha\pi\right)^{1/4}e^{-\alpha(x-q)^2/2+i p x/\hbar+i\gamma},\quad j=\frac pm|\psi|^2$}
Here $\alpha>0$ and $q,p,\gamma$ are real functions of time. Differentiation in position gives $\psi_x=[-\alpha(x-q)+ip/\hbar]\psi$. The standard <probability current> is therefore
$$
j=\frac\hbar m\operatorname{Im}(\overline\psi\psi_x)=\frac pm\sqrt{\frac\alpha\pi}\,e^{-\alpha(x-q)^2}.
$$
For a <coherent state> moving in a quadratic <Hamiltonian>, the centre follows the classical equation $\dot q=p/m$, so the current is the normalized <normal distribution>[Gaussian probability density] transported at its centre velocity. The factor $\sqrt{\alpha/\pi}$ is essential: omitting it changes the current by a dimensional constant and violates the <continuity equation> for the normalized density. A spatially constant phase does not affect the current.
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