= Madelung equations
{c}
{title2=$\psi=R e^{iS}$}
{wiki}
Writing the <wavefunction> in modulus and dimensionless <quantum phase> gives a <probability continuity equation> and a <Hamilton-Jacobi equation> with a <quantum potential>:
$$
\partial_t(R^2)+\nabla\cdot\left(R^2\frac\hbar m\nabla S\right)=0,\qquad
\hbar S_t+\frac{\hbar^2}{2m}|\nabla S|^2+V-\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}=0.
$$
The phase equations hold locally away from <wavefunction> nodes. They follow by equating real and imaginary parts of the <Time-dependent Schrodinger equation>.
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