Madelung equations

Remarks

The imaginary part of the Schrödinger equation gives the continuity equation, and the real part gives the quantum Hamilton–Jacobi equation $\partial_t S + \abs{\nabla S}^2 / (2m) + V + Q = 0$, whose gradient is the momentum equation. The momentum equation is the Euler equation of a potential flow with the extra force $-\nabla Q / m$. Example: the plane wave $\psi = e^{i (k x - \omega t)}$ with $V = 0$ has $\rho = 1$, $S = \hbar (k x - \omega t)$ and $\mathbf{u} = (\hbar k / m) \mathbf{e}_x$. Then $Q = 0$, and both equations hold since $\rho$ and $\mathbf{u}$ are constant. The Hamilton–Jacobi equation gives $\omega = \hbar k^2 / (2m)$, the free dispersion relation. Non-example: at a zero of $\psi$ the phase is not defined, so the equations do not apply there. Around such a zero the circulation of $\mathbf{u}$ is a multiple of $2\pi\hbar/m$. To recover the Schrödinger equation from a solution of the Madelung equations, one needs this quantisation condition on the circulation.