Time-dependent Schrödinger equation ( $i ℏ \partial_{t} ψ = H ψ$ )

i ℏ \frac{\partial ψ}{\partial t} = H ψ .

Madelung equations ( $ψ = R e^{i S}$ )

\partial_{t} (R^{2}) + \nabla \cdot (R^{2} \frac{ℏ}{m} \nabla S) = 0, ℏ S_{t} + \frac{ℏ ^{2}}{2 m} ∣\nabla S ∣^{2} + V - \frac{ℏ ^{2}}{2 m} \frac{\nabla ^{2} R}{R} = 0.

The phase equations hold locally away from wavefunction nodes. They follow by equating real and imaginary parts of the Time-dependent Schrödinger equation.

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