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Madelung equations 2026-10-07

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 (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 Schrodinger equation.

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Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 57 / 1 / a / ii / Solution Created 2026-10-03 Updated 2026-10-07

In Bohmian mechanics the particle has a definite position

X (t)

at every time. Its wavefunction obeys the usual autonomous wave equation, while its actual position follows the guidance equation

\dot{X} (t) = v (X (t), t), v = \frac{ℏ}{m} \nabla S = \frac{j}{∣ ψ ∣ ^{2}} .

Here

j

is the probability current. The Born rule is the quantum-equilibrium choice of initial position distribution

ρ = ∣ ψ ∣^{2}

; quantum equilibrium equivariance ensures that this distribution persists because it obeys the same probability continuity equation as the wave amplitude. The guidance equation fixes the initial velocity as well as subsequent velocities: the second-order equation below does not permit an independent arbitrary initial velocity.

Define the quantum potential

Q = - \frac{ℏ ^{2}}{2 m} \frac{\nabla ^{2} R}{R} .

Taking the gradient of the real Madelung equations gives

ℏ \partial_{t} \nabla S + \nabla (\frac{ℏ ^{2}}{2 m} ∣\nabla S ∣^{2}) = - \nabla (V + Q) .

On any smooth phase patch

\nabla \times v = 0

, so

\nabla (∣ v ∣^{2} /2) = (v \cdot \nabla) v

. Along the actual path, differentiation is the material derivative

D / D t = \partial_{t} + v \cdot \nabla

. Therefore

\frac{d ( m X ˙ )}{d t} = m \frac{D v}{D t} = - \nabla (V + Q)_{X (t)} .

This is the Bohmian mechanics Newton form: the classical force is supplemented by the amplitude-dependent quantum potential. Neither division by

R

nor a smooth phase is justified at a wavefunction node, so the derivation applies on nonzero-amplitude regions. A nonzero circulation around a node is compatible with the locally curl-free guidance equation.

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Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 57 / 1 / a / i / Solution Created 2026-10-03 Updated 2026-10-07

Use the dimensionless quantum phase

S

, so that

ψ = R e^{i S}

with

R \geq 0

. Work locally away from wavefunction nodes, where

R

and

S

are differentiable. Differentiating the wavefunction for substitution in the Time-dependent Schrodinger equation gives:

\partial_{t} ψ = e^{i S} (R_{t} + i R S_{t}), \nabla^{2} ψ = e^{i S} [\nabla^{2} R - R ∣\nabla S ∣^{2} + i (2\nabla R \cdot \nabla S + R \nabla^{2} S)] .

Cancel

e^{i S}

in the time-dependent equation and equate real and imaginary parts. The resulting Madelung equations are

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

The second equation is a Hamilton-Jacobi equation for the action

ℏ S

, with an additional quantum potential. To see the meaning of the first, multiply it by

2 R

and set

ρ = R^{2}

. It becomes the probability continuity equation

\partial_{t} ρ + \nabla \cdot (ρ \frac{ℏ}{m} \nabla S) = 0.

Thus the probability density is transported by the velocity field

ℏ\nabla S / m

. The Madelung equations are a local rewriting of the linear wave equation; their apparent nonlinearity comes from expressing a complex wavefunction in modulus and phase variables.

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