Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 57 / 1 / a / i

Table of contents
- Solution i

Solution

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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