Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 84 / 1 / i

Table of contents
- Solution i

Solution

Treat

ψ

and its complex conjugate as independent variables in the functional derivative. Integrating the gradient variation by parts, with vanishing boundary terms, gives

δ E = \int [- \frac{ℏ ^{2}}{2 m} \nabla^{2} ψ + V_{0} ∣ ψ ∣^{2} ψ + W_{0} ∣ ψ ∣^{4} ψ] δ ψ^{*} d x + complex conjugate .

The factors follow from

\partial_{ψ^{*}} ∣ ψ ∣^{4} = 2∣ ψ ∣^{2} ψ

and

\partial_{ψ^{*}} ∣ ψ ∣^{6} = 3∣ ψ ∣^{4} ψ

. Thus the cubic-quintic Gross–Pitaevskii equation is

i ℏ \partial_{t} ψ = - \frac{ℏ ^{2}}{2 m} \nabla^{2} ψ + V_{0} ∣ ψ ∣^{2} ψ + W_{0} ∣ ψ ∣^{4} ψ .

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