A. Zee, Quantum Field Theory in a Nutshell · Problem V.1.2

Field Theory Redux

Physics — QFT

Problem

To confine the superfluid in an external potential

W(\vec{x})

we would add the term

-W(\vec{x})\phi^\dagger(\vec{x}, t)\phi(\vec{x}, t)

to (1). Derive the corresponding equation of motion for

\phi

. The equation, known as the Gross-Pitaevskii equation, has been much studied in recent years in connection with the Bose-Einstein condensate.