---
tag: F5EAWFMC8A
kind: concept
name: Gross–Pitaevskii equation
msc: [35Q55, 35Q40]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Gross–Pitaevskii_equation
same_as:
  wikidata: Q910667
  wikipedia: Gross–Pitaevskii equation
---

## Summary

The Gross–Pitaevskii equation describes the ground state and the dynamics of a Bose–Einstein condensate.
It is a [[nonlinear Schrödinger equation|GEJHRXC176]] with a cubic term, which models the interaction between the particles, and an external trapping potential.
Its unknown is a single complex wave function for the whole condensate.
It can be derived from the many-body quantum mechanics of bosons in a suitable limit of many weakly interacting particles.

## Explanations

### L1 human

At very low temperatures a gas of bosonic atoms can collapse into one shared quantum state, a Bose–Einstein condensate.
Then a single wave function describes all the atoms at once.
That wave function feels the trap that holds the gas and also the repulsion or attraction between the atoms, which depends on how dense the gas is.
The Gross–Pitaevskii equation is the Schrödinger equation with this density-dependent extra term.