Gross-Pitaevskii Equation: The mean-field description of a BEC is iℏ∂ψ/∂t = [−ℏ²∇²/2m + V(r) + g|ψ|²]ψ
where g = 4πℏ²a/m (a = scattering length) controls contact interactions. In the ground state,
ψ(r,t) = φ(r)e^{−iμt/ℏ}, giving the time-independent GPE: μφ = [−∇²/2 + ω²r²/2 + gN|φ|²]φ
(in dimensionless units with φ normalized to 1). Two limiting regimes:
Non-interacting (gN→0): φ is a Gaussian (harmonic oscillator ground state), μ = ℏω/2.
Thomas-Fermi (gN≫1): kinetic energy negligible, density profile n(r) = (μ − V(r))/gN
— an inverted parabola out to R_TF = √(2μ/ω²). The chemical potential scales as μ ~ (gN)^{2/5}.
Use the slider to explore the crossover from Gaussian to Thomas-Fermi regimes.