An axially symmetric (for instance, harmonic) confining potential is commonly used for the study of vortices in BEC. To determine ϕ ϕ ( ρ ρ , z ) , the energy of ψ ψ ( r → → ) must be minimized, according to the constraint ψ ψ ( r → → ) = ϕ ϕ ( ρ ρ , z ) e i ℓ ℓ θ θ . This is usually done computationally, however, in a uniform medium, the following analytic form demonstrates the correct behavior, and is a good approximation: