The Madelung equations are derived by writing the wavefunction in polar form: ψ ψ ( x , t ) = 1 m ρ ρ m ( x , t ) e i ℏ ℏ S ( x , t ) , and substituting this form into the Schrödinger equation i ℏ ℏ ∂ ∂ ∂ ∂ t ψ ψ ( x , t ) = [ − − ℏ ℏ 2 2 m ∇ ∇ 2 + V ( x , t ) ] ψ ψ ( x , t ) .