The non-relativistic Hamiltonian for a quantized charged particle of mass m in a classical electromagnetic field is (in cgs units) H = 1 2 m ( p − − q A c ) 2 + q ϕ ϕ where A is the three-vector potential and φ is the scalar potential. This form of the Hamiltonian, as well as the Schrödinger equation Hψ = iħ∂ψ/∂t, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation A → → A ′ = A + ∇ ∇ Λ Λ ϕ ϕ → → ϕ ϕ ′ = ϕ ϕ − − 1 c ∂ ∂ Λ Λ ∂ ∂ t ψ ψ → → ψ ψ ′ = U ψ ψ H → → H ′ = U H U † † , where U = exp ⁡ ⁡ ( i q Λ Λ ℏ ℏ c ) and Λ = Λ(x, t) is the gauge function.