This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function. Pauli had introduced the 2 × 2 sigma matrices as pure phenomenology — Dirac now had a theoretical argument that implied that spin was somehow the consequence of incorporating relativity into quantum mechanics. On introducing the external electromagnetic 4-potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form (in natural units ħ = c = 1) [ − − i γ γ μ μ ( ∂ ∂ μ μ + i e A μ μ ) + m ] ψ ψ = 0 where γ γ μ μ are the gamma matrices (known as Dirac matrices) and i is the imaginary unit. A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by i, have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the units restored: ( ( m c 2 − − E + e ϕ ϕ ) c σ σ ⋅ ⋅ ( p − − e c A ) − − c σ σ ⋅ ⋅ ( p − − e c A ) ( m c 2 + E − − e ϕ ϕ ) ) ( ψ ψ + ψ ψ − − ) = ( 0 0 ) . so ( E − − e ϕ ϕ ) ψ ψ + − − c σ σ ⋅ ⋅ ( p − − e c A ) ψ ψ − − = m c 2 ψ ψ + − − ( E − − e ϕ ϕ ) ψ ψ − − + c σ σ ⋅ ⋅ ( p − − e c A ) ψ ψ + = m c 2 ψ ψ − −