Using all of the above information to plug into the Dirac equation results in E [ ϕ ϕ χ χ ] = [ m I σ σ → → ⋅ ⋅ p → → σ σ → → ⋅ ⋅ p → → − − m I ] [ ϕ ϕ χ χ ] . This matrix equation is really two coupled equations: ( E − − m ) ϕ ϕ = ( σ σ → → ⋅ ⋅ p → → ) χ χ ( E + m ) χ χ = ( σ σ → → ⋅ ⋅ p → → ) ϕ ϕ