Here, α = ( ϵ 1 + ϵ 2 ) 2 {\displaystyle \alpha ={\frac {(\epsilon _{1}+\epsilon _{2})}{2}}} and δ = ( ϵ 1 − ϵ 2 ) 2 {\displaystyle \delta ={\frac {(\epsilon _{1}-\epsilon _{2})}{2}}} are real numbers. The matrix σ 0 {\displaystyle \sigma _{0}} is the 2 × {\displaystyle \times } 2 identity matrix and the matrices σ k {\displaystyle \sigma _{k}} ( k = 1 , 2 , 3 ) {\displaystyle (k=1,2,3)} are the Pauli matrices. This decomposition simplifies the analysis of the system, especially in the time-independent case where the values of α , β , γ {\displaystyle \alpha ,\beta ,\gamma } and δ {\displaystyle \delta } are constants.