Hamiltonian Operator and Eigenvalue Equation

Abstract

The Dirac Hamiltonian operator [Dirac (1928a, b)] for a one-electron atomic system is

$${{\rm H}_D} = c\alpha \bullet p + {c^2}\beta + V(r)$$

(1)

where c is the velocity of light, p is the momentum operator (p = -i∇), and the operators α and β are defined by

$$a = \left( {\begin{array}{*{20}{c}} 0&{{\sigma ^P}}\\ {{\sigma ^P}}&0 \end{array}} \right)\beta = \left( {\begin{array}{*{20}{c}} I&0\\ 0&{ - I} \end{array}} \right)$$

(2)

I is the two-dimensional unit matrix and the σ^P denote the Pauli matrices

$$\begin{array}{*{20}{c}} {\sigma _{x}^{P} = \left( {\begin{array}{*{20}{c}} 0 & 1 \\ 1 & 0 \\ \end{array} } \right)} & {\sigma _{y}^{P} = \left( {\begin{array}{*{20}{c}} 0 & { - i} \\ i & 0 \\ \end{array} } \right)} & {\sigma _{z}^{P} = \left( {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & { - 1} \\ \end{array} } \right)} \\ \end{array} $$

(3)

V(r)is the central potential and r is the position vector of the electron.