Vector Potential Correction and Current Density

As shown in the previous section, several magnetic effects, such as the spin-orbit effect are included in the relativistic effect. To consider the effect of an external magnetic field, however, the vector potential. A, should be incorporated in the momentum operator (Jensen 2006), 

Since the term in parentheses in the imaginary term on the left-hand side is derived as 

The extended magnetic term on the left-hand side, which is called the Zeeman interaction term, leads to the Zeeman effect, which is a splitting of spectral lines under the influence of a magnetic field. Meanwhile, it is easily proven for non-hybridized electron spins in the nonrelativistic case that the Pauli spin matrix, a, in Eq. (6.55) is just twice the spin operator, s, in Eq. (2.91). The Zeeman interaction term is, therefore, written as 

let us consider the kinetic energy in the first term on the left-hand side of Eq. (6.101). The square of the generalized momentum operator. A, is derived as 

Fortunately, the Zeeman interaction term in Eq. (6.102) can be assembled with the inner product of the vector potential and the momentum in Eq. (6.105) by using the magnetization density operator, m, corresponding to the magnetic dipole moment, m, whose expectation value is the reverse sign of the first energy derivative in terms of magnetic field, as 

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