How does the spin appear

It will be shown that the Dirac equation for the free electron in an external electromagnetic field is leading to the spin concept. Thus, in relativistic theory, the spin angular momentum appears in a natural way, whereas in the non-relativistic formalism it was the subject of a postulate of quantum mechanics, p. 25. 

If the vector it had numbers as its components, the last term would have had to be zero, because the vector product of two parallel vectors would be zero. This, however, need not be true when the vector components are operators (as it is in our case). Since tt = p - A, then 17-71) = ir and (irx it) = curM. To check this, we will obtain the last equality for the x components of both sides (the proof for the other two components looks the same). Let the operator (77 x tt) act on an arbitrary function f(x, y, z). As a result we expect the product of / and the vector iqj curl A. Let us see  

This is what we expected to get. From the Maxwell equations (p. 962), we have curl A = H, where H represents the magnetic field intensity. Let us insert this into the Dirac equation (valid for kinetic energy much smaller than 2moc, see eq. (3.60)) 

In the last parenthesis, beside the kinetic energy operator (first term), there is a strange second term. The term has the appearance of the interaction energy —M H of a mysterious magnetic dipole moment M with magnetic field H (cf. interaction with magnetic field, p. 659). The operator of this electronic dipole moment M = = —p-BW, where fis stands for the Bohr magneton equal to 

It is exactly twice as much as we get for the orbital angular momentum and the corresponding orbital magnetic dipole (hence the anomalous magnetic spin moment of the electron), see eq. (12.53). 

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