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# 11.5: Electron Spin

According to Equation (1132), the relativistic Hamiltonian of an electron in an electromagnetic field is

 (1185)

Hence,

 (1186)

where use has been made of Equations (1118) and (1119). Now, we can write

 (1187)

for , where

 (1188)

and

 (1189)

Here, 0 and denote null and identity matrices, respectively, whereas the are conventional Pauli matrices. Note that , and

 (1190)

It follows from (1186) that

 (1191)

Now, a straightforward generalization of Equation (508) gives

 (1192)

where and are any two three-dimensional vectors that commute with . It follows that

 (1193)

However,

 (1194)

where is the magnetic field strength. Hence, we obtain

 (1195)

Consider the non-relativistic limit. In this case, we can write

 (1196)

where is small compared to . Substituting into (1195), and neglecting , and other terms involving , we get

 (1197)

This Hamiltonian is the same as the classical Hamiltonian of a non-relativistic electron, except for the final term. This term may be interpreted as arising from the electron having an intrinsic magnetic moment

 (1198)

In order to demonstrate that the electron's intrinsic magnetic moment is associated with an intrinsic angular momentum, consider the motion of an electron in a central electrostatic potential: i.e., and . In this case, the Hamiltonian (1185) becomes

 (1199)

Consider the component of the electron's orbital angular momentum,

 (1200)

The Heisenberg equation of motion for this quantity is

 (1201)

However, it is easily demonstrated that

 (1202) (1203) (1204) (1205)

Hence, we obtain

 (1206)

which implies that

 (1207)

It can be seen that is not a constant of the motion. However, the -component of the total angular momentum of the system must be a constant of the motion (because a central electrostatic potential exerts zero torque on the system). Hence, we deduce that the electron possesses additional angular momentum that is not connected with its motion through space. Now,

 (1208)

However,

 (1209) (1210) (1211) (1212)

so

 (1213)

which implies that

 (1214)

Hence, we deduce that

 (1215)

Since there is nothing special about the direction, we conclude that the vector is a constant of the motion. We can interpret this result by saying that the electron has a spin angular momentum , which must be added to its orbital angular momentum in order to obtain a constant of the motion. According to (1198), the relationship between the electron's spin angular momentum and its intrinsic (i.e., non-orbital) magnetic moment is

 (1216)

where the gyromagnetic ratio takes the value

 (1217)

As explained in Section 5.5, this is twice the value one would naively predict by analogy with classical physics.