Hence,

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

(1187) |

for , where

(1188) |

and

Here,

(1190) |

It follows from (1186) that

(1191) |

Now, a straightforward generalization of Equation (508) gives

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

(1193) |

However,

(1194) |

where is the magnetic field strength. Hence, we obtain

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

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

(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.