# 5.5: Magnetic Moments

Consider a particle of electric charge and speed performing a circular orbit of radius in the - plane. The charge is equivalent to a current loop of radius in the - plane carrying current . The magnetic moment of the loop is of magnitude and is directed along the -axis. Thus, we can write

(458) |

where and are the vector position and velocity of the particle, respectively. However, we know that , where is the vector momentum of the particle, and is its mass. We also know that , where is the orbital angular momentum. It follows that

(459) |

Using the usual analogy between classical and quantum mechanics, we expect the above relation to also hold between the quantum mechanical operators, ** ** and , which represent magnetic moment and orbital angular momentum, respectively. This is indeed found to the the case.

Spin angular momentum also gives rise to a contribution to the magnetic moment of a charged particle. In fact, relativistic quantum mechanics predicts that a charged particle possessing spin must also possess a corresponding magnetic moment (this was first demonstrated by Dirac--see Chapter 11). We can write

where \(g\) is called the *gyromagnetic ratio*. For an electron this ratio is found to be

(461) |

The factor 2 is correctly predicted by Dirac's relativistic theory of the electron (see Chapter 11). The small correction , derived originally by Schwinger, is due to quantum field effects. We shall ignore this correction in the following, so

for an electron (here, \(e> 0\)).

### Contributors

- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)