$$\require{cancel}$$

# 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

 (460)

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

 (462)

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