Magnetic Moments

Consider a particle of electric charge $q$ and speed $v$ performing a circular orbit of radius $r$ in the $x$ -$y$ plane. The charge is equivalent to a current loop of radius $r$ in the $x$ -$y$ plane carrying current $I=q\,v/2\pi\, r$ . The magnetic moment $\mu$ of the loop is of magnitude $\pi\, r^2\, I$ and is directed along the $z$ -axis. Thus, we can write

$$\mbox{\boldmath$\mu$} = \frac{q}{2}\, {\bf x} \times {\bf v},$$ (458)

where ${\bf x}$ and ${\bf v}$ are the vector position and velocity of the particle, respectively. However, we know that ${\bf p} = {\bf v} /m$ , where ${\bf p}$ is the vector momentum of the particle, and $m$ is its mass. We also know that ${\bf L} = {\bf x}\times{\bf p}$ , where ${\bf L}$ is the orbital angular momentum. It follows that

$$\mbox{\boldmath$\mu$} = \frac{q}{2\,m} \,{\bf L}.$$ (459)

Using the usual analogy between classical and quantum mechanics, we expect the above relation to also hold between the quantum mechanical operators, $\mu$ and ${\bf L}$ , 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

$$\mbox{\boldmath$\mu$} = \frac{q}{2\,m} \left({\bf L} + g \,{\bf S}\right),$$ (460)

where $g$ is called the gyromagnetic ratio. For an electron this ratio is found to be

$$g_e = 2\left( 1 + \frac{1}{2\pi} \frac{e^2}{4\pi \,\epsilon_0\,\hbar \,c} \right).$$ (461)

The factor 2 is correctly predicted by Dirac's relativistic theory of the electron (see Chapter 11). The small correction $1/(2\pi\, 137)$ , derived originally by Schwinger, is due to quantum field effects. We shall ignore this correction in the following, so

$$\mbox{\boldmath$\mu$} \simeq - \frac{e}{2\,m_e} \left({\bf L} + 2 \,{\bf S}\right)$$ (462)

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