124 Magnetic Moment

A vector property that is used to describe how strongly a particle or current loop couples to a magnetic field, where a magnetic dipole is a localized circulating current or a spin.

The magnetic energy. A magnetic moment \(\boldsymbol{\mu}\) in a field \(\mathbf{B}\) has energy \(-\boldsymbol{\mu}\cdot\mathbf{B}\). This principle is used to compute the Zeeman shift of a level.

The magnetic energy is

\[ U = -\boldsymbol{\mu}\cdot\mathbf{B} \]

where

\(U\) is the magnetic energy.
\(\boldsymbol{\mu}\) is the magnetic moment.
\(\mathbf{B}\) is the magnetic field.

The orbital moment. Orbital motion produces a moment proportional to \(\mathbf{L}\). This principle is used to write the orbital magneton coupling.

The orbital magnetic moment is

\[ \boldsymbol{\mu}_{L} = -\dfrac{e}{2m_{e}}\mathbf{L} \]

where

\(\boldsymbol{\mu}_{L}\) is the orbital magnetic moment.
\(e\) is the elementary charge.
\(m_{e}\) is the electron mass.
\(\mathbf{L}\) is the orbital angular momentum.

The spin moment. Spin produces a moment with a \(g\)-factor near \(2\). The \(g\)-factor is the dimensionless factor that converts spin to magnetic moment. This principle is used to write the spin magnetic moment.

The spin magnetic moment is

\[ \boldsymbol{\mu}_{S} = -g_{s}\dfrac{e}{2m_{e}}\mathbf{S} \]

where

\(\boldsymbol{\mu}_{S}\) is the spin magnetic moment.
\(g_{s}\) is the electron spin \(g\)-factor.
\(\mathbf{S}\) is the spin.

124.1 References

Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — magnetic moments and the Zeeman effect.
Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — magnetic dipole moment.

123 Lagrangian

125 Measurement