A magnetic moment is what turns an angular momentum into something a laboratory can grip. The field does not change the levels’ existence; it makes them separate enough to see.

§8.1 asserted that a component of angular momentum comes only in integer or half-integer multiples of $\hbar$, and admitted the claim was surprising. This section makes it visible. The route is magnetism: a moving charge is a magnet, a magnet in a field has an orientation energy, and if the angular momentum is quantized then so is that energy — which you can see, because a non-uniform field will sort the atoms in space according to it.

A classical magnet is an orbiting charge

The simplest magnetic moment in classical physics is a charged particle going round in a circle. Its moment turns out to be proportional to its angular momentum:

The derivation is three lines, and worth doing because every quantum formula in this section is a modification of it.

Quantum magnets: the same formula, three corrections

In quantum physics magnetic moments are still proportional to angular momenta, but the angular momenta are now fuzzy vectors — so only $\mu_z$ has a definite value. Three cases matter, and each modifies Eq. (8.7) differently.

Electron spin carries an extra factor of 2:

Electron orbital motion does not:

A whole atom gets a factor $g$ that interpolates between them:

The factor $g$ in Eq. (8.10) is the Landé g-factor landé g-factor The dimensionless factor g in an atom's magnetic moment, g = 1 + [j(j+1) − l(l+1) + s(s+1)]/2j(j+1). It equals 2 when the magnetism is due to electron spin alone and 1 when due to orbital motion alone. defined in ch. 8 — open in glossary , and for an atomic state with quantum numbers $j$, $l$ and $s$ it is

which takes the value $g = 2$ when the magnetism is all spin and $g = 1$ when it is all orbital.

Two natural units

Equations (8.8)–(8.10) all carry the same combination, so it gets a name — the Bohr magneton bohr magneton μ_B = eħ/2m_e = 9.274×10⁻²⁴ J T⁻¹, the natural unit for magnetic moments associated with electrons. defined in ch. 8 — open in glossary :

Protons and neutrons are composite — they contain quarks and gluons — so their moments are not given by any clean formula. They are measured:

which names a second unit, the nuclear magneton nuclear magneton μ_N = eħ/2m_p = 5.05×10⁻²⁷ J T⁻¹, the natural unit for nuclear magnetic moments — smaller than the Bohr magneton by the mass ratio m_p/m_e ≈ 1836. defined in ch. 8 — open in glossary :

Magnetic energy: a continuum, or a ladder

A classical moment in a field has orientation energy

and taking $\mathbf B$ along $z$,

Classically $\mu_z$ can be anything between $+\mu$ and $-\mu$, so the energy is a continuum between $-\mu B$ and $+\mu B$. Quantum mechanically $\mu_z$ is quantized, so the energy is too. Using Eq. (8.10), an atomic state with quantum numbers $j$ and $m_j$ has $E_{\mathrm{mag}} = m_j\,g\mu_B B$, giving $2j+1$ levels:

The book’s Fig. 8.2 draws the ladder for three values of $j$:

The Stern–Gerlach experiment

The Stern–Gerlach experiment stern–gerlach experiment Passing atoms through a non-uniform magnetic field, so each is deflected by a force −μ_z ∂B/∂z and the beam splits into 2j+1 separate beams. Direct evidence that a Cartesian component of angular momentum is quantized; silver atoms split into two, so their ground state has j = ½. defined in ch. 8 — open in glossary turns that energy ladder into a picture. A uniform field would only twist the atoms; the trick is a non-uniform one: if $B$ varies with $z$, the energy $E_{\mathrm{mag}} = -\mu_z B(x,y,z)$ varies with position, and a position-dependent energy is a force,

Each atom is pushed along $z$ by an amount proportional to its own $\mu_z$. A classical beam would smear into a continuous band, because $\mu_z$ could be anything from $-\mu$ to $+\mu$. A quantum beam splits into $2j+1$ separate beams.