🧠 Remember: 🍎 magnetic moment
$$ \vec{\mu}=I\,\vec{A} $$
where $I$ is the current and $\vec A$ is an area vector
🍎 Classically if a current is flowing around a loop $r(\theta)$ then the area of that loop is given by the integral
$$ A=\int_0^{2\pi} \frac12\,r^2\,\mathrm{d}\theta =\int_0^T \frac12\, r^2\,\frac{\mathrm{d}\theta}{\mathrm{d}t}\,\mathrm{d}t $$
where $T$ is the period of the rotation
🍎 we know $r^2\,\frac{\mathrm{d}\theta}{\mathrm{d}t}=L/m_e$ where $L$ is the angular momentum so
$$ \vec{A} = \frac{T}{2m_e}\,\vec{L} $$
🍎 The current flowing around the nucleus is
$$ I=\frac{-e}{T} $$
🍎 Therefore the magnetic moment of the electron in an orbital with angular momentum $\vec L$ is
$$ \vec{\mu} = I\,\vec{A} = \frac{-e}{T}\,\frac{T}{2m_e}\,\vec{L} $$
🍇 In quantum $\vec \mu$ becomes an operator. Since $\vec \mu \propto \vec L$ we can’t measure all components of $\vec \mu$ simultaneously so the $z$ component is given by
$$ \widehat{\mu}_z=-\frac{e}{2m_e}\,\widehat{L}z\qquad\text{with eigenvalues}\;-\frac{e\hbar}{2m_e}\,m\ell $$
The value $m$ related to $\ell$ will be called $m_\ell$ from now on
🍇 We define the quantum unit of magnetic moment
$$ \mu_B=\frac{e\hbar}{2m_e}=9.27\times10^{-24}\,\mathrm{A\,m^2} $$
called the Bohr magneton
A beam of atoms is fired through an inhomogeneous magnetic field, created by a shaped permanent magnet. A neutral particle would not be bent by a constant magnetic field, but in a gradient $\partial B_z/\partial z$ a particle with magnetic dipole moment $\vec \mu$ is bent by a force
$$ F_z=\mu_z \frac{\partial B_z}{\partial z} $$
🍎 for atoms of a given total angular momentum $|\vec L|$, one would expect a range of results with $L_z$ between $\pm |\vec L|$.
🍇 Instead a set of spots was observed, corresponding to $L_z$ values $\hbar m_\ell, |m_\ell| \le \ell$
🪗 Results:
Angular momentum is quantized
Variants of this experiment are possible, for example by making several measurements along different axis one can test their incompatibility
The details of the magnetic moments measured did not fit the pattern predicted by $\mu_z=-\mu_B L_z /\hbar$, it was found that even electrons in the $s$ orbit ($\ell=0)$ have a magnetic moment. This lead to the discovery of intrinsic angular momentum or “spin”
All electrons have a spin quantum number $s=\frac 12$
$$ S^2=s(s+1)\hbar^2 \qquad S_z=m_s\hbar \qquad m_s=-s,\ldots s\, $$
the magnetic moment is given by
$$ \mu_z = -g\,\mu_B\,m_s $$
where $g$ is the gyromagnetic ratio with a value of $g=2.0023\ldots(10 \text{ signifcant figures}) \approx 2$
🗒️ Note: A variant of the Stern-Gerlach experiment used an electric field to remove the Lorentz force on an electron and was able to measure the spin of free electrons directly
With $s=\frac 12$ we have $m_s = \pm \frac 12$ we call these spin-up $m_s=+\frac 12= +=\,\uparrow$ and spin-down $m_s=-\frac 12= -=\,\downarrow$. We define the spin wavefunctions $\chi_\pm$ to be the eigenfunctions of $\widehat S_z$
$$ \widehat{S}z\,\chi\pm = \pm\frac{\hbar}2\,\chi_\pm\, $$