Spin & Quantum Numbers

Think of an apartment building. To find one person you need their full address: which building, which floor, which apartment, and which of the two roommates. Every electron in an atom has its own four-part address too — the four quantum numbers — and the universe enforces one strict rule: no two electrons in the same atom may share the exact same address.

The four parts each answer a simple question. The principal number $n$ = 1, 2, 3, … says which "floor" (shell) the electron lives on — bigger $n$ means a bigger, higher-energy orbit. The orbital number $\ell$ says what shape the room is: $\ell=0$ is a round $s$ room, $\ell=1$ a dumbbell $p$ room, then $d$, $f$. The magnetic number $m_\ell$ says which way that room points in space. And spin $m_s$ is the electron's own built-in two-way switch: up or down. A neat consequence: floor $n$ can hold exactly $2n^2$ electrons — so the second shell holds 8, which is precisely why the second row of the periodic table has 8 elements.

The deeper rules pin down the allowed values. For a given floor, $\ell$ can only run from $0$ up to $n-1$, and for a given shape $m_\ell$ runs in integer steps from $-\ell$ to $+\ell$ — that's $2\ell+1$ orientations. The angular momentum isn't $m_\ell\hbar$ in length; its true magnitude is $|\vec{L}|=\sqrt{\ell(\ell+1)}\,\hbar$, while only its shadow on the $z$-axis, $L_z=m_\ell\hbar$, is sharp. Spin behaves the same way: $|\vec{S}|=\tfrac{\sqrt{3}}{2}\hbar$ but $S_z=\pm\tfrac{1}{2}\hbar$. Stack these with the Pauli exclusion principle — no repeated address — and the entire periodic table falls out.

Try this in the sim above: (1) Set $n=2$ and drag the $\ell$ slider — it refuses to pass $\ell=1$, because $\ell$ can never reach $n$; then sweep $m_\ell$ and watch the $p$-orbital swing to point along different axes. (2) Switch to Zeeman Splitting, set $\ell=2$, and raise $B$: the single level fans into $2\ell+1=5$ evenly spaced lines. (3) Switch to Stern-Gerlach — the beam always splits into exactly two spots, never a smooth smear. That clean two-way split is spin being stuck at $\pm\tfrac{1}{2}$.

Quantum numbers $(n,\ell,m_\ell,m_s)$ uniquely specify each electron state in an atom. They emerge from solving the Schrödinger equation in spherical coordinates plus the introduction of intrinsic spin from relativistic quantum mechanics (Dirac equation).

Number	Meaning	Allowed values
$n$	Principal — energy & size	1, 2, 3, …
$\ell$	Orbital angular momentum	0, 1, …, $n-1$
$m_\ell$	$L_z$ projection	$-\ell$ to $+\ell$
$m_s$	Spin projection	$\pm\frac{1}{2}$
$E_n$	Energy of level $n$	$-13.6/n^2$ eV
$L^2$	Angular momentum²	$\ell(\ell+1)\hbar^2$
$g_s$	Electron $g$-factor	≈ 2.0023

Step 1 — Separation of variables

Write $\psi(r,\theta,\phi) = R(r)Y_\ell^{m_\ell}(\theta,\phi)$. The angular part gives:

$$\hat{L}^2 Y_\ell^{m_\ell} = \ell(\ell+1)\hbar^2 Y_\ell^{m_\ell}, \qquad \hat{L}_z Y_\ell^{m_\ell} = m_\ell\hbar Y_\ell^{m_\ell}$$

Step 2 — Energy quantization

$$E_n = -\frac{m_e e^4}{2(4\pi\epsilon_0)^2\hbar^2 n^2} = -\frac{13.6\,\text{eV}}{n^2}$$

Step 3 — Spin from the Dirac equation

Dirac (1928) showed that demanding Lorentz invariance for a spin-½ particle introduces an intrinsic angular momentum $S$ with:

$$|\vec{S}| = \sqrt{s(s+1)}\,\hbar = \frac{\sqrt{3}}{2}\hbar, \qquad S_z = \pm\tfrac{1}{2}\hbar$$

Step 4 — Magnetic moment & Zeeman effect

where $\mu_B = e\hbar/2m_e \approx 9.274\times10^{-24}$ J/T is the Bohr magneton.

Step 5 — Pauli exclusion & periodic table

No two electrons in an atom share all four quantum numbers. This forces electrons into successive shells, producing the periodic structure of the elements.

Mapping to the simulation

Sliders set $n,\ell,m_\ell$; the canvas renders the corresponding orbital probability density and spin orientation. The Stern-Gerlach mode shows beam splitting from spin; Zeeman shows energy levels splitting linearly with $B$; Larmor shows $\vec{S}$ precessing at frequency $\omega_L$.