Foundations — Spin — intrinsic angular momentum

This page is a dictionary you can see. The parent note Spin throws many symbols at you at once ($\vec{S}$, $\hslash$, $m_s$, ${\mu }_B$, commutators…). Here we earn every one of them, from absolutely nothing, each built on the one before.

0 — What "angular momentum" even means

Before any quantum symbol, we need the ordinary idea of spinning motion having "amount."

Look at the figure: the burnt-orange arrow sits along the axis of turning, pointing the way a right-handed screw would advance. That arrow is a vector — it has a length (how much) and a direction (which axis). We write it $\vec{L}$ for orbital motion, and later $\vec{S}$ for spin.

Why the topic needs it: spin is an angular momentum, so it is an arrow. Everything — its magnitude, its tilt, its projection — is a statement about this arrow.

1 — Components and the $z$-axis: $S_x,S_y,S_z$

An arrow in space can be described by its shadows on three perpendicular rulers: the $x$, $y$, and $z$ directions.

Why $z$ and not $x$? Nothing physical prefers $z$; it is just the direction we choose to measure along (in Stern–Gerlach it's the direction of the field gradient). Once chosen, $S_z$ is "the up-down amount of spin."

Why the topic needs it: the crucial spin fact — that the arrow can never fully align with $z$ — is a statement that $S_x$ and $S_y$ can't both be zero, so $|\vec{S}|$ is always bigger than $S_z$. You literally cannot see that without components.

2 — Planck's constant $\hslash$: the "chunk size" of the quantum world

Nature does not let angular momentum take just any value. It comes in chunks, and the size of one chunk is set by a constant.

Why divide by $2\pi$? A full turn is $2\pi$ radians. Angular momentum is naturally measured "per radian," so $h/2\pi$ is the version that fits rotation cleanly. That's why $\hslash$, not $h$, shows up everywhere spin lives.

Why the topic needs it: every spin quantity is measured in units of $\hslash$. $S_z=m_s\hslash$ says "$S_z$ is $m_s$ chunks of $\hslash$." Without $\hslash$ you can't state how big spin is.

3 — Quantum numbers: $s$ and $m_s$ (plain counting labels)

Since spin comes in chunks, we can count them. The counters are pure numbers called quantum numbers.

Look at the figure: the deep-teal $z$-axis has only the two allowed heights marked (the arrow's tip must land on one of them). It cannot sit anywhere in between — that is what "quantized" means. Notice the arrow is tilted, never flat-up: its full length $|\vec{S}|$ (plum) is longer than either allowed height.

Why the topic needs it: "how many spots in Stern–Gerlach?" is answered by counting the allowed $m_s$: there are $2s+1$ of them. Two spots ⇒ $2s+1=2$ ⇒ $s=\frac{1}{2}$. This counting IS the discovery of spin.

4 — Angle from the axis: $\theta$, $\cos$, and $\arccos$

The tilt of the arrow is a genuine angle, and we measure it with trigonometry — so we need one right triangle.

Drop the arrow $\vec{S}$ (length $|\vec{S}|$) and its $z$-shadow $S_z$. They form a right triangle: the arrow is the slanted long side (hypotenuse), and $S_z$ is the vertical side sitting against the angle $\theta$ we measure from the $z$-axis.

Why cosine and not sine or tangent? We are comparing the piece along $z$ (adjacent to $\theta$) with the whole arrow (hypotenuse). "Adjacent over hypotenuse" is precisely $\cos$. Sine would compare the sideways piece; tangent needs two legs, not a leg-and-hypotenuse — cosine is the exact tool for "how much of the arrow points up the axis."

Why the topic needs it: the parent's Example 2 computes the electron's tilt, $\theta =\arccos (1/\sqrt{3})\approx 54.7^{\circ }$. That whole result is just "read the triangle, then undo the cosine."

5 — Magnitude versus projection: why $\sqrt{s(s+1)}$, not $s$

This is the single most confused pair of symbols in the topic, so we separate them clearly.

Why $s(s+1)$? Because $|\vec{S}{|}^2=S_x^2+S_y^2+S_z^2$ collects all three shadows. The "$+s$" hidden inside $s(s+1)$ is the leftover contribution of $S_x,S_y$ that can never vanish. Using bare $s$ (or $m_s$) would pretend those sideways shadows are zero — the classic error.

6 — Magnetic moment: $\vec{\mu }$, $e$, $m_e$, $g_s$, and ${\mu }_B$

Spin matters physically because a spinning charge acts like a tiny bar magnet. That magnet's strength is the magnetic moment.

Why the minus sign? The electron's charge is negative, so its magnet arrow points opposite to its spin arrow. That's why "spin up" gives ${\mu }_z\approx -{\mu }_B$.

Why the topic needs it: the beam splits because this magnet feels a force in a field that changes with position. Two values of ${\mu }_z$ ⇒ two forces ⇒ two spots.

7 — The reading key for the deeper machinery

The parent note also writes $[S_x,S_y]=i\hslash S_z$ and $S_{\pm }=S_x\pm i{S}_y$. You do not need to master these here — later Deep Dives build them — but you should be able to read the symbols:

Why the topic needs it: these are the "first principles" from which $\sqrt{s(s+1)}$ and the $2s+1$ count are derived rather than memorised.

Prerequisite map

Equipment checklist

Test yourself — you should be able to answer each before reading the main Spin note.

Connections

• Parent: Spin — the topic this page prepares you for.
• Orbital Angular Momentum — same vector/component machinery, integer $m$.
• Quantum Numbers — where $s$ and $m_s$ sit among $n,\ell ,m_{\ell }$.
• Stern-Gerlach Experiment — where the $2s+1$ count becomes visible spots.
• Bohr Magneton — the unit ${\mu }_B$ built here.
• Dirac Equation — explains why $g_s\approx 2$.
• Zeeman Effect · Fine Structure · Pauli Exclusion Principle — downstream uses.