Angular Momentum and Spin

Why does the periodic table have the shape it has — two elements in the first row, then eight, then eight again? Why does a hospital MRI scanner flip the tiny magnets inside your hydrogen atoms to build a picture of your knee? Why, when you send a beam of silver atoms through a carefully shaped magnet, does it split into exactly two spots and nothing in between? Every one of these facts comes back to a single, deeply quantum idea: angular momentum — the physics of spinning and orbiting — does not come in a smooth range of values. It is quantised, locked to a ladder of allowed rungs.

In classical mechanics a spinning top or an orbiting planet can carry any amount of angular momentum, pointing in any direction, and you can measure all three of its components as precisely as you like. In the quantum world all three of those freedoms are broken at once. The magnitude is quantised; the direction is quantised (only a handful of tilts are allowed — a fact with the wonderful name space quantisation); and you are forbidden, in principle, from knowing all three components together. On top of the angular momentum of orbital motion there sits a second, stranger kind — spin, an angular momentum a particle carries just for being what it is, even sitting perfectly still. This page builds the whole picture, from the hydrogen atom's orbiting electron to the two-spot experiment that first revealed spin.

Orbital angular momentum: a quantised length and a quantised tilt

An electron bound in an atom carries orbital angular momentum \vec{L} from its motion around the nucleus. Solving the quantum problem (the eigenvalues of the operators \hat{L}^2 and \hat{L}_z) turns up two integers. The first is the orbital quantum number l = 0, 1, 2, \dots, which fixes the length of the vector:

|\vec{L}| = \sqrt{l(l+1)}\,\hbar.

Note carefully what this is not: it is not l\hbar. For l = 2 the length is \sqrt{6}\,\hbar \approx 2.449\,\hbar, strictly bigger than 2\hbar. That little gap — the difference between \sqrt{l(l+1)} and l — turns out to matter enormously, as the graph below shows: the true magnitude always lies above the naive guess.

The second integer is the magnetic quantum number m (often written m_l), which fixes the component of \vec{L} along whichever axis you choose to call z:

L_z = m\,\hbar, \qquad m = -l, -l+1, \dots, -1, 0, 1, \dots, l-1, l.

Count those allowed values of m: they run from -l to +l in integer steps, which is exactly 2l + 1 of them. So the vector is allowed only 2l+1 distinct tilts relative to the z axis. Nature has taken the continuous compass of classical directions and replaced it with a short list of permitted angles. This is space quantisation.

The vector-cone picture: why you can't know all three components

Here is where the quantum world truly departs from the classical. The operators for the three components \hat{L}_x, \hat{L}_y, \hat{L}_z do not commute — [\hat{L}_x, \hat{L}_y] = i\hbar\,\hat{L}_z, and cyclic permutations. Two quantities whose operators fail to commute cannot both have definite values at once. You may know the length |\vec{L}| and one component (by convention L_z) simultaneously — but the moment you pin those down, the other two components L_x and L_y become completely uncertain.

The standard way to draw this is the vector cone. Picture the angular-momentum vector as fixed in length \sqrt{l(l+1)}\,\hbar and fixed in height L_z = m\hbar, but with its x and y parts smeared right around a circle — so the tip sweeps a cone about the z axis. Each allowed m is one cone, at one fixed tilt. Reveal the figure below for l = 2: five vectors of identical length \sqrt{6}\,\hbar, sitting at five discrete heights.

Look at the top vector. Its L_z = 2\hbar is the largest projection allowed, yet the vector's true length is \sqrt{6}\,\hbar \approx 2.449\hbar — so even at maximum tilt it leans over, never standing bolt upright along the axis. If it could align perfectly with z, then L_x = L_y = 0 exactly and we would know all three components — which the commutation relations forbid. The cone is the geometry of that prohibition.

Spin: an angular momentum with no moving parts

Everything so far came from an electron moving around a nucleus. But experiments in the 1920s forced a second kind of angular momentum onto the stage — one the electron carries intrinsically, even when it is not orbiting anything at all. It is called spin, written \vec{S}, and it obeys the same algebra as orbital angular momentum but with one shocking twist: its quantum number is a half-integer.

For an electron the spin quantum number is s = \tfrac{1}{2}. Feed that into the very same formulae:

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

The number of orientations is 2s + 1 = 2: just two states, universally nicknamed spin up (S_z = +\tfrac{1}{2}\hbar) and spin down (S_z = -\tfrac{1}{2}\hbar). That factor of two is the hidden doubling behind the periodic table: each spatial orbital holds two electrons, one of each spin, which is the content of the Pauli exclusion principle. Halve the spin, and the neat "two, eight, eight" pattern of the elements dissolves.

Spin is a genuine angular momentum: it contributes to the total, it responds to magnetic fields, it is conserved. What it is not is a little ball physically rotating — that picture, tempting as it is, breaks under the slightest scrutiny (see the "Watch out!" box below). Spin is a fundamentally quantum property with no classical machine behind it.

The Stern–Gerlach experiment: seeing space quantisation with your own eyes

In 1922 Otto Stern and Walther Gerlach sent a beam of silver atoms through a strong, inhomogeneous magnetic field — a field that is stronger at one pole than the other, so that a tiny atomic magnet feels not just a twist but a net force, deflected up or down in proportion to its magnetic moment's z-component. Then they caught the atoms on a plate and looked at where they landed.

Classical physics is unambiguous about what should happen. The atomic magnets leave the oven pointing every which way, so their z-components should form a continuous spread from fully up to fully down. The beam should smear out into a single continuous vertical band — a blur. That is not what appears on the plate.

Instead the beam splits into two sharp, separate spots — one deflected up, one down, and a conspicuous gap in the middle where classical physics predicted the most atoms of all. The magnetic moment's z-component takes only two values. A silver atom's angular momentum comes entirely from a single unpaired outer electron, and that electron has spin s = \tfrac{1}{2}, with exactly 2s + 1 = 2 allowed orientations. The two spots are spin up and spin down, made visible. It is the most direct demonstration of space quantisation ever devised — and the reveal below traces a beam being sorted into its two quantum outcomes.

Worked examples

Example 1 — counting orientations. A d-electron has l = 2. How many distinct orientations (values of m) does its orbital angular momentum have, and what is its magnitude? The number of orientations is

2l + 1 = 2(2) + 1 = 5,

namely m = -2, -1, 0, 1, 2. The magnitude is

|\vec{L}| = \sqrt{l(l+1)}\,\hbar = \sqrt{2\cdot 3}\,\hbar = \sqrt{6}\,\hbar \approx 2.449\,\hbar.

Example 2 — the largest tilt. For that same l = 2 electron, what is the largest possible z-component, and does the vector ever point straight up the axis? The maximum is m = l = 2, so L_z^{\max} = 2\hbar. But the full length is \sqrt{6}\,\hbar \approx 2.449\hbar, which is larger, so even at maximum tilt the vector leans over by an angle

\cos\theta_{\min} = \frac{L_z^{\max}}{|\vec{L}|} = \frac{2}{\sqrt{6}} \approx 0.816 \;\Longrightarrow\; \theta_{\min} \approx 35.3^\circ.

It never reaches \theta = 0. No orbital vector ever lies along its axis.

Example 3 — counting spin states. How many spin states does an electron (s = \tfrac{1}{2}) have, and what is the length of its spin vector?

2s + 1 = 2\cdot\tfrac{1}{2} + 1 = 2, \qquad |\vec{S}| = \sqrt{\tfrac{1}{2}\cdot\tfrac{3}{2}}\,\hbar = \frac{\sqrt{3}}{2}\,\hbar \approx 0.866\,\hbar.

Two states — spin up and spin down — which is precisely why the Stern–Gerlach beam split in two.

Example 4 — a full subshell. Including spin, how many electrons fit in the l = 1 (p) subshell? There are 2l + 1 = 3 orbital orientations, and each holds 2s + 1 = 2 spins, so

2(2l+1) = 2\cdot 3 = 6 \text{ electrons}.

Six — matching the six p-block columns of the periodic table.

Trap one — the spinning ball. The word "spin" invites you to imagine the electron as a tiny charged sphere twirling on its axis. Do the arithmetic and it collapses: to produce the observed spin angular momentum, a point-like electron (radius experimentally under 10^{-18}\,\text{m}) would have to whirl its surface far faster than the speed of light. Worse, the electron has no measurable size at all. Spin is an intrinsic quantum property — as fundamental as charge or mass — not a description of internal motion. It is called spin only because it adds to angular momentum like a spin would.

Trap two — the magnitude. It is desperately tempting to write the length of the angular-momentum vector as l\hbar. It is \sqrt{l(l+1)}\,\hbar, which is always strictly larger for l \ge 1. This is not pedantry: the whole reason the vector can never align with an axis is that its length \sqrt{l(l+1)}\,\hbar exceeds its largest possible projection l\hbar. Confuse the two and the vector-cone picture — and the uncertainty at its heart — quietly disappears.

Nothing stops you from measuring L_x — you simply turn your apparatus (say a Stern–Gerlach magnet) to point along x instead of z. What happens is subtle and beautiful: the measurement gives you a definite L_x, but in doing so it destroys the definite L_z you had before. An atom prepared as "spin up along z", sent through an x-magnet, splits 50/50 into x-up and x-down; and if you then re-measure z, it too is again 50/50. The information about one component is not hidden, it genuinely does not exist alongside the other. This is non-commutation made experimental, and it is the beating heart of quantum weirdness — the same principle that powers a quantum computer's qubit.