Measuring a Spin

A spin-½ has only two possible outcomes along any axis you choose: $+1$ or $-1$. Not a projection, not a magnitude — two values, whichever direction you point the apparatus. Everything strange about quantum measurement is already visible in that one fact.

A state is a direction

Every pure state of a two-level system can be written

and conversely every $(\theta, \varphi)$ is a state. The correspondence runs both ways, so a spin state is a direction in space — which is what licenses drawing one as an arrow at all.

Note where $|u\rangle$ and $|d\rangle$ ended up. They are orthogonal — they exclude each other absolutely — and they sit antipodal on the sphere, not perpendicular. Angles in Hilbert space are half angles in real space, and that factor of two is the whole reason a half-angle appears in the formula above.

The equator is $\theta = 90°$: every state there is an equal superposition of $|u\rangle$ and $|d\rangle$, differing only by the phase $\varphi$.

What a measurement returns

Point an apparatus along some axis $\hat{n}$, at angle $\theta$ from where the spin points. Rewriting the state in that apparatus's own basis costs nothing but a relabelling,

so a single reading is $+1$ with probability $\cos^2(\theta/2)$ and $-1$ with probability $\sin^2(\theta/2)$. At $\theta = 0$ that is certainty; at $\theta = 180°$, certainty of the opposite sign; at $\theta = 90°$ both are $\tfrac{1}{2}$ — a fair coin, from an apparatus working perfectly.

One electron, four apparatuses

Send a single electron through a line of them, each measuring along its own axis, without disturbing the spin in between.

Read the dials in order: $+1$, $+1$, $-1$, $+1$.

The first takes an arbitrary incoming state and returns $+1$; the spin is left pointing along $+z$, in the state $|u\rangle$. Measuring the same thing again returns the same answer — that is the second apparatus, and it is why measurement can be used to prepare a state. Point the apparatus along $+z$, keep only the electrons that read $+1$, and you have a source of $|u\rangle$.

The third is the interesting one. Its needle points along $-z$, and the dial reads $-1$ — but the spin comes out still pointing up, still $|u\rangle$. The reading reversed; the state did not. Reversing the apparatus reverses what counts as positive, nothing more.

The fourth turns the needle to $+x$, where $\theta = 90°$, and for the first time the outcome is not forced.

The average, not the reading

$\langle \sigma_n \rangle = \cos\theta$ is a statement about an ensemble, not about any electron. To see it you have to prepare, measure once, and discard — over and over.

The re-preparation is the experiment, not stage dressing. Reuse the electron that just came out and it is already an eigenstate of $\sigma_n$: every subsequent press returns the same value, and the average runs to $\pm 1$ instead of $\cos\theta$.

Which closes the loop with the sphere: $\cos\theta$ is the projection of the measured axis onto $z$ — the vertical component of the needle drawn on each dial. The apparatus never displays it. No single electron ever displays it. It exists only in the average of many, each of which said nothing but $+1$ or $-1$.