Measuring Spin Along z

Spin-1/2 is the simplest quantum system that cannot be explained by classical physics. An electron (or any other spin-1/2 particle) has exactly two possible outcomes when its spin is measured along any fixed axis: it registers either spin-up ( $+\hbar/2$ ) or spin-down ( $-\hbar/2$ ). There is no in-between value.

The spin-state space

The eigenstates of the $z$ -component of spin, $\hat{S}_z$ , form an orthonormal basis for the two-dimensional Hilbert space of spin-1/2:

$\hat{S}_z |{+z}\rangle = +\frac{\hbar}{2}|{+z}\rangle, \qquad \hat{S}_z |{-z}\rangle = -\frac{\hbar}{2}|{-z}\rangle.$

Any spin state can be expanded in this basis:

$|\psi\rangle = \alpha\,|{+z}\rangle + \beta\,|{-z}\rangle,$

where $\alpha$ and $\beta$ are complex numbers satisfying the normalization condition $|\alpha|^2 + |\beta|^2 = 1$ .

The Born rule for spin

When a $z$ -measurement is performed on the state $|\psi\rangle$ , the Born rule gives the probabilities directly:

$P\!\left(+\tfrac{\hbar}{2}\right) = |\alpha|^2, \qquad P\!\left(-\tfrac{\hbar}{2}\right) = |\beta|^2.$

Because $|\alpha|^2 + |\beta|^2 = 1$ , the two probabilities always add up to 1 — one of the two outcomes must occur. After the measurement the state collapses to whichever eigenstate was found: if the result is spin-up, the post-measurement state is $|{+z}\rangle$ .

A worked example

Suppose a particle is prepared in

$|\psi\rangle = \frac{3}{5}|{+z}\rangle + \frac{4i}{5}|{-z}\rangle.$

Check normalization first:

$\left|\frac{3}{5}\right|^2 + \left|\frac{4i}{5}\right|^2 = \frac{9}{25} + \frac{16}{25} = \frac{25}{25} = 1. \checkmark$

The probability of measuring spin-up is

$P\!\left(+\tfrac{\hbar}{2}\right) = \left|\frac{3}{5}\right|^2 = \frac{9}{25} = 0.36,$

and the probability of measuring spin-down is

$P\!\left(-\tfrac{\hbar}{2}\right) = \left|\frac{4i}{5}\right|^2 = \frac{16}{25} = 0.64.$

Notice that the phase factor $i$ in the spin-down amplitude does not affect its probability — only the modulus matters for measurement.

Connection to the Stern–Gerlach experiment

The Stern–Gerlach apparatus is precisely a device that performs an $\hat{S}_z$ measurement: a non-uniform magnetic field along $z$ deflects the particle up or down depending on which eigenstate it collapses into. Sending an ensemble of identically prepared particles through the apparatus and counting deflections is how $|\alpha|^2$ and $|\beta|^2$ are measured in practice.

Try it

This is a numerical exercise — your code should return a number. A spin-1/2 particle is prepared in $|\psi\rangle = \tfrac{3}{5}|{+z}\rangle + \tfrac{4i}{5}|{-z}\rangle$ . What is the probability of measuring spin-up along $z$ ?

Run your code to see the quantum state.