Measuring Spin Along x

The $S_x$, $S_y$, and $S_z$ operators for a spin-$\tfrac{1}{2}$ particle are given by the Pauli matrices:

$S_i = \frac{\hbar}{2}\,\sigma_i, \qquad i \in \{x, y, z\}.$

The $S_z$ eigenstates $|{\uparrow}\rangle$ and $|{\downarrow}\rangle$ form a convenient basis. When a particle is in one of these states and we measure $S_x$ instead, the result is no longer certain — the $S_z$ eigenstates are not eigenstates of $S_x$.

Eigenstates of $S_x$

The $S_x$ operator has matrix representation

$S_x = \frac{\hbar}{2}\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}.$

Solving $S_x |v\rangle = \lambda |v\rangle$ gives eigenvalues $\lambda = \pm\hbar/2$ and, after normalisation, the eigenstates expressed in the $z$-basis:

You can verify these directly: acting $S_x$ on $|{+}\rangle_x$ gives $\tfrac{\hbar}{2}|{+}\rangle_x$, and on $|{-}\rangle_x$ gives $-\tfrac{\hbar}{2}|{-}\rangle_x$.

Predicting x-measurement outcomes

Suppose the particle has been prepared in $|{\uparrow}\rangle$ (spin-up along $z$). Invert the two relations above to express $|{\uparrow}\rangle$ in the $S_x$ eigenbasis:

By the Born rule, the probability of each outcome is the squared modulus of the corresponding amplitude:

Both outcomes are equally likely — a $50{:}50$ split. This is the defining signature of a state that is an eigenstate of one spin component being measured along a perpendicular component.

Bloch sphere picture

On the Bloch sphere, $|{\uparrow}\rangle$ sits at the north pole and $|{\downarrow}\rangle$ at the south pole, while $|{+}\rangle_x$ points along $+x$ and $|{-}\rangle_x$ along $-x$. Measuring $S_x$ for a state on the $z$-axis is geometrically equivalent to projecting a vector that is perpendicular to the measurement axis — the two projections are equal, explaining the 50:50 outcome directly.

Try it

This is a numerical exercise — your code should return a number. A spin-$\tfrac{1}{2}$ particle is prepared in $|{\uparrow}\rangle_z$ and a Stern–Gerlach device is oriented along $x$. Compute the probability of obtaining $S_x = +\hbar/2$.

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