Spin Observables (Preview)

Among all physical observables in quantum mechanics, spin is special: it is intrinsically quantum, has no classical analogue, and yet its mathematics is among the simplest — a two-dimensional complex vector space. This makes it the perfect testing ground for the operator formalism developed in this module.

Spin-1/2 as a two-level system

An electron (or any spin-1/2 particle) can register only two outcomes when its spin component along any axis is measured: $+\hbar/2$ ("spin up") or $-\hbar/2$ ("spin down"). Dividing out the overall factor $\hbar/2$ , the scaled measurement outcomes are simply $+1$ and $-1$ .

A spin state lives in a two-dimensional complex Hilbert space spanned by $|0\rangle$ (spin up along $z$ ) and $|1\rangle$ (spin down along $z$ ). This space is isomorphic to the qubit state space, so every spin-1/2 problem can be mapped directly onto a qubit — and conversely, every qubit can be thought of as a spin-1/2 particle.

The three Pauli operators

The spin components along the three Cartesian axes are proportional to the Pauli operators $\sigma_x$ , $\sigma_y$ , $\sigma_z$ (also written $X$ , $Y$ , $Z$ ):

X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

Each matrix is written in the $\{|0\rangle, |1\rangle\}$ basis. Each is Hermitian ( $\sigma^\dagger = \sigma$ ) so, as required for observables, all eigenvalues are real. You can verify this by inspection: $Z$ is already real and symmetric; $X$ is real and symmetric; and $Y$ has $Y_{12} = -i$ with $Y_{21} = i = (-i)^*$ , so $(Y^\dagger)_{12} = Y_{21}^* = -i = Y_{12}$ .

The physical spin-component operators are related to the Paulis by

\hat{S}_x = \tfrac{\hbar}{2}\,X, \quad \hat{S}_y = \tfrac{\hbar}{2}\,Y, \quad \hat{S}_z = \tfrac{\hbar}{2}\,Z.

The scaled factor $\hbar/2$ ensures the eigenvalues come out as $\pm\hbar/2$ , matching experiment.

Eigenvalues and eigenstates

Each Pauli matrix has eigenvalues $+1$ and $-1$ . The corresponding eigenstates are:

| Operator | Eigenvalue | Eigenstate | |----------|-----------|------------| | $Z$ | $+1$ | $|0\rangle$ | | $Z$ | $-1$ | $|1\rangle$ | | $X$ | $+1$ | $|{+}\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ | | $X$ | $-1$ | $|{-}\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$ | | $Y$ | $+1$ | $|{+i}\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + i|1\rangle)$ | | $Y$ | $-1$ | $|{-i}\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle - i|1\rangle)$ |

These can each be verified by matrix-vector multiplication. For example, check that $X|{+}\rangle = |{+}\rangle$ :

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix} = |{+}\rangle. \checkmark

The Pauli algebra

The Pauli operators obey a compact set of commutation and anti-commutation relations. Their squares are all the identity:

X^2 = Y^2 = Z^2 = I.

Their pairwise products cycle according to

XY = iZ, \quad YZ = iX, \quad ZX = iY,

and reversing the order introduces a minus sign (e.g. $YX = -iZ$ ). This means the Paulis do not commute: $[X, Y] = XY - YX = 2iZ \neq 0$ .

Non-commutativity is physically significant: it implies that spin components along different axes are incompatible observables — measuring $S_x$ then $S_z$ gives a different distribution than measuring $S_z$ then $S_x$ . No classical picture of a spinning ball can reproduce this.

Try it

A spin-1/2 particle modelled as a qubit starts in $|0\rangle$ (the $+1$ eigenstate of $Z$ ). Prepare the $+1$ eigenstate of $X$ instead, which is $|{+}\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ . Apply the single gate that rotates the $Z$ -eigenbasis into the $X$ -eigenbasis.

Run your code to see the quantum state.