Pauli Matrices in Historical Context

Pauli matrices entered quantum mechanics because electron spin required a two-valued internal degree of freedom whose components do not behave like commuting classical numbers. They are not merely a convenient table of matrices; historically, they made Pauli’s nonrelativistic theory of the magnetic electron calculable.

Pauli Matrices develops the algebra. Here the two-component electron state and magnetic coupling explain why these matrices entered physics.

Two-Valued Degree of Freedom

By 1925, Pauli’s exclusion principle and anomalous spectra pointed toward a two-valued electron property beyond ordinary orbital quantum numbers. After Uhlenbeck and Goudsmit proposed electron spin, this two-valuedness was interpreted as spin projection for a spin-$1/2$ particle.

A spin-$1/2$ state needs a two-dimensional internal state space. In a position representation, the wavefunction becomes a two-component object:

$$\psi(\mathbf x) = \begin{pmatrix} \psi_+(\mathbf x)\\ \psi_-(\mathbf x) \end{pmatrix}.$$

The two components are not two different particles and not two spatial coordinates. They are components in an internal spin basis. A measurement along one axis can be definite while a measurement along another axis is not.

Matrix Representation

To represent spin components in a two-dimensional state space, Pauli used three Hermitian two-by-two matrices. In the standard $S_z$ basis, they are

$$\sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \qquad \sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \qquad \sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$$

The physical spin operators are

$$S_i = \frac{\hbar}{2}\sigma_i.$$

This normalization gives the correct spin-$1/2$ eigenvalues:

$$S_i \quad\text{has outcomes}\quad \pm\frac{\hbar}{2}.$$

The important point is that the three matrices do not commute. They encode the fact that different spin components cannot all be simultaneously sharp.

Magnetic Coupling

The matrices became physically meaningful through magnetic interactions. For an electron, using $e>0$ and the convention

$$\boldsymbol\mu_s = -g_e\mu_B\frac{\mathbf S}{\hbar},$$

the spin magnetic Hamiltonian is

$$H_{\mathrm{spin}} = -\boldsymbol\mu_s\cdot\mathbf B = g_e\mu_B \frac{\mathbf S\cdot\mathbf B}{\hbar}.$$

For spin-$1/2$ this becomes

$$H_{\mathrm{spin}} = \frac{g_e\mu_B}{2} \boldsymbol\sigma\cdot\mathbf B.$$

Thus a magnetic field couples directly to a linear combination of Pauli matrices. A uniform field selects the spin component appearing in the Hamiltonian and drives unitary precession; it does not automatically perform a measurement. A field gradient plus a resolved spatial detector can convert that component into a recorded outcome.

This made Pauli’s theory a workable nonrelativistic description of spin in magnetic fields. It connected two-valued spectra, magnetic moments, and the matrix language that had already become central in quantum mechanics.

Modern Spin-1/2 Formalism

In modern notation, the Pauli matrices satisfy

$$[\sigma_i,\sigma_j] = 2i\sum_k\epsilon_{ijk}\sigma_k.$$

Therefore the spin operators satisfy

$$[S_i,S_j] = i\hbar\sum_k\epsilon_{ijk}S_k.$$

This is the angular momentum algebra for a spin-$1/2$ representation. A spin component along a unit vector $\hat{\mathbf n}$ is represented by

$$S_{\hat{\mathbf n}} = \frac{\hbar}{2} \hat{\mathbf n}\cdot\boldsymbol\sigma.$$

Changing the apparatus orientation changes the matrix being measured. That is the formal version of the Stern–Gerlach lesson: spin “up” is not an absolute direction attached to the electron independently of measurement context.

The same matrices now appear throughout two-level physics: magnetic resonance, qubits, avoided crossings, two-band models, and effective Hamiltonians. Their historical origin in electron spin remains the simplest way to understand why they are so ubiquitous.

Common Misconceptions

• Pauli matrices are not the Pauli exclusion principle.
• The matrices $\sigma_i$ are dimensionless; physical spin operators are $S_i=\hbar\sigma_i/2$.
• The sign convention for $\sigma_y$ matters.
• A two-component spinor is not an ordinary two-dimensional spatial vector.
• Pauli matrices are not classical components of a tiny spinning body.
• A Hamiltonian proportional to $\boldsymbol\sigma\cdot\mathbf B$ depends on magnetic-moment and charge-sign conventions.
• Knowing the matrices by heart is not the same as understanding why noncommuting spin components were needed historically.

Exercises

1. Why do Pauli matrices need to be two-by-two matrices rather than ordinary numbers in the spin-$1/2$ theory?

Solution

A spin-$1/2$ state has two internal components. Spin observables must act on that two-dimensional internal state space. Ordinary numbers could only multiply both components uniformly; they could not represent different spin components, spin flips, or noncommuting measurements along different axes.

2. What are the eigenvalues of $S_z=\hbar\sigma_z/2$ in the standard convention?

Solution

The matrix $\sigma_z$ has eigenvalues $+1$ and $-1$. Therefore $S_z=\hbar\sigma_z/2$ has eigenvalues

$$\frac{\hbar}{2}, \qquad -\frac{\hbar}{2}.$$

3. Using $[\sigma_x,\sigma_y]=2i\sigma_z$, compute $[S_x,S_y]$.

Solution

Since $S_i=\hbar\sigma_i/2$,

$$[S_x,S_y] = \frac{\hbar^2}{4} [\sigma_x,\sigma_y] = \frac{\hbar^2}{4} 2i\sigma_z = i\hbar S_z.$$

4. Prepare $\lvert+z\rangle=(1,0)^T$ and measure spin along $\hat{\mathbf n}=(\sin\theta,0,\cos\theta)$. Use the projector $P_+=(I+\hat{\mathbf n}\cdot\boldsymbol\sigma)/2$ to find the positive-outcome probability.

Solution

The upper-left entry of $P_+$ gives

$$P(+\hat{\mathbf n})=\frac{1+\cos\theta}{2}=\cos^2\frac\theta2.$$

It equals one for the same axis, one half for a perpendicular axis, and zero for the reversed axis. The smooth axis dependence coexists with only two possible outcomes for each fixed axis.

References

• P. A. M. Dirac, “The Quantum Theory of the Electron,” Proceedings of the Royal Society A 117, 610-624, 1928, DOI: 10.1098/rspa.1928.0023.
• M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed., American Institute of Physics, 1989.
• W. Pauli, “Zur Quantenmechanik des magnetischen Elektrons,” Zeitschrift für Physik 43, 601-623, 1927.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• G. E. Uhlenbeck and S. Goudsmit, “Spinning Electrons and the Structure of Spectra,” Nature 117, 264-265, 1926, DOI: 10.1038/117264a0.
• B. L. van der Waerden, ed., Sources of Quantum Mechanics, Dover, 1968.