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.

The canonical formal page is Pauli Matrices. This page explains why those matrices appeared in the first place.

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. The direction of the field selects the measured or energetically preferred spin component.

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.

Cross-Links

• Spin and Discrete Outcomes
• Electron Spin
• Magnetic Moments
• Pauli Exclusion Principle
• Space Quantization
• What Spin Is and Is Not
• Spin-1/2 Hilbert Space
• Pauli Matrices
• Spin Rotations
• Spin and Pauli Matrix Conventions
• Pauli Matrix Table
• Spin in Magnetic Field Hamiltonian

References

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

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. In one sentence, distinguish Pauli matrices from the Pauli exclusion principle.

Solution

Pauli matrices represent spin-$1/2$ operators on a two-dimensional internal Hilbert space, while the Pauli exclusion principle states that identical fermions cannot occupy the same complete one-particle state.