Pauli Matrices

The Pauli matrices are the standard matrix representatives of spin-$1/2$ components. In the $S_z$ basis,

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

Spin operators are

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

Algebra

The Pauli matrices satisfy

$$\sigma_i\sigma_j = \delta_{ij}I +i\sum_k\epsilon_{ijk}\sigma_k.$$

Therefore

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

and

$$\{\sigma_i,\sigma_j\}=2\delta_{ij}I.$$

The symmetric-product meaning and Pauli-vector dot-product form are developed in Anticommutators.

Multiplying by $\hbar/2$ gives the spin commutator

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

Eigenvalues and Measurements

Each Pauli matrix has eigenvalues $\pm1$. The corresponding spin component $S_i$ has eigenvalues

$$\pm\frac{\hbar}{2}.$$

For a unit vector $\hat{\mathbf n}$,

$$\sigma_{\hat n} = \hat{\mathbf n}\cdot\boldsymbol\sigma$$

also has eigenvalues $\pm1$. This operator represents measurement of spin along the direction $\hat{\mathbf n}$, up to the factor $\hbar/2$.

The corresponding ideal spin-analyzer projectors are used in Stern–Gerlach Revisited.

Expanding Two-by-Two Hamiltonians

Any Hermitian two-by-two Hamiltonian can be written as

$$H=c_0I+\mathbf b\cdot\boldsymbol\sigma,$$

with real $c_0$ and real vector $\mathbf b$. The vector $\mathbf b$ determines the preferred axis in spin space.

This form is used for spin in a magnetic field, two-level atoms, avoided crossings, and qubit Hamiltonians.

For the magnetic-field spin convention, see Spin in Magnetic Fields.

The Pauli matrix $\sigma_y$ also enters the standard spin-$1/2$ time-reversal operator $\Theta=-i\sigma_yK$, where the complex conjugation operator is essential.

Common Mistakes

• Confusing dimensionless $\sigma_i$ with physical spin $S_i$.
• Losing the factor $\hbar/2$.
• Reversing the sign convention in $\sigma_y$.
• Treating Pauli matrices as commuting numbers.
• Assuming the displayed basis is always the energy basis.

Cross-Links

• Spin-1/2 Hilbert Space
• Spin as Intrinsic Angular Momentum
• Bloch Sphere
• Spin Measurements
• Stern–Gerlach Revisited
• Bloch Sphere for Density Operators
• Spin Rotations
• Spin in Magnetic Fields
• Time Reversal for Spin-1/2 Particles
• From SU(2) Spinors to Lorentz Spinors
• Pauli Matrix Identity Index
• Pauli Matrix Table
• Two-Level System Hamiltonian

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. Verify that $[\sigma_x,\sigma_y]=2i\sigma_z$.

Solution

Using $\sigma_x\sigma_y=i\sigma_z$ and $\sigma_y\sigma_x=-i\sigma_z$,

$$[\sigma_x,\sigma_y] = i\sigma_z-(-i\sigma_z) = 2i\sigma_z.$$

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

Solution

Since $\sigma_z$ has eigenvalues $+1$ and $-1$, $S_z$ has eigenvalues $+\hbar/2$ and $-\hbar/2$.