Pauli Matrices and Spin-1/2

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

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

Pauli Products and Commutators

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 the Pauli commutator by $(\hbar/2)^2$, then replacing $\sigma_k=2S_k/\hbar$, 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 coefficients can be recovered without diagonalizing:

$$c_0=\tfrac12\operatorname{Tr}H,\qquad b_i=\tfrac12\operatorname{Tr}(H\sigma_i).$$

For $\mathbf b\ne0$, its direction determines the energy eigenbasis. Since $(\mathbf b\cdot\boldsymbol\sigma)^2=|\mathbf b|^2I$, the energies are $c_0\pm|\mathbf b|$. At $\mathbf b=0$, the Hamiltonian is proportional to the identity and selects no axis.

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.

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. Find the energies and energy projectors of $H=E_0I+a\sigma_x+b\sigma_z$, where $a,b$ are real and not both zero.

Solution

Let $R=\sqrt{a^2+b^2}$. The energies are $E_0\pm R$ and the projectors are

$$P_\pm=\frac12\left[I\pm\frac{a\sigma_x+b\sigma_z}{R}\right].$$

The identity $(a\sigma_x+b\sigma_z)^2=R^2I$ verifies both $P_\pm^2=P_\pm$ and $HP_\pm=(E_0\pm R)P_\pm$. The splitting is $2R$, not $R$.

References

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