3.2 Commutation relations for Pauli matrices

对照

An important set of relations satisfied by Pauli matrices are their commutation relations:

$$[{\sigma }_j,{\sigma }_k]\equiv {\sigma }_j{\sigma }_k-{\sigma }_k{\sigma }_j=2i\sum _{l=1}^3{ϵ}_{jkl}{\sigma }_l\text{\hspace{1em}(3.4)}$$

where ${ϵ}_{jkl}$ satisfies ${ϵ}_{123}=1$ , is antisymmetric under permutation of two of its subscripts, and vanishes if two of the subscripts take the same value. More explicitly, this says:

$$[{\sigma }_1,{\sigma }_2]=2i{\sigma }_3,[{\sigma }_2,{\sigma }_3]=2i{\sigma }_1,[{\sigma }_3,{\sigma }_1]=2i{\sigma }_2$$

These relations can easily be checked by explicitly computing with the matrices. Putting together equations 3.2 and 3.4 gives a formula for the product of two Pauli matrices:

$${\sigma }_j{\sigma }_k={\delta }_{jk}1+i\sum _{l=1}^3{ϵ}_{jkl}{\sigma }_l$$

While physicists prefer to work with the self-adjoint Pauli matrices and their real eigenvalues, the skew-adjoint matrices

$$X_j=-i\frac{{\sigma }_j}{2}$$

can instead be used. These satisfy the slightly simpler commutation relations

$$[X_j,X_k]=\sum _{l=1}^3{ϵ}_{jkl}{X}_l$$

or more explicitly

$$[X_1,X_2]=X_3,[X_2,X_3]=X_1,[X_3,X_1]=X_2\text{\hspace{1em}(3.5)}$$

The non-triviality of the commutators reflects the non-commutativity of the group. Group elements $U\in SU(2)$ near the identity satisfy

$$U\simeq 1+{ϵ}_1{X}_1+{ϵ}_2{X}_2+{ϵ}_3{X}_3$$

for ${ϵ}_j$ small and real, just as group elements $z\in U(1)$ near the identity satisfy

$$z\simeq 1+iϵ$$

The $X_j$ and their commutation relations can be thought of as an infinitesimal version of the full group and its group multiplication law, valid near the identity. In terms of the geometry of manifolds, recall that $SU(2)$ is the space $S^3$ . The $X_j$ give a basis of the tangent space ${\mathbf{R}}^3$ to the identity of $SU(2)$ , just as $i$ gives a basis of the tangent space to the identity of $U(1)$.

Figure 3.1: Comparing the geometry of $U(1)$ as $S^1$ to the geometry of $SU(2)$ as $S^3$.

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上一节：3.1 The two-state quantum system · 下一节：3.3 Dynamics of a two-state system

附录：约定与记号

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正文：英文 · 原著转录

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原书 PDF · 印刷页 30、31

来源版本：2025-10-20

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