# 问题：

Let $v$ be any real, three-dimensional unit vector and $θ$ a real number. Prove that $\exp (i \theta \vec{v} \cdot \vec{\sigma})=\cos (\theta) I+i \sin (\theta) \vec{v} \cdot \vec{\sigma}$, (2.58)

where $\vec{v} \cdot \vec{\sigma} \equiv \sum_{i=1}^{3} v_{i} \sigma_{i}$.

This exercise is generalized in Problem 2.1 on page 117.

$Pauli$矩阵的指数

## 解答

$\exp (i \theta (v_{1} \sigma_{1}+v_{2} \sigma_{2}+v_{3} \sigma_{3}))=\cos (\theta) I+i \sin (\theta) (v_{1} \sigma_{1}+v_{2} \sigma_{2}+v_{3} \sigma_{3})$；

$(v_{1} \sigma_{1}+v_{2} \sigma_{2}+v_{3} \sigma_{3})=\left(\begin{array}{cc}v_{3} & v_{1}-i v_{2} \\ v_{1}+i v_{2} & -v_{3}\end{array}\right)$;

$(\vec{a} \cdot \vec{\sigma})=\sum_{i j} a_{i} \sigma_{i} b_{j} \sigma_{j}=\sum_{i j} a_{i} b_{j} \delta_{i j} I+i \vec{\sigma} \cdot(\vec{a} \times \vec{b})$;

$\exp (i \theta \vec{v} \cdot \vec{\sigma})$

$=\sum \frac{(-1)^{n}(\theta \vec{v} \cdot \vec{\sigma})^{2 n}}{(2 n) !}+i \sum \frac{(-1)^{n}(\theta \vec{v} \cdot \vec{\sigma})^{2 n+1}}{(2 n+1) !}$

$=\sum \frac{(-1)^{n} \theta^{2 n}}{(2 n) !}+i(\vec{v} \cdot \vec{\sigma}) \sum \frac{(-1)^{n} \theta^{2 n+1}}{(2 n+1) !}$

$=\cos (\theta)+i \sin (\theta)(\vec{v} \cdot \vec{\sigma})$.

#### 参考

[1]www.qtumist.com