# 算符和对易关系

- 产生湮灭算符的**反对易**关系：
  - $[a_p^\dagger,a_q^\dagger]_+=a_p^\dagger a_q^\dagger+a_q^\dagger a_p^\dagger=0$
  - $[a_p,a_q]_+=0$
  - $[a_p^\dagger,a_q]_+=\delta_{pq}$
  
- 占据数算符$\hat{N}_p=a_p^\dagger a_p$
  - $\hat{N}_p$是厄密(Hermitian)的，$\hat{N}_p={\hat{N}_p}^\dagger$
  - $\hat{N}_p$是幂等(idempotent)的，$\hat{N}_p={\hat{N}_p}^2$
  - **对易**关系：
    - $[\hat{N}_p,a_q^\dagger]=\delta_{pq}a_q^\dagger$
    - $[\hat{N}_p,a_q]=-\delta_{pq}a_q$
  - 当有一个串(string) $\hat{X}\equiv a_p^\dagger a_qa_ra_p\cdots$，$[\hat{N}_p,\hat{X}]=N_p^X\hat{X}$，这里$N_p^X$是$\hat{X}$中p位产生算符的数目减去湮灭算符的数目
  
- 粒子数算符$\hat{N}=\sum_{p=1}^m\hat{N}_p$，$[\hat{N},\hat{X}]=N^X\hat{X}$，这里$N^X$是$\hat{X}$中所有产生算符的数目减去湮灭算符的数目

- 激发算符$\hat{X}_q^p=a_p^\dagger a_q$
  
- 正则对易子：

  $$
  [\hat{r}_\alpha,\hat{p}_\beta]=i\delta_{\alpha\beta}\hat{N}
  $$

  此式仅在完备基组下成立，因为非完备基组，算符矩阵元素的乘积不等于乘积的算符矩阵的元素。详见[Explicit text](./others.md)

## 降秩 (Rank Reduction)

可以利用一些对易关系降低算符的秩，一个升/降算符的秩是$1/2$.

$$
[\hat{A}, \hat{B}\hat{C}]=[\hat{A}, \hat{B}]\hat{C}+\hat{B}[\hat{A}, \hat{C}]=[\hat{A}, \hat{B}]_+\hat{C}-\hat{B}[\hat{A}, \hat{C}]_+
$$

$$
[\hat{A}, \hat{B}\hat{C}]_+ = [\hat{A}, \hat{B}]\hat{C}+\hat{B}[\hat{A}, \hat{C}]_+=[\hat{A}, \hat{B}]_+\hat{C}-\hat{B}[\hat{A}, \hat{C}]
$$

Jacobi identity

$$
[\hat{A}, [\hat{B}, \hat{C}]] + [\hat{C}, [\hat{A}, \hat{B}]] + [\hat{B}, [\hat{C}, \hat{A}]] = 0
$$