轨道角动量本征函数——球谐函数
1. 角动量对易关系
在研究角动量相关问题时，一些矢量算符与角动量的对易关系非常有用，如下表所示：
| 对易关系 | 表达式 |
| — | — |
| ([\hat{J}i, \hat{T}_j]) | (i\hbar\hat{T}_k\epsilon{ijk}) |
| ([\hat{T}\pm, \hat{J}_z]) | (\mp\hbar\hat{T}\pm) |
| ([\hat{T}\pm, \hat{J}_x]) | (\pm\hbar\hat{T}_z) |
| ([\hat{T}\pm, \hat{J}y]) | (i\hbar\hat{T}_z) |
| ([\hat{T}\pm, \hat{J}\pm]) | (0) |
| ([\hat{T}\pm, \hat{J}\mp]) | (\pm2\hbar\hat{T}_z) |
| ([\hat{T}_z, \hat{J}\pm]) | (\pm\hbar\hat{T}_\pm) |
| ([\hat{J}, [\hat{T}_1 \cdot \hat{T}_2]]) | (0) |
| ([\hat{J}, \hat{T}^2]) | (0) |
| ([\hat{J}^2, [\hat{J}^2, \hat{T}]]) | (2\hbar^2 [\hat{J}^2 \hat{T} + \hat{T} \hat{J}^2] - 4\hbar^
)
