Commutator Bracket | Quantum Mechanics

The commutator bracket of two operators A and B is defined as:

[$\hat{A,}\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}.$

Example:

(1) $\left[\hat{x},\frac{d}{dx}\right]\psi$ = $\hat{x}$$\frac{d\psi}{dx}-\frac{d}{dx}(\hat{x}\psi)$

=$x\frac{d\psi}{dx}-x\frac{d\psi}{dx}-\psi$

=$-\psi$

$\Longrightarrow\left[\hat{x},\frac{d}{dx}\right]$ = $-1$

(2) $\left[\hat{x},\hat{p_{x}}\right]\psi$ = $\hat{x}\hat{p_{x}}\psi-\hat{p_{x}}\hat{x}\psi$

= $-i\hbar x\frac{\partial\psi}{\partial x}+i\hbar\frac{\partial}{\partial x}(x\psi)$

= $i\hbar\psi$

$\Longrightarrow\left[\hat{x},\hat{p_{x}}\right]$ = $i\hbar$

Similarly,

$\left[\hat{y},\hat{p}_{y}\right]$ = $i\hbar$

and

$\left[\hat{z},\hat{p}_{z}\right]=i\hbar$

Properties of the commutator bracket:

1. $\left[\hat{A,}\hat{B}\right]=-\left[\hat{B},\hat{A}\right]$

2. $\left[\hat{A,}\hat{B}+\hat{C}+\hat{D}+…..\right]=\left[\hat{A},\hat{B}\right]+\left[\hat{A},\hat{C}\right]+…..$

3. $[\hat{A},\hat{B}]^{\dagger}$ = $[\hat{B}^{\dagger},\hat{A}^{\dagger}]$

4. $\left[\hat{A,}\hat{B}\hat{C}\right]=\left[\hat{A},\hat{B}\right]\hat{C}+\hat{B}\left[\hat{A},\hat{C}\right]$

5. $\left[\hat{A,}\left[\hat{B},\hat{C}\right]\right]+\left[\hat{B},\left[\hat{C},\hat{A}\right]\right]+\left[\hat{C},\left[\hat{A},\hat{B}\right]\right]=0$

6.$\left[\hat{A,} ƒ(\hat{A})\right]=0$

7. $\left[ƒ(\hat{A}),G(\hat{A})\right]=0$

8. $\left[ƒ(\hat{A}),G(\hat{B})\right]=0$ only if $\left[\hat{A},\hat{B}\right]=0$

9. $\left[\hat{A},\hat{B}^{n}\right]=n\hat{B}^{n-1}\left[\hat{A},\hat{B}\right]$

10. $\left[\hat{A}^{n},\hat{B}\right]=n\hat{A}^{n-1}\left[\hat{A},\hat{B}\right]$