Commutator

The commutator $[\hat{A}, \hat{B}]$ is defined as follows:R4(97)

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

Two operators $\hat{A}$ and $\hat{B}$ are said to commute when $[\hat{A}, \hat{B}]$ = 0.

Exercise 7: Prove that $[\hat{x}, -i\hbar \frac{\partial}{\partial x}] = i \hbar.$

Hint: Prove that $[\hat{x}, -i\hbar \frac{\partial}{\partial x}] \psi(x) = i \hbar \psi(x),$ where $\psi(x)$ is a function of $x$.