Exploring Quantum Physics - Basic commutators

The goal of this post is to show the basic commutator relationships. Recall that the position operator is simply multiplying with the position, and the momentum operator is $-i\hbar\frac{\partial}{\partial x}$. Now we work in 3 dimensional space.

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

Now we wanted to know the commutation relationship between position and momentum operator, we have

$\begin{eqnarray*} [\hat{x_m}, \hat{p_n}] &=& \hat{x_m}\hat{p_n} - \hat{p_n}\hat{x_m} \\ [\hat{x_m}, \hat{p_n}] \psi &=& x_m(-i\hbar)\frac{\partial}{\partial x_n}\psi - (-i\hbar)\frac{\partial}{\partial x_n}x_m\psi \\ &=& (-i\hbar)(x_m\frac{\partial}{\partial x_n}\psi - \frac{\partial}{\partial x_n}x_m\psi) \\ \end{eqnarray*}$

In this form, it is obvious that if $m \ne n$, then we can simply pull $x_m$ out from the partial derivative at the latter term and the whole thing cancel out. So let focus on the case when $m = n$, the expression becomes:

$\begin{eqnarray*} [\hat{x_m}, \hat{p_m}] \psi &=& (-i\hbar)(x_m\frac{\partial}{\partial x_m}\psi - \frac{\partial}{\partial x_m}x_m\psi) \\ &=& (-i\hbar)(x_m\frac{\partial}{\partial x_m}\psi - (\psi + x_m\frac{\partial}{\partial x_m}\psi)) \\ &=& (-i\hbar)(-\psi) \\ [\hat{x_m}, \hat{p_m}] &=& i\hbar \\ \end{eqnarray*}$