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*} $