Problem

Determine the commutator of the quantum mechanical operators for x and p_x.

Solution:

The commutator of 2 operators  and B is:

$$\begin{eqnarray*}[{\hat {A}},{\hat {B}}]& = & {\hat {A}}{\hat {B}}- {\hat {B}}{\hat {A}} \end{eqnarray*}$$

$$\begin{eqnarray*}[{\hat {x}},{\hat {p_{x}}}]={\hat {x}}{\hat {p_{x}}}- {\hat {p_{x}}}{\hat {x}} \end{eqnarray*}$$

$$\begin{eqnarray*} {\hat {x}} & = & x \\ {\hat {p_{x}}} & = & -i{\hbar}\frac{d}{dx} \end{eqnarray*}$$

If f (x) be a function of x

$$\begin{eqnarray*} ({\hat {x}}{\hat {p_{x}}}-{\hat {p_{x}}}{\hat {x}})\;f & = & (... ...} + i{\hbar}f+ i\hbar x\frac{df}{dx}) \\ & = & (i{\hbar}) f \\ \end{eqnarray*}$$

Hence

$$\begin{eqnarray*} ({\hat {x}}{\hat {p_{x}}}-{\hat {p_{x}}}{\hat {x}}) & = & i{\hbar} \end{eqnarray*}$$