Linear Operators

Consider a new wave function

$$\psi'({\bf r}) = A \psi({\bf r})$$ (1.9)

where A is an operator. If

$$A[c_{1} \psi_{1}+ c_{2} \psi_{2}] = c_{1} A\psi_{1}+ c_{2}A \psi_{2}$$ (1.10)

then A is called a linear operator. Consider the following operators

$$A \equiv - i \hbar {\partial \over \partial x}$$ (1.11)

and

$$B \equiv x$$ (1.12)

then

$$(A B - B A) \psi = -i \hbar {\partial \over \partial x} x \psi - x \hbar {\partial \over \partial x} \psi= - i \hbar \psi.$$ (1.13)

Operator A, B do not commute.