Position and momentum operators are canonical conjugates

The Position Operator and Momentum Operator are Canonical conjugates We explicitly calculate $[X,P]\psi$ for a generic $\psi$:

$$\begin{array}{rll}[X,P]\psi & =(XP-PX)\psi & \text{As matrix multiplication distributes over matrix addition}\\ & =XP\psi -PX\psi & \text{Plugging in the definitions of Position X and Momentum P.}\\ & =X(-i\hslash \frac{\partial \psi }{\partial x})-P(x\psi )& \\ & =-xi\hslash \frac{\partial \psi }{\partial x}-(-i\hslash \frac{\partial }{\partial x}(x\psi ))& \text{As (fg)′=f′g+fg′}\\ & =-i\hslash x\frac{\partial \psi }{\partial x}+i\hslash x\frac{\partial \psi }{\partial x}+i\hslash \psi \frac{\partial x}{\partial x}& \text{Cancel repeated terms}\\ & =i\hslash \psi \frac{\partial x}{\partial x}& \text{As ∂x∂x=1}\\ & =i\hslash \psi & \\ & \Downarrow & \text{As the above holds ∀ψ}\\ [X,P]& =i\hslash & \end{array}$$