Calculate the following commutators:
a) [x, i d/dx ]
b) [x, d/dy ]
c) [-d^2/dx^2, x ]
The best way I know to calculate these are to operate on an arbitrary function.
a) for example:
$\displaystyle \left [ x, i \frac{d}{dx} \right ] = x \left ( i \frac{d}{dx} \right ) - \left ( i \frac{d}{dx} \right ) x$
So operate this on some function $\displaystyle f(x)$:
$\displaystyle \left [ x, i \frac{d}{dx} \right ] f(x) = x \left ( i \frac{d}{dx} \right ) f(x) - \left ( i \frac{d}{dx} \right ) ( x f(x) )$
$\displaystyle = ixf^{\prime}(x) - if(x) - ixf^{\prime}(x)$
(Remember we need to use the product rule on the last term.)
$\displaystyle = -if(x)$
Thus
$\displaystyle \left [ x, i \frac{d}{dx} \right ] f(x) =-i f(x)$
Thus
$\displaystyle \left [ x, i \frac{d}{dx} \right ] = -i $
-Dan