This answer to this question will go a long way towards improving my understanding of representations of Hilbert space in general. How do we show that the position representation of p is -i ∂/∂x? Here is what I have (using Dirac's notation of priming eigenvalues, unprimed are operators): Using ∫ |p'><p'| dp' = 1 we have <x'|p|ψ> = ∫ <x'|p'> <p'|p|ψ> dp' = ∫ <x'|p'> p' <p'|ψ> dp' Now if I write <x'|p'> = exp(ix'p'), we have = ∫ p' exp(ix'p') <p'|ψ> dp' = -i ∂/∂x' ∫ exp(ix'p') <p'|ψ> dp' = -i ∂/∂x' ∫ <x'|p'> <p'|ψ> dp' = -i ∂/∂x' <x'|ψ> = -i ∂/∂x' ψ(x') But how do we show that <x'|p'> = exp(ix'p')? It should follow directly from [x,p] = i. Anybody see how? Or is there a way to get this result more directly from [x,p] = i?