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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'

= ∫ <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

= ∫ p' exp(ix'p') <p'|ψ> dp'

= -i

*x' ∫ exp(ix'p') <p'|ψ> dp'*

= -i∂/∂

= -i

*x' ∫ <x'|p'> <p'|ψ> dp'*

= -i∂/∂

= -i

*x' <x'|ψ>*

= -i∂/∂

= -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*?