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I've often just assumed

$$\langle x|p\rangle = \frac{1}{(2\pi\hbar)^{1/2}}e^{ipx/\hbar}$$

Is there a formal way to prove this?

My guess is that, since $\langle x|x'\rangle = \delta(x'-x)$, and a summation of wavepackets is essentially a delta function, we can prove the result starting from the definition of the delta function.

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  • $\begingroup$ The only trouble is the normalization. Since neither the bra or the ket in your expression are normalizable that depends on some convention. $\endgroup$
    – lcv
    Oct 3, 2019 at 0:18
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/41880/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Oct 3, 2019 at 0:21
  • $\begingroup$ By the way, strictly speaking that is not a matrix element rather a scalar product. This also shows the problem with your question more explicitly. How to compute the scalar product between two objects which are not normalizable (hence are not in the Hilbert space)?. Obviously this cannot be done in full rigour. It can only be done in a way to be consistent with some other convention/notation (such as your inline equation for example). $\endgroup$
    – lcv
    Oct 3, 2019 at 0:24

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