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.
