How does <x|p> become exp(ipx/ℏ) in wavefunction derivation?

[Eole]

Here's a derivation of wavefunction of State Ψ in representations of coordinates and momentum
Ψ (x)=<x|Ψ >=<x|∫dp|p><p|Ψ >=∫dp<x|p><p|Ψ>=∫dp exp{ipx/h}Ψ(p)
Ψ (p)=<p|Ψ >=∫dx exp{-ipx/h}Ψ(x)


Ψ (x)=<x|Ψ >=<x|∫dp|p><p|Ψ >=∫dp<x|p><p|Ψ>=∫dp exp{ipx/h}Ψ(p)
i don't understand how ∫dp<x|p><p|Ψ> become ∫dp exp{ipx/h}Ψ(p)
Could you please tell me the drivation of this formula?

and another question is why Ψ (x) could be denoted as <x|Ψ >?

 

[MalleusScientiarum]

You can prove (and I challenge you to do so) that $$\langle x | p \rangle = \exp \{i p x/ \hbar\}$$

Try doing it by testing the action of $$\hat{p} |x \rangle$$ and using the completeness relation.

 

[quantumdude]

Eole,

You've got another version of this exact same thread in the Homework Section. Please do not post multiple threads for the same topic.