I've been trying to derive the result of the following dot product in position space: $\langle p|x\rangle$.
The information I'm supposed to be using:
i) Any state can be written as a linear combination of the basis vectors: $|p\rangle=\sum_i a_i|i\rangle$
ii) The dot product between two functions is given by: $\langle f|g\rangle=\int_{-\infty}^{\infty}f(x)^*g(x)dx$
iii) The eigenstates of the operator $\hat{x}$ are $|x\rangle$.
iv) I must use the eigenstates of p and x: $\langle p|x\rangle =\int_{-\infty}^{\infty}\psi^*_p(x)\psi_x(x)dx$
v) The representation of the momentum operator is $\hat{p}=-i\hbar\frac{\partial}{\partial x}$
vi) Functions can be represented as $\psi(x)=\langle x|\psi\rangle$.
Well, I'm pretty much struggling to start to solve this problem and it seems that my lack of understanding regarding the representation of an eigenstate in a base might have to do with it. What I thought first was:
Using (iv) and (vi) I got
$\langle p|x \rangle =\int_{-\infty}^{\infty}\langle \psi_p|x\rangle ^*\langle x'|\psi\rangle dx{} dx'$ , where $|x\rangle \langle x'|$ should be an operator. However, I'm stuck from this point forward. I don't know how to write the eigenstates in a way that helps my attempt. What is wrong with my approach?
