$$\langle p | x \rangle=e^\frac{px}{i\hbar}$$ can pretty easily be derived from wave mechanics. But how can it be derived without resorting to it? I've seen it be derived from the translation operator $\hat T$. It is shown that for an infinitesimal displacement $\epsilon$ the translation operator in the position eigenbasis is: $$\langle x |\hat T |\psi\rangle=\left( 1+\dfrac{\partial}{\partial x}\right)\langle x | \psi\rangle=\left( 1+\dfrac{\partial}{\partial x}\right)\psi_x$$ In which $\psi_x$ is the state vector's representation in position space, such that $$|\psi\rangle=\int^{\infty}_{-\infty}\psi_x |x\rangle dx$$ And it then assumes that because momentum generates translation, $\frac{\partial}{\partial x}=\frac{i}{\hbar}\hat p$.
Now this leaves me rather unsatisfied. Is there any way to prove the form of the momentum operator, and from it prove $$\langle p | x \rangle=e^\frac{px}{i\hbar}$$ using, for example, the matrix representation of quantum mechanics?
