Eigenstates of position and momentum operators in QM

Question

In Griffiths pages 103-105 "Introduction to Quantum Mechanics" 2nd editiion he states that the eigenfunctions of the position and momentum operators are $$g_y(x) = \delta(x-y)$$ where the eigenvalue equation is $$xg_y(x) = yg_y(x)$$

and for the momentum operator it is $$f_p(x) = \frac{1}{\sqrt{2 \pi \hbar}}e^{\frac{i p x}{\hbar}}$$ where the eigenvalue equation is $$\frac{\hbar}{i}\frac{d}{dx}f_p(x) = p f_p(x).$$

In other literature, this is not stated, why is this the case? Are these in fact the eigenfunctions of the position and momentum operators?

Answer 1

These relations are found in every book on QM, but the usual notation is $$ X|x\rangle=x|x\rangle $$ and $$ P|p\rangle=p|p\rangle $$

To go from these equations to the ones you've written, you just have to project them into the position basis $|x'\rangle$ (and use $\langle x'|x\rangle=\delta(x-x')$ and $\langle x'|p\rangle\sim\exp[ipx]$).

Edit Important: see ACuriousMind's comment below.