More information on the momentum eigenstate in the position basis

[hmparticle9]

The following sentence appears in my book (Introduction to Quantum Mechanics by Griffiths (3rd Edition)) :

"$\langle x | p \rangle$ is the momentum eigenstate (with eigenvalue p) in the position basis." Page 122.

I am not sure why this is the case. I have done some digging... on a previous page (114) it says

"The momentum space wave function $\Phi(p ,t)$ is the $p$ component in the expansion of $| S(t) \rangle$ in the basis of momentum eigenfunctions:
$$\Phi(p ,t) = \langle p | S(t) \rangle$$
(with $| p \rangle$ standing for the eigenfunction of $\hat{p}$ with eigenvalue $p$)"

So is it convention that $| p \rangle$ is the eigenfunction/eigenstate of $\hat{p}$? If that is the case then that is one problem solved.

The second problem is, why does application of $\langle x|$ to $| p \rangle$ mean that we are taking $| p \rangle$ with respect to the position basis?

From doing some further reading it seems as though that my second question could have the same answer as the first.

 

[Nugatory]

The text inside a ket is merely a label, so in general we need to know what convention is being used to assign labels to know what is meant by $|whatever\rangle$.

There are some conventions that are so commonly used and generally accepted that authors often won't bother stating them explicitly. You've just figured two of them out: unless the context tells us otherwise, $|p\rangle$ is usually the eigenket of $\hat{p}$ with eigenvalue $p$ and likewise for $|x\rangle$ and $\hat{x}$.

The expression $\langle x|\psi\rangle$ is a number, so for a fixed $|\psi\rangle$ and variable $x$ it maps values of $x$ to numbers - which makes it a function of $x$, the position-space representation of $|\psi\rangle$.