I had a homework problem in my intro QM class, basically asking me to find which of a given set of functions were eigenfunctions of the momentum operator, $\hat{p_x}$. For example,
$$ \psi_1 = Ae^{ik(x-a)} $$ which is an eigenfunction of $\hat{p_x}$, with eigenvalue of $\hbar k$. I had another function:
$$ \psi_2 = A\cos(kx) + iA\sin(kx) $$ which is also an eigenfunction of $\hat{p_x}$, with eigenvalue of $\hbar k$.
Now this maybe a basic question, but I am aware that $p=\hbar k$, so both eigenvalues are just the momentum, $p$. But is it the case that for every one-dimensional function that I can think of: if that function is an eigenfunction of $\hat{p_x}$, the corresponding eigenvalue will be $\hbar k$? I feel like this sort of makes sense, but I can't quite see why. Can anyone perhaps elaborate on this?