Non-normalizable eigenfunctions

Question

I was reading Shankar's Principles of Quantum Mechanics when on page 65, he starts talking about infinite spaces and operators in them. He introduces an operator $K$, which in the $x$ basis takes the form $K=-iD$ with $D = d/dx$ in this basis. He then solves the eigenvalue and vector problem for this operator and finds that in the $x$ basis, the eigenkets $|k \rangle$ take the form
$$\psi_k(x)= \langle x|k\rangle = Ae^{ikx}.$$
However, he later shows that such functions cannot be normalized, since
$$\langle k|k'\rangle = \int_{-\infty}^{\infty}\langle x|k' \rangle \langle k | x \rangle dx=\int_{-\infty}^{\infty}e^{i(k-k')x}dx=2\pi\delta(k-k').$$
My question then is, if $\psi_k(x)$ is not normalizable, then how can $|\psi_k(x)|^2$ represent a probability density? and what is the physical interpretation for $|\psi_k(x)|^2$ in these cases?

Answer 1

Although an absolute notion of probability density $P(x)=|\psi(x)|^2$ does not make sense for a non-normalizable wavefunction $\psi(x)$, there is still a relative/comparative notion of probability density by considering ratios $|\psi(x_1)|^2~:~ |\psi(x_2)|^2$ between two positions $x_1$ and $x_2$.

This relative probability interpretation is utilized when studying scattering in QM, such as, reflection/transmission in 1D, cf. this related Phys.SE post.