How does the wave function of free particle $\psi(x,t)=A\exp \{ i(kx-\omega t)\}$ satisfy normalisation condition?

Question

I am confused about the wave function of a free particle

$$\psi(x,t)=A\exp \{ i(kx-\omega t)\}$$

How does this satisfy the normalization condition? Since this corresponds to a plane wave, what meaning does the probability have?

Answer 1

It does NOT satisfy the usual normalization condition. The plane wave has the same probability density everywhere. The normalization of plane waves requires the introduction of $\delta$-functions.