Normalization problem of periodic wave function

Question

So, I want to normalize the eigen wavefunctions of the momentum operator ($-i\hbar \frac{\partial}{\partial x}\psi(x)=p\cdot\psi(x)$ where $p$ is a real number).
The solution is $\psi(x)_p=C\cdot e^{\frac{i}{\hbar}px}$ where $C$ is an imaginary or real number and $p$ is real, and now the function is of course not normalizeable. Now I've searched a little and they always wrote something like $\int_{-\infty}^{\infty}\psi(x)_p\psi_{p'}^*(x)dx=2\pi\delta(p-p')$ and $\phi(x)= \sum_{p,p'}\psi_p(x)\int_{-\infty}^{\infty}\phi(x')\psi_{p'}(x')dx'$ (I'm not sure where $\phi$ is from), and that you get the Fourier transformation from that, and that my example is not normalizeable in the classical sense.
So I guess I can't get a value for C, since that would be the classical way. But how can I normalize this function then?

Answer 1

The momentum operator $P$ doesn't have a normalizable eigenfunction. One writes the eigenfunction as $$\chi(x)=\frac{1}{\sqrt{2\pi}}e^{ikx}$$ The factor comes in here. If we consider the finite boundary, say $L$. Then

$$\chi_n(x)=\frac{1}{\sqrt{L}}e^{ik_nx}$$ The completeness required: $$\sum_{n=-\infty}^\infty \frac{1}{L}e^{ik_nx}e^{-ik_{n'}x}=\delta (x-x').\ \ \ \ x,x'\in [-L/2,L/2]$$

Now let $L\rightarrow \infty $ then $$\sum_{n=-\infty}^\infty\left\{\cdots \right\}\rightarrow \int dn \{\cdots \}=\int dk\left( \frac{dn}{dk}\right)\{\cdots\}$$ where $$\frac{dn}{dk}=\frac{L}{2\pi}$$ therefore, $$\int \frac{1}{2\pi} e^{ik_nx}e^{-ik_{n'}x} \ dk=\delta (x-x')$$

where we reorganize $$\chi_n(x)=\frac{1}{\sqrt{2\pi}} e^{ik_nx}$$ as desired.