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?


1 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.

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    $\begingroup$ Okay thank you :) but can you actually work with this equation ? (use it for experiments for example) and is there some kind of "substitute" equation that is normalizeable ? Because in the literature that I've read they always ignored the normalization problem or said that one can normalize the function in a different way $\endgroup$ Nov 20, 2021 at 14:38
  • $\begingroup$ @Young Kindaichi, Have you missed out a $dk$ in the last integral? $\endgroup$ Nov 20, 2021 at 14:44
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    $\begingroup$ @petermafai They are actually not the real physical states because of uncertainty in momentum. But they are easy to work with. There are different methods like box normalization, can be used to normalize these. Or you can also superpose them to make a packet which is normalizable and physically possible. $\endgroup$ Nov 20, 2021 at 16:22
  • $\begingroup$ @BrendanDarrer I think not :) $\endgroup$ Nov 20, 2021 at 16:24
  • $\begingroup$ @Young Kindaichi, Yes, because I edited it. It was just a small typo you made! :) $\endgroup$ Nov 20, 2021 at 16:42

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