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enter image description here

Hello,

I just have a question about this passage; specifically, I do not understand why the result of the inner product (the integral of u_k* and u_k') being the delta function defies conventional normalization. I was under the impression that this is the expected result, due to the orthogonality of eigenfunctions. If you could explain a little bit about normalization of the momentum wave function, I would really appreciate it. I have also attached a link of the full text below:

https://ocw.mit.edu/courses/nuclear-engineering/22-02-introduction-to-applied-nuclear-physics-spring-2012/lecture-notes/MIT22_02S12_lec_ch2.pdf

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    $\begingroup$ Welcome to Physics! Please do not post images of texts you want to quote, but type it out instead so it is readable for all users and so that it can be indexed by search engines. For formulae, use MathJax instead. $\endgroup$ – ZeroTheHero Jun 14 '18 at 20:00
  • $\begingroup$ I think you might be confusing the Kronecker delta with the Dirac delta. $\endgroup$ – probably_someone Jun 14 '18 at 20:01
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For a set of orthogonal wavefunctions, the usual normalization condition is the Kronecker delta:

$$\int u_k u_{k'}^* = \delta_{kk'}$$

which is $1$ when $k=k'$ and zero otherwise.

Here we have the Dirac delta, which is infinite when $k=k'$ and zero otherwise. Conceptually this is because the momentum eigenfunctions (plane waves) do not fall off at infinity, so they have infinite area.

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