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In quantum mechanics, there a lot of emphasis on $L^2$ spaces since Hilbert spaces describe states in quantum mechanics, so we have

$$ \langle \psi | \psi \rangle = \int |\psi^2(x)|\, dx$$

Even though technically the position operator $\mathbf{x}$ and momentum operator $\mathbf{p} = -i\hbar \frac{d}{dx}$ are not bounded, so maybe wave functions should just be a Banach space or something... However at least we can write $\langle x | p \rangle = e^{ipx}$ and things mostly work out.

Do physicist ever use the other $L_p$ norms? Especially $p \neq 0,1,2,\infty$... we could have:

$$ ||\psi||^p = \int |\psi|^p \, dx $$

Maybe these types of averages are not as natural as the mathematicians would think.

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    $\begingroup$ The other $L^p$ spaces are completely uninteresting for QM because they are not Hilbert spaces. $\endgroup$ – ACuriousMind Dec 20 '15 at 0:02
  • $\begingroup$ They may however occur in some variational problems (e.g. in continuum mechanics, although those will typically be Sobolev spaces not $L^p$-spaces, as you need derivatives of the occurring functions). $\endgroup$ – Sebastian Riese Dec 20 '15 at 1:31
  • $\begingroup$ In the framework of rigged Hilbert space, things like $|x\rangle$ and $|p\rangle$ are legitimate objects. See en.wikipedia.org/wiki/Rigged_Hilbert_space $\endgroup$ – higgsss Dec 20 '15 at 6:18
  • $\begingroup$ If you consider operators, $p$-norms are very natural, especially the trace norm (for density operators and the like) and the operator norm. $\endgroup$ – Norbert Schuch Dec 22 '15 at 11:54
  • $\begingroup$ Related: physics.stackexchange.com/q/41719/2451 $\endgroup$ – Qmechanic Mar 25 '16 at 22:44

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