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Why do we omit solutions that do not converge at $\pm\infty$ from the physical Hilbert space, what is the argument for us being allowed to do so?

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    $\begingroup$ Who says that we are removing these solutions (solutions to what?), and in what context? Perhaps also have a look at physics.stackexchange.com/q/331976/50583 and its linked questions. $\endgroup$ – ACuriousMind Jul 28 '19 at 16:18
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    $\begingroup$ I second the questions of the previous comment, but it's worth noting that in most situations where you're dealing with functions and you want to make them converge at $\pm \infty$, the ones that do not converge aren't even in the Hilbert space, as this space is usually $L^2$ or some extension of it to tempered distributions $\endgroup$ – user2723984 Jul 28 '19 at 16:21
  • $\begingroup$ @ACuriousMind Shankar when solving problems in his book Principles of quantum mechanics. I can give plenty of examples from there, the most recent one that kinda made me ask why( he probably did explain it somewhere but I have since forgotten where and can't find it) is on page 192. imgur.com/a/nB3grCe $\endgroup$ – gyzgyz123 Jul 28 '19 at 16:22
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We are only interested in solutions $\psi(x)$ which are normalizable, i.e. $$ \int_{-\infty}^{+\infty} |\psi(x)|^2 \text{d}x = 1.$$

If $\psi(-\infty)$ or $\psi(+\infty)$ diverge, then this normalization would not be possible.


Edit:

Actually the above is nearly true (for practical physical purposes), but not completely true (in strict mathematical sense). There are some pathological counter-examples, where $\psi(x)$ is normalizable, but $\psi(-\infty)$ and $\psi(+\infty)$ are divergent. See question "Normalizable wavefunction that does not vanish at infinity" and its answers.

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  • $\begingroup$ That makes sense. Thank you. $\endgroup$ – gyzgyz123 Jul 28 '19 at 16:23
  • $\begingroup$ Hmm... ok we are interested in those solutions but why aren't they part of the hilbert space, or by definition only normalizable solutions are part of it? $\endgroup$ – gyzgyz123 Jul 28 '19 at 16:25
  • $\begingroup$ Actually I answered my question from here:physics.stackexchange.com/q/307998 $\endgroup$ – gyzgyz123 Jul 28 '19 at 16:30

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