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Free particle energy eigenfunctions are $A\exp{[i(Et-\textbf{p}\cdot\textbf{r})/\hbar]}$ are non-normalizable. To normalize them one introduces a procedure called 'box normalization' where one imposes periodic boundary condition (PBC). But isn't imposing PBC quite an ad hoc procedure for a free particle?

Qmechanic
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  • More on boundary conditions: https://physics.stackexchange.com/q/87132/2451 , https://physics.stackexchange.com/q/220554/2451 , https://physics.stackexchange.com/q/215672/2451 , https://physics.stackexchange.com/q/323776/2451 and links therein. – Qmechanic Sep 06 '17 at 08:31
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    In what situation is "one" introducing this box normalization? To what end? The question is phrased as if this procedure should be absolutely familiar to the reader, but I've never seen anyone introduce a box normalization without some motivation (like "we only care about a finite about of space in this case" or "for ease of calculation, we'll take the infinite volume limit in the end"). Completely generically, there is no reason to introduce any such normalization at all for the free particle - the energy eigenfunctions simply are not normalizable and do not represent physical states. – ACuriousMind Sep 06 '17 at 08:58
  • The statements in this question are highly subfield-dependent. There might be places that have good reasons to use box normalization, but it's perfectly possible (and in my experience, the norm) to go years without seeing it used in anger. Delta-function normalization is much more convenient for a wide range of applications. – Emilio Pisanty Sep 06 '17 at 09:10

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I'm not sure in what context OP is asking, but the standard heuristic logic goes as follows: A physics experiment is typically conducted within a compact laboratory space, and it is assumed that the outcome does not depend significantly on what is outside this arena if it is big enough, i.e. it should not depend on what boundary conditions (BCs) that we impose.

If it doesn't depend on the choice of BCs, then pick the BCs that make the calculation easiest to perform, e.g. periodic BCs, even if physically implausible.

Qmechanic
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