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Could you give an explicit example of the behavior of a wave function near zero, so that the wave function is continuous, but its magnitude squared is not?

Do the highlighted words constitute legitimate and correct use of a and the? My reasoning is that a wave function is mentioned for the first time, so it is indefinite. On the other hand, the behavior of... is definite, since it is a specific/definite behavior (that of the wave function). However, an explicit example of... seems to fall it in both categories - it is specific/definite in the sense that it refers to the behavior, but it is one of many possible examples of such a behavior.

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The sentence is perfectly correct. The use of a / an here (in both cases) means one case of whatever, with the implication that more than one exist or could exist. i.e. there are many examples but I only want you to give one; there are multiple wave functions and I want you to give one (or maybe a typical example representing all in this context). Replacing "a wave function" with "the ..." would be stating that there is only one.

"the behaviour" means the whole behaviour, although in this case "an ... example" qualifies this. The same meaning would be conveyed by "Could you give me a behaviour of a wave function" but less clearly.

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It is correct usage. It mentions an example (any example) of a wave function that is continuous but has non-continuous magnitude squared (any such wave function). And finally, with regard to that example wave function, it requests the one and only behavior near zero.

Compare to “an example of the name of a girl that’s 12 letters long.” We would usually say instead, “an example of a girl’s name…” because this version with the genitive is simpler. But in the passage you cite, the joint resctriction on continuity (of both the function itself and the square of its magnitude) makes the genitive way less of an option.

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