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Same thing here. No matter where the particles are,
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you have to get the same energy.
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So Ψ has to be such that
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when you calculate the kinetic energy for it,
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changes in that kinetic energy, in different positions,
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exactly compensate for the changes in potential energy.
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When you've got that,
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then you've got a correct Ψ;maybe.
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It's also important that Ψ remain finite,
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that it not go to infinity.
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And if you're a real mathematician,
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it has to be single valued;