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I have a question regarding how to extract probability from an overlap integral. Specifically, I am calculating the probability of a particle in a bound state in a delta potential $V=-\alpha \delta(x)$ ($\alpha$ is positive) remaining in a bound state after $V$ changes strength to $V=-\beta \delta(x)$. I am able to calculate the overlap integral of the bound state wavefunctions of these two $V$'s just fine; I am just wondering if I need to square the value of the overlap integral to find the probability of remaining in a bound state or if we just keep the value of the overlap integral as our probability. And if we do have to square the value of the overlap integral, why is that?

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You do need to square it. State overlaps are probability amplitudes and you need to (mod) square them to get the corresponding probabilities. This is the Born rule and it is a fundamental postulate of quantum mechanics, so unfortunately there's little to say about "why" other than "because."

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