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Why |psi|^2 instead of psi is the probability density

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An obvious objection against psi instead of |psi|^2 as a probability density is that psi is not always positive, it can even be complex. Probabilities must always be positive, of course. Now, one can object by saying that you can multiply a wavefunction by a suitable (complex) number, so that it becomes positive. However, this can only work locally, not globally. E.g. it is known that the wavefunction for a bound particle must change sign. The higher the energy of the bound state, the more zeroes the wavefunction will have. So, however you normalize the wavefunction, there must be regions were it is negative and thus the integral over some interval where psi is negative ...

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