How do we experimentally find the probability distribution of quantum particles, i.e the one obtained by mod squaring the solution of the Schrödinger equation? For example, to study the case of an infinite potential well, how do we proceed in the lab?
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We cannot, in a strict sense, measure the wave function $\psi(x)$ or its square $\lvert \psi(x)\rvert^2$. This is because one cannot infer the "true" probabilitity distribution of a random variable by making finitely many measurements on it. The only thing you can do is place your hope in the law of large numbers and expect that, if you prepare many identical quantum states all with the same state/wavefunction and then measure their position, you can then fit a probability distribution to the results that approximates the "true" wavefunction. However, it is unclear for what purpose this would be of any use. Note also that you cannot do these repeated measurement on a single particle/state, since the first measurement will already change it.
If you are willing to relax your notion of "measuring the wavefunction" to "measuring the state", then this is the active field of quantum tomography. The basic idea is that we don't attempt to measure $\psi(x)$, but a complete set of commuting observables, whose values uniquely determine the state - and crucially, this measurement can be carried out with finitely many measurement, and the state is not changed further after the first measurement.
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