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what experiments do you know, where the wavefunction of a particle (or better its probability distribution) is investigated? Has the prob. distribution for systems like (spherical) potential well, harmonic oscillator, hydrogen atom ever been measured directly?

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The wave nature appearing as a probability density distribution is experimentally very evident in the single-electron-ataa time double slit experiment:

dblslit

Individual electrons are dots on the screen, and the build up over time gives the probability density distribution that clearly shows interference effects of a wave nature.

There are experiments with Hydrogen atoms that accumulate measurements of the orbitals of the atom.

h2orb

Examples of four atomic hydrogen states. The middle column shows the experimental measurements, while the column at right shows the time-dependent Schrödinger equation calculations — and they match up rather nicely.

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