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Have the energy eigenfunctions in position space ever been experimentally tested for the simplest system undergraduates encounter when learning quantum mechanics, the square well? If not, what is the best example of the experimental verification of position space energy eigenstates or Schrodinger evolution, for simple systems? Almost always the 2-slit experiment is mentioned, or tests of hydrogen such as this one (for which the precision of theory-experiment agreement is quite low), but are there any other tests that test the probability distributions that undergraduates typically derive when learning quantum mechanics?

You may wish to consider the exeriment by Crommie, Lutz and Eigler (Nature 363, 524 (1993)), who look at standig waves between step edges in a surface two-dimensional electron gas using scanning tunneling microscopy. Other experiments by the same IBM group have shown the standing waves in quantum corrals. Generally, STM images of patterned surface 2D electron gases taken at cryogenic temperatures have shown beautiful images of the 'wave functions' (better, the probability distributions, or even better, the local density of states) of confined systems within the last 30 years.

Another technique uses electron tunneling in semiconductor heterostructures to image the wave functions of quantum dots, as shown in this paper, for example (there are more papers in this direction by the same and other groups). These states are the bound states of so-called 'artificial atoms' (another name for few-electron quantum dots).

I found additional experiments such as this one using photoelectron spectroscopy to image the wave functions of adsorbed molecules.

Although these experiments may not be exactly what you are looking for, because they are rather complicated to explain in detail to undergraduates, they may be used as a 'teaser' to make them curious about learning advanced methods to eventually understand them.