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The infinite square well (and variations) are some of the best-studied systems in quantum mechanics and are often used as the starting point for any quantum mechanical education, as the Schrodinger equation is easy to solve for such a system.

I'm looking for specific research papers that describe experimental results that affirm the particle-in-a-box theory because, while the theory is beautiful, it is nearly always presented to students without any discussion of experimental confirmations. Obviously, any real system is going to be more like the non-infinite square well. I'm well acquainted with the other experimental results that affirm other introductory aspects of quantum mechanics, such as the double-slit for affirming particle-wave duality, the stern-gerlach experiment for spin, and various interferometers for superposition, but have yet to come across any for particle-in-a-box theory. The best that I am aware of is Bose-Einstein condensate experiments, but those are typically focused on affirming that bosons can indeed occupy the same state and are not so much what I am looking for. The simplest practical system that comes to mind to demonstrate what I'm looking for would be something along the lines of a Jaynes-Cummings model single-atom cavity experiment where the atom is repeatedly found in the most likely predicted locations.

Is this question misguided? Are there actually no such experiments and we only know the theory to be true because more sophisticated developments of the theory align with experiments and extrapolate that the basics of the theory must therefore also be true?

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    $\begingroup$ An Optical cavity in a laser is an almost perfect square well for photons. Not because of a lower potential inside, but because of nearly perfectly reflecting mirrors. They typically produce Gaussian Beams. Does that example work? $\endgroup$
    – mmesser314
    Aug 6, 2023 at 16:45
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    $\begingroup$ Quantum dots would be another example (the article also mentions quantum wells and quantum wires). $\endgroup$ Aug 6, 2023 at 16:45

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Search for papers/ videos on GaAs/AlGaAs heterostructures. It is possible to construct quantum wells by combining certain types of semiconductors with different band gaps. For example, you can build a system made of three thin layers, of which top and bottom are AlGaAs layers while the middle layer is made of GaAs. The differences in band gaps, in addition to the physical structure, will essentially create a quantum well for electrons in the GaAs layer. This is an example system, there are tons of literature on this and other types of systems/ configurations.

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A very common example used in introductory physical chemistry is modeling the $\pi$ electrons in a conjugated polyene like $\beta$ carotene. When you model the electrons this way, you get a surprisingly decent estimate of the electronic structure and occupations for the $\pi$ system, with some drawbacks related to the fact that a finite well with “soft” potential edges does a far better quantitative job. This might not satisfy the constraints of your question, as it is not much of a research grade example, but I figured I would leave it as food for thought nonetheless.

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