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?