At this point, we’ve more or less exhausted the number of interesting examples for one-dimensional quantum systems. Since we live in a three-dimensional world, let’s go there to see if quantum physics has anything still hidden up its sleeve.
Fortunately, nature is full of three-dimensional systems to explore. The most instructive problem is that of the simplest atom–one electron attached to one proton, that is–the hydrogen atom. If we turn to the Schroedinger equation yet again and insert the potential energy of the electron due to the presence of the proton, we should be able to catch a glimpse of three-dimensional, electron wave functions. Of course, as a nifty by-product, we should be able to find the atomic energy levels, too.
For convenience, we’ll assume the proton is at a fixed position in space, which is more or less true anyway, and then reference all possible locations of the electron relative to that position.
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