The original Bohr model of the hydrogen atom had a positively charged nucleus orbited by a negatively charged electron. Electrical attraction curved the path of the electron much as gravity closes the orbit of a planet or moon.
There is a difference. A moon can orbit a planet at a wide variety of altitudes and speeds. Electrons only orbit atoms at specific energy levels. Why these specific levels? The standard answer is, "don't ask." There are perfectly good equations that predict observations. Questioning the meaning of the equations results in headaches.
But the energy levels just happen to be such that the electron returns to whence it came some integer number of periods later. The electron returns to the same point it was before in the same phase. The time that has passed is an integer multiple of the electron's period. The distance traveled is an integer multiple of its wavelength. Might we be seeing single particle interference again? Electrons have the highest probability of orbiting an atomic nucleus at energy levels where they reinforce themselves one orbit later?
The interferometer experiment had shades of gray between the white D and black E. Electron orbitals are much more binary. There is a high probability of observing an orbit at certain well known and quite specific energy levels, but zero chance of observing an electron in between. Why gray interferometers but black and white atoms?
The electron is not interfering only with its next orbit, but with the one after that, and the next one, and the next, and... In an interferometer, there is not only a bright point at D with zero path difference, or F with one wavelength path difference, but also at H with twice the wavelength difference and J with three times the wavelength. Particles do not only interfere with where they might be on period in the future or path, but also where they might be two periods or three periods or N periods in the future or past.
Thus, when calculating the odds of an electron being observed at a specific energy level, one must consider the interference effects of many times around the nucleus. If the energy level is just slightly off an optimal integer multiple of the wavelength, the first and second orbit might positively interfere, but the first and seventeenth orbit might be totally opposite in phase. This implies the second and eighteenth orbits would also be totally out of phase, as would be the third and nineteenth. The chances of observing such an orbit would be zero. Due to self interference, the chances of observing any orbit save one where the orbital path is an integer multiple of the wavelength is zero.
So we have an electron spinning around the nucleus, interfering with itself every time around. Is that all it is interfering with? Not necessarily. There are many specific paths that an electron might take at a given energy level. All of them are interfering with one another. To work up the probability of observing an electron at any given point, one has to work out the interference pattern resulting from all possible paths with a given energy level, without forgetting the future versions of the paths, nor the future versions of the paths.
Electron orbitals have unusual shapes. What causes these shapes? The standard answer is, "don't ask." One might, however, set up an interference pattern between all possible paths of an electron at a given energy level, then calculate the odds of observing said electron at any given point.
Id try it, but the math would give me a headache.