This calculation agrees with experimentally measured spectral lines, but why would we expect it to be true, even if we accept that the electron moves according to the Schrodinger equation?
Your puzzlement arises because you are putting the cart in-front of the horse. The cart is the theoretical model of quantum mechanics and the horse is the data. As your question is migrated from math.SE one can understand this orientation, which is dominant also here.
The whole theoretical package of Quantum Mechanics did not arrive by a seemingly holy inspiration ( as some physical theories having to do with apples is said to have been), but was a slow accumulation of observations that forced physicists to think outside of the box of the mathematics used in classical mechanics and thermodynamics.
It started with the table of elements, the photoelectric effect, the black body radiation, the spectral lines in atomic spectra. All these could not be squeezed within the classical models. Bohr tried with his model.
The photoelectric effect forced thinking into light as particles, (once more , as Newton had proposed particles), the photons.
Then it was known and expected in classical electromagnetism that an accelerating electron would lose energy in the form of radiation into light,( so photons come into any radiation). This would be a continuous spectrum. Classical mechanics and classical electromagnetism could not produce the spectral lines, because by the classical equations the electron should fall on the nucleus emitting a continuous spectrum in the field of the protons, not the distinct spectra lines which were observed . So Bohr postulated that the electron was staying into orbits with specific energy and could only lose energy in photons ( the classical expectation) in quantized steps. This explained the phenomena mathematically by fitting series to the spectral lines, but was not satisfactory because it gave no framework for the other observations listed above, of forced states , quantized states for energy changes in the atomic micro framework.
After all, there's no particular reason for an electron to be in an eigenstate.
I explained the particular reason, if it were not in a stable orbit there would not be spectral lines to be observed and we would not have atoms, and be here discussing this
in the physical form we have.
What would make people think it was anything more than a (very suggestive) coincidence?
The postulates of Quantum Mechanics imposed on the mathematical solution of the Schrodinger equation brought logic and a causal path to the random efforts for a theoretical framework, outside the box of classical theories. So the appropriation of the differential equation now called "Schrodinger equation" to interpret the data was not a coincidence but a great think outside the box of classical theories. By imposing the physical postulates on the interpretation of the solutions, the fortuitous fits of the Bohr model series could be understood as derived from a formal mathematical physical theory.