Hydrogen atoms have only one electron. This makes it easier to try to understand how they work. When atoms are heated, they can give off light. The familiar yellow street lights heat sodium into a gas which emits this distinctive yellow light which drains things of colour.
We can look at the colour of the light given off by atoms by splitting it into a spectrum with a prism. Light from the sun produces a continuous band of colour. Light from a sodium lamp has a bright yellow line and some other faint green and red lines.
The hydrogen atom also has a spectrum consisting of sharp lines, but the main lines are not visible to the eye and have to be seen using a camera. Light comes in photons. The colour of a photon depends on its energy. Red photons have less energy than blue photons.
In the sun and planet model of the atom, a photon is emitted when an electron falls from a higher orbit to a lower one. The amount of energy in a photon depends on the orbits involved. We can measure the differences in energy levels of the orbits by studying the spectrum of the light emitted by the atoms.
The structure of the hydrogen atom is determined by the number of strands of magnetic flux which wrap around the electron's orbit. The atom has least energy and the tightest orbit when only one strand of magnetic flux wraps its orbit.
The amount of energy stored in a magnetic field depends on two things, the number of stands of magnetic flux in the field and by the electric current which generates it. If we put together the laws of electromagnetism and the laws of planetary motion, we can calculate the exact energy levels of the hydrogen atom.
The laws were known more than 50 years before the research into atomic spectra, what was not known was the fact that magnetic flux comes in strands. We had the concept of lines of force, but no one had ever made a magnetic field with just one or two strands of flux. To do that, we need liquid helium and superconductors.
Tiny little bar magnets can be made from short lengths of very thin fuse wire cooled to superconducting temperatures. Fuse wire is made of copper and coated with tin. The tin becomes superconducting and traps magnetic flux from the earth's magnetic field. The number of strands captured depends on the diameter of the wire and its direction relative the earth's magnetic field. Once formed, the strength of those bar magnets can be measured and the amount of flux in them can be calculated.
These ideas give us a simple theory of the hydrogen atom based on the laws of classical physics.