First two equations are for the cathode ray experiment, which sets magnetic force equal to the centripetal force. The cathode rays then became known as electrons. The third is the oil drop experiment, in which the electron's electric force is equal to its weight. Simple and elegant.
Scientists then became interested in why elements only emit certain wavelengths. They came up with various mathematical models for the hydrogen atom, in which the Balmer series describes visible wavelengths, the Lyman series UV, and the Paschen series IR. All three combined shows the emission spectrum for hydrogen. Rydberg's constant R is 1.0974 E7 /m.
The orbit radius then became of interest. The second step of the derivation is electric and centripetal force. The Bohr radius is the inner orbit of a hydrogen atom.
The energy of light emitted is the energy it takes for an electron to jump levels. The quantum condition is interesting where the angular momentum L is.. quantized. The integer n refers to the orbit level, or principal quantum number. The most amazing thing is probably how the angular momentum reduces to an integer and some constants.
The total energy is the kinetic energy minus the potential energy. Substitute v from angular momentum and r from radius. The ground state of a hydrogen atom is 13.6eV. The second set of equations describe the wavelength associated to an electron skipping energy levels, as Balmer, Lyman, and Paschen had previously attempted.
To explain the quantized energy levels, the Broglie wavelength of an electron must fit the circumference of its orbit by an integer n. The equation can then derive angular momentum as in the quantum condition, thus verifying its legitimacy.
Except that physics likes to be counter intuitive. It continues to boggle our minds and turn our brains inside out..
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