Bohr made the major innovation of hypothesising the quantization of angular momentum in units of
where
This means we can write
We can use the ordinary rules of classical Newtonian mechanics to derive the equation giving the differences in the energy levels of the electrons in the Hydrogen atom.
We can equate the Coulomb force to the force required to keep the electron in orbit around the nucleus.
From this we can obtain
(1)
and
(2)
The total energy of the electron in it's orbit is the sum of the kinetic and potential energies: Use of (2) gives
This means an energy change of
where
is the frequency and
is the wavelength of the associated emitted or absorbed photon. Substitute (1) to give
Use this and divide by
to get
is called the Rydberg constant and is given the symbol
It has the value