Bohr postulated that the electron in the hydrogen atom could only exist stably in orbits that had angular momentum equal to an integer multiple of
where
called Planck's constant.
Mathematically, for an assumed circular orbit, the angular momentum
(1)
The centripetal force is given by mv^2 over r and this is equal to the electrostatic force between the proton and electron.
Substitute the expression for v from (1) into this and rearrange to obtain
The energy levels are given by the sum pf the kinetic and potential energies.
So
Substitution of the values of
and
into this expression gives the formula
We can derive the Rydberg formula by considering the difference between two energy levels.
Dividing by
and taking the
sign inside the brackets gives
which becomes the Rydberg formula by putting