# How did DeBroglie's hypothesis account for the fact that the energy in a hydrogen atom is quantised?

Jan 16, 2015

Bohr assumed that electrons move in an orbit around the central nucleus and only certain orbits are allowed.

The electron can be considered as a standing wave. This means that only an integral number of wavelengths can fit into a circular orbit. So we can write:

$n \lambda = 2 \pi r$

$n$ is an integer

$\lambda$ = wavelength of electron

$r$ = radius of orbit.

The wavelength of the electron is given by the de Broglie expression:

$\lambda = \frac{h}{m v}$

Where:

$h$ = the Planck Constant

$m$ = mass of electron

$v$ = velocity of electron

Substituting for $\lambda$ into the 1st equation we get:

$\frac{n h}{m v} = 2 \pi r$

The angular momentum of the electron = $m v r$ so rearranging we get:

$m v r = \frac{n h}{2 \pi}$

This is an important result in that it tells us that the angular momentum of the electron can only take integral values of $\frac{h}{2 \pi}$ I.e it is quantised.