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

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

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Answer 1 · 2015-01-16T18:33:58

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:

$n λ = 2 π r$ $nlambda =2pir$

$n$ $n$ is an integer

$λ$ $lambda$ = wavelength of electron

$r$ $r$ = radius of orbit.

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

$λ = \frac{h}{m v}$ $lambda =(h)/(mv)$

Where:

$h$ $h$ = the Planck Constant

$m$ $m$ = mass of electron

$v$ $v$ = velocity of electron

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

$\frac{n h}{m v} = 2 π r$ $(nh)/(mv)=2pir$

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

$m v r = \frac{n h}{2 π}$ $mvr = (nh)/(2pi)$

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

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

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