Why are orbital radii quantized in the hydrogen atom?

[Physicsiscool]

What concepts lead to the quantination of radii in the hydrogen atom, for example: r = r(o) * n squared?

 

[masudr]

The energy and angular momentum operators have discrete spectra when applied to the system of a hydrogen atom. And angular momentum being momentum * radius means that the radius is quantised.

To understand why the operators are quantised, you have to go all the way back to how we choose quantum operators, and that's down to satisfying the classical correspondence limit (in the limit as masses and momentums become large, results predicted by QM tends to results predicted by classical mechanics) as well as satisfying the canonical quantum mechanical relations (pq - qp = -ih etc).

To understand what these operators work on, you should be familiar with the Hilbert space. You should also understand Schrödinger's equation - that's what we use to derive radii levels.

 

[Leo32]

I read a text which showed this quantisation to come foreward from 2 assumptions:

- the electrical attraction between electron/proton keeping the electron in a circular path around the proton
- the fact that the length of the path needs to be a multiple of the brogly wavelength

It's too early in the morning at work to dash out the formula's from my head, but it should give you an idea.

Greetz,

Leo

 

[dextercioby]

The ideas of quantization are much more deeper than to picture all sorts of de Broglie waves and how they fit into trajectories,concepts which have nothing to do with the QM.This description is basically taught at high-school level and it's for the mass,not for the ones who are interested in going under he surface.I remember my 12-th grade manual,it was so stupid,when talking Bohr Hydrogen atom,it mentioned de Broglie's quantizing condition...I threw it away.Thankfully me and the teacher were lot smarter than the authors...

 

[Physicsiscool]

This has been very helpful. Thank you!