Excited electrons and spectral lines

[berdan]

Homework Statement

Well,the problem is such : There are two hydrogen atoms.In each atom electrons were excited to the same energy level.Lifetime of one of the excited electrons is t1,when lifetime of other is t2.

The question is : Which on of them had more wider spectral line,when during descend they emmited photons.

Homework Equations

The regular hv=E2-E1=-13.6(1/(N1)^2-1/(N2)^2) [EV]

The Attempt at a Solution

Well,I said that if one lifetime is bigger,then one of them descended directly to its original level (less time),and the one with bigger lifetime is the one that passed throught several other energy levels before returning to original.Thus,the one that passed throught several different orbits emmited several spectral lines ,when the one that returned "faster" to its original level has less "wider" spectral line.


I might be badly mistaken (in my assumption that longer "lifetime" means ,it passed throught several different orbits before returning back to its original).

 

[collpitt]

Hello!

I think that the electron having a shorter lifetime would show wider spectral lines.

I think you have to change your approach towards the problem. Try to use the uncertainty principle to get your answer.
Do look at the proof that I have provided.

Spoiler

From the uncertainty principle,

ΔEΔt ~ \hbar (considering E tto be ground state)
and
ΔE ~ hΔ\nu
so,

Δ\nu ~ 1/(2\piΔt)
This equation signifies that for a short lifetime the range of frequencies would be more, thus implying a wider spectral band.

 

[berdan]

Well,I've just found the slide that talks about it,and you are absolutely correctous,sir.
He did used the uncertainty principles to get something close to this...Man,I never going to get this quantum stuff :(.

Thank you many many times.