# Lineshape function

1. Oct 20, 2004

### George Isaac Aziz

Hi all!

I have a problem understanding the essence of the lineshape function. This function is supposed to describe the linewidth of the emission from a two level system, i.e. practically, the discrete levels of stationary states are not exactly discrete. I thought this be the case when we include the interaction of the electromagnetic wave in the Hamiltonian of the problem. In other words, when we have an isolated atom, the states turn out to be stationary, there is no way for an electron to leave the stationary state if it already exists there as dictated by the time evolution of the wavefunction. In this case, the energy levels turn out to be discrete and everything is OK. I thought that when we include an electromagnetic wave of the correct frequency in the Hamiltonian of the problem, that is the system is no more isolated but interacting with radiation, we should be able to calculate the spread in the discrete energy levels due to the uncertainty principle directly from the Schrodinger equation. Unfortunately this does not turn out to be the case and we have to introduce this spread in an adhoc fashion. I have tried to look for a derivation from first principles to this problem, but I have failed. Does anybody know a solution?

2. Oct 20, 2004

### Claude Bile

Are you referring to natural lifetime broadening or just broadening in general?

Claude.

3. Oct 21, 2004

### George Isaac Aziz

Yes I was referring to natural lifetime broadening. I know that in gases the Doppler broadening is more effective, however it is the natural lifetime broadening that I am concerned with

4. Oct 21, 2004

### vanesch

Staff Emeritus
It is quite a subtle issue, and you need to use the full quantum treatment of the EM field, coupled to the atom, in order to understand spontaneous emission. You can find an explanation in section 15.5 of "optical coherence and quantum optics" by Mandel and Wolf. I have to say I don't understand all the details myself yet (need some more time to study it).

cheers,
Patrick.

5. Oct 21, 2004

### lalbatros

Hello

I think you could benefit from reading the following reference:

Quantum Mechanics (2 vol. set) by Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe
Chapter XIII : Approximation methods for time-dependent problems
Complements of chapter XIII, DXIII: Decay of a discrete state resonantly coupled to a continuum of final states

In this chapter the authors consider:

• the system has a 1 discrete state (say an atom in an excited state and the EM field in the ground state)
• in addition, the system has a continuous range of state (say the atom in ground state and the EM field in some excited states)
• the hamiltonian is not time dependent and assumes the form: $$H = H_{discrete} + H_{continuum} + V_{interaction}$$
• the interaction couples only the discrete level and the continuum

This simple model shows quite clearly the salient features of the question you asked.
I just read it myself and I have these comments:

• there is no exact analytical solution to this problem, although it is the simplest possible
• there is a convincing good approximation to the solution
• this solution predicts a line width for the transition probabilities to the continuous spectrum,
directly linked to the decay rate of the excited state (known effect for atomic transitions)
• this solution also predicts a displacement of the line as compared to the energy gap between excited and ground atomic state (also known effect for atomic transitions)
• the continuous part of the spectrum is, of course, the basic reason for the above consequences
• One key point in the derivation is related to the Fourier transform of the Heaviside step function.
This math contains physics of the decay!
• The (slow) decay of the exited state to the ground state produces a width of the emitted spectrum.
This is simply related to the Fourier transform of a decaying signal.
This is why the shape, finally.

If you find additional references (on the web), please let me know, I am interrested.

Note that the problem you asked is of a wider interrested.
It relates to other fields of physics, like statistical mechanics.

Last edited: Oct 22, 2004
6. Oct 22, 2004

### George Isaac Aziz

Thanks guys I'll check out the references you gave me and get back to you. I hope that I find a derivation that includes the continuum corresponding to the discrete eigenvalues.

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