# What is the average force for this process

1. Aug 5, 2010

### ani4physics

Hi all. I have been thinking about this problem for quite a while now. Still kinda clueless. We know that when we apply electromagnetic radiation on an atom/molecule, then depending on the frequency of the radiation the atom/molecule makes electronic transition. We know how to get the average energy absorbed per unit time by the atom. But what is the average force acting on the electrons during that process. I know the average force can be obtained using Ehrenfest theorem. But how do I apply it for the transition process. Any help greatly appreciated.

2. Aug 6, 2010

### The_Duck

Classically if you have an electron with charge e in an electric field E, the force on the electron is Ee. If the electric field is oscillating like a sine wave, then the average force over time will be zero. This is what we expect: if the average force was nonzero, then the electron would gain momentum and zoom off somewhere else. This is not what happens; the electron's (average) position is not affected by the oscillating field.

3. Aug 6, 2010

### ani4physics

Hey thanks very much for the reply. I am still confused. If the electron makes a transition from ground to excited state, the average value of the momentum is changed. And I guess the average value of the position operator is also changed (please correct me if I am wrong). So is the average force zero?

4. Aug 6, 2010

### alxm

'Force' in the classical sense (F=ma). Has no useful meaning in quantum mechanics, because position and momentum don't commute.

If you want to study electronic transitions in atoms, you have to use qm. Or, if you think qm makes use of classical forces, you've misunderstood basic quantum mechanics. On an unrelated note, I don't see how Ehrenfest's theorem is relevant here. And the average momentum for any bound electronic state is zero.

I don't know what you're trying to calculate, but you can't understand electron-photon interactions until you first understand properly how it works without photons around. The line of questioning suggests you don't.

5. Aug 7, 2010

### ani4physics

Hey thanks very much. Yeah I understand that classical concept of force is not correct for electronic transitions. However, we can define an operator for force ( for example, if an electric field acts on the molecule, the operator for the external force on electrons can be represented by the charge density of the electrons and the external field). My question is, what is the average value of that operator when the electron makes a transition.
Also, regarding Ehrenfest's theorem: If the molecule interacts with a time-dependent perturbation, the average force on the electrons can be calculated by taking the expectation value of the force operator w.r.t the time-dependent wave function, right? So the time-derivative of the average value of the momentum is the average force. Please correct me if I am wrong.

Thanks so much for the great suggestions so far. Really appreciate it.

6. Aug 7, 2010

### alxm

No, that's the point - you can't define an operator for force on an electron in any meaningful way, because position and momentum do not commute. You can define a potential, but that's not the same thing.

The theory of electronic transitions is well established, and nowhere does anyone ever use a 'force operator'. What would you even do with this force, should you be able to calculate it?

This is only true in the classical limit. Besides, <p> doesn't change during an electronic transition. <p^2> does, and <L> does.

Last edited: Aug 7, 2010
7. Aug 7, 2010

### ani4physics

What I am trying to calculate is this: we know that when the atom interacts with the radiation field, it absorbs energy from the radiation. we can calculate the rate of energy transfer from the transition rate : dE/dt = transition rate * (En - E0)) summed over all excited states. so, this is the power absorbed from the radiation source by the atom. I am trying to see if we can derive a force * displacement analog for this result. Of course we can't get a classical result for this. But is there a way to get a quantum mechanical version for this?
Once again, thanks a lot for the great suggestions.

8. Aug 7, 2010

### ani4physics

Let me write the problem in a different way: Suppose the atom is interacting with an external time-dependent perturbation. So under this condition, the average value of the electronic momentum is <P> = <psi(t)[P]psi*(t)>, where P is the electronic momentum operator and psi(t) can be obtained from time-dependent perturbation theory. So, d<P>/dt = d<psi(t)[P]psi*(t)>/dt. Is this the average force acting on the electrons in presence of the perturbation? If so, what will be the d<P>/dt when the electron makes a transition from ground to excited state?

9. Aug 8, 2010

### alxm

Then it's zero, because <p> is zero for every bound excited state. Which is physically what you'd expect anyway. If an electron had a net linear momentum it would fly off and not be bound to the atom.

10. Aug 8, 2010

### ani4physics

Ok for every bound stationary states of the atom, <p> is zero. But what about the value of <p> when the atom is perturbed. Will <p> be zero for perturbed wave function as well? because if we take <psi(t)[p]psi(t)> and calculate psi(t) by first-order time-dependent perturbation, then <p> will invlove off-diagonal elements and will not be zero. so in that case, d<p>/dt is not equal to zero.