Interaction of light with matter

[paweld]

I'm not sure what is the correct interaction hamiltonian between an electron
in an atom and electromagnetic wave (described classically). According to
me there exist two version of this hamiltonian which differ by factor 2:
$$H_{int} = \vec{E_0} \vec{r} \cos(\omega t)$$
or
$$H_{int} = 2 \vec{E_0} \vec{r} \cos(\omega t) = \vec{E_0} \vec{r} \exp( \textrm{i} \omega t ) + \vec{E_0} \vec{r} \exp(- \textrm{i} \omega t )$$.
Which one is correct?

 

[johng23]

You can specify whatever form of the field you want. Some people specify the field to be 2Ecos(wt) so they can write it as an exponential without carrying the factor of two around throughout the calculation. As long as you know what conventions you are using you should be able to get the right answer.

 

[paweld]

Are you sure that it's only convention. It looks like the amplitude of the field was E not 2E.

 

[johng23]

Well you'd have to show the specific example to clarify why a particular author used the factor of 2, but the fact is that the dipole moment operator is just E.r , no matter how you define the field (doesn't need to be a simple sinusoid at all). The 2 must come from something in the definition of the field.

 

[paweld]

Somebody has told my that the factor 2 comes from quantum nature of electromagnetic
field and if I want to obtain correct answer treating radiation classically then I should
include the factor 2.

 

[paweld]

I wonder what is the answer if we assume that the coupling is:
$$H_{int} = \hat{\vec{E}} \vec{r}$$
where $$\hat{\vec{E}}$$ is the electromagnetic field opperator (derivative of potential opperator
introduced in QED) and the field is initially in coherent state in one of its modes.
Is it possible to recover "classical" result.