The Jaynes-Cummings Hamiltonian

[kof9595995]

This Hamiltonian popped up when I was reading an article, as a reference(wikipedia): http://en.wikipedia.org/wiki/Jaynes–Cummings_model#cite_note-1
I don't understand why the Hamiltonian $\hat H_{atom}$ and $\hat H_{int}$ look the way they are. Usually we we just take a classical Hamiltonian and "operatorize" it, but I fail to see the classical counterpart for $\hat H_{atom}$ and $\hat H_{int}$

 

[Cthugha]

This Hamiltonian popped up when I was reading an article, as a reference(wikipedia): http://en.wikipedia.org/wiki/Jaynes–Cummings_model#cite_note-1
I don't understand why the Hamiltonian $\hat H_{atom}$ and $\hat H_{int}$ look the way they are. Usually we we just take a classical Hamiltonian and "operatorize" it, but I fail to see the classical counterpart for $\hat H_{atom}$ and $\hat H_{int}$

$\hat H_{atom}$ is just a generic two-level system. Therefore in the easiest case you get just two energy levels at $E_{1/2}=\pm\hbar \omega$. The interaction Hamiltonian just describes the interaction between the bosonic light field and this two-level system. The energy-conserving terms of $\hat E \hat S$ describe the destruction of a photon combined with the simultaneous excitation of the two-level system and the creation of a photon combined with the simultaneous transition of the two-level system from the excited to the ground state.

 

[kof9595995]

The energy-conserving terms of $\hat E \hat S$ describe the destruction of a photon combined with the simultaneous excitation of the two-level system and the creation of a photon combined with the simultaneous transition of the two-level system from the excited to the ground state.

I can't see the physics you said form $\hat E \hat S$. Actually I did find something more elaborate on this: http://uncw.edu/phy/documents/Shafer499Talk.pdf
But I don't understand the content on page 27, i.e. why are those four equations the defining properties of a dipole operator, the author gave a handwaving reason "The dipole operator is responsible for “moving” the atom between energy levels.", but I don't really see why it has to be the way it is.
EDIT: I think I understand now, the author is probably referring to dipole transition.