Help me interpret what my professor meant?

[Matterwave]

Homework Statement

3. Degenerate Jaynes-Cummings model The Hamiltonian is
$$\hat{H}=\frac{1}{2}\hbar\omega_{0}\hat{\sigma}_{z}+\hbar\omega\hat{a}\dag\hat{a}+\frac{1}{2}\hbar\Omega i(\hat{\sigma}_{+}\hat{a}\dag-\hat{\sigma}_{-}\hat{a})$$ (8)

(a) Find the probability of the atom being in the ground state at time t when the initial state
of the system is |−, n>, where |±> are the two atom states and |n> is the n photon state.
(b) Do the same for the semiclassical case when the coupling constant in the interaction
term and the photon operators are replaced by the electrical dipole moment times the
appropriate electric field of the same magnitude as the electric field fluctuation of the n
photon state where n is macroscopic.
(c) Compare the results of the two cases.So, I have no problem solving part a, but as far as part b, I'm confused on what he means. I tried to just replace $$\Omega \rightarrow 2\mu_e\varepsilon(\omega)$$, and then replace the creation operator as sqrt(n) and the annihilation operator as sqrt(n), and solve for the non-composite state Hamiltonian. The problem with that is I get trivially the same exact solution as I got in part a because everything is done in analogy, and all I did was replace a constant with another constant, so I'm confused on what my professor is trying to ask. I will talk with him tomorrow to ask him, but in the mean time I was hoping you guys could help.

EDIT: Here $$\varepsilon(\omega)=\sqrt{\frac{2\hbar\omega}{\epsilon_0}}$$ so that $$\frac{1}{2}\epsilon_0\varepsilon^2(\alpha\alpha*)=\hbar\omega n$$

 

[Matterwave]

To explain better what I attempted to do for part b, I simply changed my Hamiltonian to:

$$\hat{H}=\frac{1}{2}\hbar\omega_0\hat{\sigma}_{z}+\hbar\omega n+\hbar i\mu_e\varepsilon(\omega)(\hat{\sigma}_{+}\sqrt{n}-\hat{\sigma}_{-}\sqrt{n})$$

And then my basis states are |+> and |-> where now I have gotten rid of the composite state problem by treating the electric field as external.