 Solution

The operator $|e⟩⟨g|\otimes A$ annihilates one field quantum and excites the atom. Its adjoint $|g⟩⟨e|\otimes A^{†}$ de-excites the atom and creates one field quantum. These are the two exchange processes in the Jaynes-Cummings model.

Unitary time evolution requires the Hamiltonian to be Hermitian. Since the common coefficient $\frac{1}{2}\hslash (\Omega -\omega )$ is real, the two terms must be adjoints with equal coefficients. Therefore