= Solution
The operator $|e\rangle\langle g|\otimes A$ annihilates one field quantum and excites the atom. Its adjoint $|g\rangle\langle e|\otimes A^\dagger$ 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 $\tfrac12\hbar(\Omega-\omega)$ is real, the two terms must be adjoints with equal coefficients. Therefore
$$
\boxed{\beta=1}.
$$
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