To demonstrate the limitations due to the series of approximations in the Jaynes-Cummings Hamiltonian, we compare it to three other two-level system Hamiltonians to isolate the effect of a given approximation on the accuracy of the polaritonic eigenspectrum. Namely, we will compare it to the Rabi Hamiltonian, the PF Hamiltonian without PDM (Eq. 56), and introduce a two-level Pauli-Fierz Hamiltonian without DSE,