  We show that semi-classical states adapted to the symmetry of the Hamiltonian
are an excellent approximation to the exact quantum solution of the ground and
first excited states of the Dicke model. Their overlap to the exact quantum
states is very close to 1 except in a close vicinity of the quantum phase
transition. Furthermore, they have analytic forms in terms of the model
parameters and allow us to calculate analytically the expectation values of
field and matter observables. Some of these differ considerably from results
obtained via the standard coherent states, and by means of Holstein-Primakoff
series expansion of the Dicke Hamiltonian. Comparison with exact solutions
obtained numerically support our results. In particular, it is shown that the
expectation values of the number of photons and of the number of excited atoms
have no singularities at the phase transition. We comment on why other authors
have previously found otherwise.
