  Exciton-polariton propagation in a quantum well, under centre-of-mass
quantization, is computed by a variational self-consistent microscopic theory.
The Wannier exciton envelope functions basis set is given by the simple
analytical model of ref. [1], based on pure states of the centre-of-mass wave
vector, free from fitting parameters and "ad hoc" (the so called additional
boundary conditions-ABCs) assumptions. In the present paper, the former
analytical model is implemented in order to reproduce the centre-of-mass
quantization in a large range of quantum well thicknesses (5a_B < L < inf.).
The role of the dynamical transition layer at the well/barrier interfaces is
discussed at variance of the classical Pekar's dead-layer and ABCs. The Wannier
exciton eigenstates are computed, and compared with various theoretical models
with different degrees of accuracy. Exciton-polariton transmission spectra in
large quantum wells (L>> a_B) are computed and compared with experimental
results of Schneider et al.\cite{Schneider} in high quality GaAs samples. The
sound agreement between theory and experiment allows to unambiguously assign
the exciton-polariton dips of the transmission spectrum to the pure states of
the Wannier exciton center-of-mass quantization.
