  We consider an exponentially expanding, flat,
Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe filled with a free
Schroedinger field. The probability fluid of the latter is used to mimic the
cosmological fluid (baryonic plus dark matter), thus providing the matter
density and pressure terms in the corresponding Friedmann-Lemaitre equations.
We first obtain the eigenfunctions of the Laplacian operator on flat FLRW
space. A quantum operator qualifying as a cosmological constant is defined to
act on the Schroedinger field. We then compute the matrix representing the
cosmological constant in the basis of Laplacian eigenfunctions. For an estimate
of the orders of magnitude involved it suffices to determine the expectation
values of this operator. The expectation value that best fits the
experimentally measured value of the cosmological constant allows us to
identify the quantum state of the Schroedinger field that best represents the
matter contents (baryonic and dark) of the current Universe. Finally, the
operator inverse (modulo dimensional factors) to the one representing the
cosmological constant provides a measure of the gravitational Boltzmann entropy
of the Universe. We compute its matrix in the basis of Laplacian eigenfunctions
and verify that the expectation values of this entropy operator comply with the
upper bound set by the holographic principle.
