  We obtain the wave functions associated to the quantum Newtonian universe
with a cosmological constant which is described by the Schr\"{o}dinger equation
and discuss some aspects of its dynamics for all forms of energy density,
namely, matter, radiation, vacuum, dark energy, and quintessence. These wave
functions of the quantum Newtonian universe are obtained in terms of the Heun's
functions and the respective energy levels are shown. We use these solutions to
investigate the expansion of the universe and found that the asymptotic
behavior for the scale factor is $R \sim \mbox{e}^{t}$ for whatever the form of
energy density is. We also analyze the behavior of the universe at early
stages.

General Relativity and Quantum Cosmology;High Energy Physics - Theory;Quantum Physics