  In this work the wave functions associated to the quantum relativistic
universe, which is described by the Wheeler-DeWitt equation, are obtained.
Taking into account different kinds of energy density, namely, matter,
radiation, vacuum, dark energy, and quintessence, we discuss some aspects of
the quantum dynamics. In all these cases, the wave functions of the quantum
relativistic universe are given in terms of the triconfluent Heun functions. We
investigate the expansion of the universe using these solutions and found that
the asymptotic behavior for the scale factor is $a(t) \sim \mbox{e}^{t}$ for
whatever the form of energy density is. On the other hand, we analyze the
behavior at early stages of the universe and found that $a(t) \sim t^{1/2}$. We
also calculate and analyze the transmission coefficient through the effective
potential barrier.
