  Using a particular form of the quantum K-essence scalar field, we show that
in the quantum formalism, a fractional differential equation in the scalar
field variable, for some epochs in the Friedmann-Lema\^itre-Robertson-Walker
(FLRW) model (radiation and inflation-like epochs, for example), appears
naturally. In the classical analysis, the kinetic energy of scalar fields can
falsify the standard matter in the sense that we obtain the time behavior for
the scale factor in all scenarios of our Universe by using the Hamiltonian
formalism, where the results are analogous to those obtained by an algebraic
procedure in the Einstein field equations with standard matter. In the case of
the quantum Wheeler-DeWitt (WDW) equation for the scalar field $\phi$, a
fractional differential equation of order $\beta=\frac{2\alpha}{2\alpha-1}$ is
obtained. This fractional equation belongs to different intervals, depending on
the value of the barotropic parameter; that is to say, when $ \omega_X \in
[0,1]$, the order belongs to the interval $1\leq \beta \leq 2$, and when $
\omega_X \in [-1,0)$, the order belongs to the interval $0< \beta \leq 1$. The
corresponding quantum solutions are also given.
