  The Friedmann-Robertson-Walker (FRW) cosmology is analyzed with a particular
potential $\rm V(\phi)=V_0 e^{-\sqrt{3} \phi}$ in the quintessence field
scenario, which emerges in the supersymmetric quantum mechanics (SUSY)
formalism. Using Hamilton's approach for a scalar field $\phi$ with standard
kinetic energy, and the Hamilton equations, we find exact solutions to the
complete set of the
  Einstein-Klein-Gordon equations without the need of the slow-roll conditions
in order to model the inflation phenomenon. We find that the solutions are in
good agreement with the inflationary conditions such as the e-folding function
${\cal N}$ which corresponds to the $\Omega$ function in the Misner
parametrization for the scale factor $\rm A(t)=e^{\Omega(t)}$ when evaluated in
$\rm \Delta t$, which is the time interval for the inflation period. The
acceleration of the scale factor was computed and it was found to be positive
for the inflation period with a range of values for the parameters of the
model. Quantum solution from the Wheeler-DeWitt equation is presented, where
the wave function in relation to the evolution of the scale factor, shows that
for this period of time at larger values of $\rm A$ and for any value of
scalar-field $\rm \varphi$, the wave function is peaked.
