Scalar Field Cosmology: Classical and Quantum Viewpoints

In this article, with an overview to the dynamics of the homogeneous and isotropic cosmology for the usual and phantom scalar fields, we investigate the model of exponentially damping fields. We obtain the extract solutions for the potential function, the scale factor of the model and the dynamics of the parameter of the equation of state. We present two proposals for the scalar field to achieve the exponential potential function in terms of time and extract Hubble's cosmological parameters, scale factor and equation of state parameter for each model. At the end of the work, we turn our attention to the quantum cosmology of the model with time-damped exponential potential and form the Wheeler-Dewitt equation with some new variables. The Wheeler-DeWitt wave packets with the Gaussian weight function are obtained for the scalar fields in different cases and draw the probability function of each one.