Quantum cosmology with conformal scalar field in the Vilenkin theory

We use Vilenkin's method to discuss quantum cosmology with conformal scalar field. The corresponding Wheeler-DeWitt equation is derived and the wave function of the universe is calculated. The matter portion of the solution has the form of eigenstates of harmonic oscillators and the spatial portion is given by the WKB method. An analysis of the cosmic wave function reveals that, when quantum effect is included, the probability density of the universe appearing at a = 0 is zero and that the most probable radius of the base state of the universe is about one Planck length. The analysis also reveals that the universe is most probably in a state of small ∼φ.