Cosmological Wave Function and Wormhole Wave Function with a Conformal Complex Scalar Field

Quantum cosmology and the quantum wormhole witha conformal complex scalar field are discussed, thecorresponding Wheeler-DeWitt equations are obtained, andthe cosmological wave functions and wormhole wave functions are calculated, respectively,with different boundary conditions. From thecosmological wave function it is found that theprobability density of the universe is zero at a = 0,while at the ground state the most probable radius is aboutthe Planck scale. It is also shown that there exist twodifferent types of universes, which can be connected bythe quantum tunneling effect, transiting from one region to another. It follows from thewormhole wave function that the most probable radius ofthe wormholes is about the Planck scale, which impliesthat the wormhole is steady due to the quantumeffect.