The Wave Function of a Collapsing Dust Sphere inside the Black Hole Horizon

We investigate the wave function of a collapsing dust sphere and the interior of the associated black hole. We first consider the wave function of purely vacuum black hole interior and discuss the relation to the corresponding classical solutions. Because of the vacuum nature, a mass scale which characterizes the black hole mass can only be introduced into the wave function by hand. Then to put a mass scale into the wave function in a natural manner, we consider a dust collapse model whose classical counterpart can be exactly solved. As a result, we obtain the wave function of the black hole interior with mass determined by the dust mass. However, we find that the wave function is meaningful only at WKB level due to the sharp boundary surface of the dust sphere.