Why should a black hole be infinitely dense? I have been listening to this on Discovery for centuries of my childhood!! That when the heavy core of a star collapses under its own gravity, it shrinks to an infinitely dense point called "singularity". However, recently I was introduced to wave mechanics and The Schrodinger equation, so when the shrinking mass gets as small as the size of a hydrogen atom, quantum mechanics should be dominating the scenario, thereby eliminating all "singularities" - there should be a great great (but finite) density in that place. Maybe I took it all wrong, so please resolve my apparent paradoxical situation.
 A: This is one of the currently unanswered questions in physics. The singularity of a black hole is a place where the spacetime curvature is very high (so general relativity is important), and where the size is very small (so quantum mechanics is important). Therefore, the general expectation is that we would need a quantum theory of gravity to tell us what happens in the singularity of a black hole. Since, at the present time, no one knows what the quantum theory of gravity is, no one can answer this question scientifically.
As far as I know, within candidate theories of quantum gravity, such as string theory, it is not known how the singularity is resolved. As a caveat, there may be special cases where the singularity can be resolved in string theory which I don't know about, or other candidate theories of quantum gravity may have something to say about resolving the singularity.
A: A black hole need not be a singularity. A mass distribution like that of the observable universe in an infinite empty space will be a black hole with an event horizon outside the distribution. The Schwarzschild radius depends on M, the mass of the hole, and the density depends on the cube of the radius. So a hole can exist with finite mass density.
It's even the question if all mass reaches the center before it's evaporated. The event horizon appears when all particles are still heading for the center, which takes about the lightspeed times the Schwarzschild radius. For the particles falling freely from the horizon.
