The most important success in the early days of quantum mechanics was the appropriate description of the hydrogen atom. It is not exaggerated to say that quantum mechanics was discovered in its final form in the 1920ies in the attempt to describe the hydrogen atom which could not be understood in terms of a classical theory.

To describe the behaviour of an electron with the help of quantum mechanics means to solve the time independent Schrödinger equation (respectively the Dirac equation in the relativistic case) with a coupled Coulomb potential and thus to find the eigenstates of the corresponding Hamiltonian. The existence of bound states within an atom has fundamental importance since it is the reason for the stability of matter. So it is suggestive to consider a similar situation, namely the behaviour of an electron within a spherical symmetric gravitational field. Since gravitation is much weaker than electromagnetism (about a factor of $10^{43}$ in case of the mass and the charge of an electron) the body being the source of the gravitational field must have a huge mass to lead to an effect of the same order of magnitude as in the electromagnetic case. The most massive objects known 