Quantum mechanical approach to small black holes

A very simple quantum model for small black holes (i.e., with the mass of an asteroid and the linear dimensions of an elementary particle) is proposed which involves extracting the main features of a small black hole from general relativity and 'translating' them into the language of quantum mechanics. Essentially, the black hole is 'simulated' by means of a suitable potential in the radial wave equation for a neutral particle in the Schwarzschild metric. The horizon of the black hole is introduced in the quantum scheme by making an analogy between the concepts of the hole's horizon and a potential well in quantum mechanics. Black-hole total scattering cross sections are derived for the general case of a neutral particle and for the limiting cases of low and high particle energy in the Schwarzschild metric as well as for the general case of a charged particle and the limiting high- and low-energy cases in the Reissner-Nordstrom metric. It is concluded that the analogy between the concepts of the horizon in general relativity and the potential well in quantum mechanics seems to be quite consistent inasmuch as the short-range potential in the total cross sections reproduces the area of the black hole, and particle-creation phenomena enter the model in a simple way when the electric field is added.