general relativity - What if a particle falls into the center of a central field?

Given a central field $U(r)$ satisfies $U(r) \rightarrow -\infty$ when $r \rightarrow 0$, then What if a particle falls into the center of a central field? Can you help me analysis this question in classical mechanics, relativistic mechanics and quantum mechanics? And, what will happen actually in experiments? Any help or suggestions will be appreciated!

Answer

If one studies how the theoretical understanding of physics has progressed we find that when infinities or infinitesimals are encountered with the prevailing at the time mathematical model, the model has reached its region of validity in describing physics.

They used to say that "nature abhors a vacuum". I would say that "nature abhors infinities and absolute zeros".

Classical mechanics and classical electrodynamics are a mathematically elegant and very successful model of the behavior of particles and radiation in nature, but we should always keep in mind the "model" part, and models have a region of validity in physics.

The infinities of your example show the need for a new mathematical description of nature and that was Quantum Mechanics. In QM central forces, as for example an electron around an atom, are no longer described by classical mechanics. The behavior is probabilistic and the electron will be trapped in an orbital, i.e. a locus of probability where it can be found if a measurement is performed.

The shapes of the first five atomic orbitals: 1s, 2s, 2px, 2py, and 2pz. The colors show the wave function phase. These are graphs of ψ(x, y, z) functions which depend on the coordinates of one electron. To see the elongated shape of ψ(x, y, z)2 functions that show probability density more directly, see the graphs of d-orbitals below.

In the S state there is a probability of finding the electron at the center of the classical potential, with no problems for the stability of the atom.

Quantum mechanical behavior sets in for small dimensions compatible with the values of h_bar, so the classical potential stops being valid when the dimensions get small enough. Edit after comment by OP:

What will happen if the electron is at the center of the classical potential? Its mechanical energy will be infinity? It still to be an electron? And, what will happen actually in experiments

The reason Quantum mechanics was postulated was in order to explain such infinities and paradoxes when taking the classical picture to the limits.

In the QM framework in the atomic dimensions there is no track or orbit for the electron in our example as there is an orbit for the moon in the 1/r attractive potential of gravity of the earth . There are only probability distributions in space of finding an electron in a particular (x,y,z) and they are called orbitals, not orbits, to avoid a confusion. Even though the potential that enters the QM equations is the same as the classical 1/r potential, the electron has a probability to be found at r=0 when it is in an orbital with 0 angular momentum ( S state); it will have the finite energy of the level of the orbital.

The orbitals are at quantized energy levels therefore one could estimate a velocity and this is finite, because the energy levels of the electrons around the atoms have finite energy levels. That was the whole point of "inventing" quantum mechanics: that the electrons did not disappear into the nucleus ( except in very rare atoms by a process called electron capture but that is another story altogether).

There have been many experiments and observations that confirm that at the micro level Quantum Mechanics reigns. For the atomic potentials the proof lies in the spectral lines which are quantized, and there exist innumerable experiments of electron nucleus scattering where , if the energy of the electron does not match one of the energies of the orbitals it will scatter and Quantum Mechanics describes that scattering with accuracy.

In contrast, in the classical mechanics framework, taking the solutions towards r=0, the head on impact would result in an explosion; this would dissipate the energy in fragments, assuming that the potential was generated by physical particles. We are discussing physics after all. There are no mass-less forces postulated in classical mechanics either, even though it is convenient to assume that one could go to r=0 in the framework of a static potential arising from the mass ( or charge). Mass takes up volume in classical mechanics.