E. Todesco, Milano Bicocca January-February 2016 Appendix B: A digression on divergences in electromagnetism and lengths in atomic physics Ezio Todesco European Organization for Nuclear Research (CERN)

E. Todesco, Milano Bicocca January-February 2016 CONTENTS Electrostatic divergence: classical radius of electron A less known divergence of magnetic field A reminder on Bohr radius Fine structure constant Compton length and its role in quantum mechanics Appendix B - 2

E. Todesco, Milano Bicocca January-February 2016 CLASSICAL RADIUS OF ELECTRON Maxwell equations present a divergence for a point charge Stored energy is proportional to the square of electric field Integrating over a sphere larger than R We obtain a divergence proportional to the size of the sphere The classical electron radius is defined by equating this energy to half of the rest mass energy Appendix B - 3

E. Todesco, Milano Bicocca January-February 2016 CLASSICAL RADIUS OF ELECTRON One can prove that the energy associated to a charge e on the surface of a sphere whose radius is r e is mc 2 /2 So the total energy associated to electric field in space and to the charges on the sphere surface is equal to the electron mass The electron classical radius is 2.8× m The scale length of Thompson scattering is the classical electron radius: Appendix B - 4

E. Todesco, Milano Bicocca January-February 2016 DIVERGENCE OF A CURRENT LOOP A less know divergence is related to the energy stored in the magnetic field We consider a loop of current of radius r L and current I The loop creates a dipole-like magnetic field whose energy is Close to the current line the field is proportional to the inverse of the distance to the current line Appendix B - 5

E. Todesco, Milano Bicocca January-February 2016 DIVERGENCE OF A CURRENT LOOP And one can approximate the integral with the contribution close to the current line where r w is the wire radius The energy diverges with the logarithm of the radius of the wire, so a point charge also has a divergence in the energy associated to the magnetic field, not only to the electric field The inductance of a loop of current depends on the wire size Appendix B - 6

E. Todesco, Milano Bicocca January-February 2016 CONTENTS Electrostatic divergence: classical radius of electron A less known divergence of magnetic field A reminder on Bohr radius Fine structure constant Compton length and its role in quantum mechanics Appendix B - 7

E. Todesco, Milano Bicocca January-February 2016 A REMINDER ON BOHR RADIUS The classical electron radius m is about four order of magnitudes smaller than the size of atoms Bohr radius can be computed from the electrostatic potential of an electron in the field created by a proton, whose energy is Plus the quantization of the angular momentum For a circular orbit Appendix B - 8

E. Todesco, Milano Bicocca January-February 2016 A REMINDER ON BOHR RADIUS Replacing the momentum using quantization, one has And minimizing the energy one finds the Bohr radius The Bohr radius is 5.3× m Appendix B - 9

E. Todesco, Milano Bicocca January-February 2016 ALPHA: THE FINE STRUCTURE CONSTANT What is rarely found in textbooks is that the ratio between the Bohr radius and the classical electron radius is ≈(137) 2, the square of an important physical constant The quantity between brackets is the inverse of the fine structure constant It has a very misleading name, in reality is the coupling constant of electromagnetism e 2 normalized through h and c Appendix B - 10

E. Todesco, Milano Bicocca January-February 2016 CONTENTS Electrostatic divergence: classical radius of electron A less known divergence of magnetic field A reminder on Bohr radius Fine structure constant Compton length and its role in quantum mechanics Appendix B - 11

E. Todesco, Milano Bicocca January-February 2016 ALPHA AND COMPTON LENGTH In between the Bohr radius and the classical electron radius one finds another very important quantity This is called reduced Compton wavelength Whose size is 3.8× m Appendix B - 12

E. Todesco, Milano Bicocca January-February 2016 ALPHA AND COMPTON LENGTH The Compton wavelength is introduced in the Compton scattering between electron and photon, treated as a collision between an electron of mass m and a photon behaving as a particle of energy hv The Compton wavelength is the wavelength of a photon having the same energy as the electron mass But the Compton radius is the fundamental length in the equations of quantum electrodynamics Appendix B - 13

E. Todesco, Milano Bicocca January-February 2016 ALPHA AND COMPTON LENGTH Schroedinger equation is usually written as But if we divide by hc/2  we obtain Appendix B - 14

E. Todesco, Milano Bicocca January-February 2016 ALPHA AND COMPTON LENGTH Dirac equation is And Klein Gordon equation is Appendix B - 15

E. Todesco, Milano Bicocca January-February 2016 SUMMARY If the charge is considered as a point, the electromagnetism has two divergences A divergence 1/r in the electrostatic energy associated to a charged sphere of radius r A divergence ln (1/r) in the magnetic energy (inductance) of a loop whose wire has a radius r Compton length r C and fine structure constant  play fundamental role in quantum electrodynamics, respectively the scale length and the coupling constant Notwithstanding their misleading names that hide their deep physical meaning Appendix B - 16

E. Todesco, Milano Bicocca January-February 2016 SUMMARY There is a stair of three lengths spaced by powers of  Bohr radius, size of the atoms m Compton radius, m Length in Compton scattering, but first of all fundamental length in quantum electrodynamics Classical electron radius m Length in Thompson scattering The fine structure constant is the perturbative parameter in quantum electrodynamics Since it is 1/100, perturbative expansions (Feynman diagrams) work Appendix B - 17