Editing Regge theory (section)

==Details<!--'Regge pole' and 'Regge poles' redirect here-->==
[[File:Chew-Frautschi plot.svg|thumb|Chew-Frautschi plot showing linear Regge trajectories.]]
The simplest example of '''Regge poles'''<!--boldface per WP:R#PLA--> is provided by the quantum mechanical treatment of the [[Coulomb potential]] <math>V(r) = -e^2/(4\pi\epsilon_0r)</math> or, phrased differently, by the quantum mechanical treatment of the binding or scattering of an electron of mass <math>m</math> and electric charge <math>-e</math> off a proton of mass <math>M</math> and charge <math>+e</math>. The energy <math>E</math> of the binding of the electron to the proton is negative whereas for scattering the energy is positive. The formula for the binding energy is the expression
:<math>E\rightarrow E_N = - \frac{2m'\pi^2e^4}{h^2N^2(4\pi\epsilon_0)^2} = - \frac{13.6\,\mathrm{eV}}{N^2}, \;\;\; m^' = \frac{mM}{M+m}, </math> 
where <math>N = 1,2,3,...</math>, <math>h</math> is the Planck constant, and <math>\epsilon_0</math> is the permittivity of the vacuum. The principal quantum number <math>N</math> is in quantum mechanics (by solution of the radial [[Schrödinger equation]]) found to be given by <math>N = n+l+1</math>, where <math>n=0,1,2,...</math> is the radial quantum number and <math>l=0,1,2,3,...</math> the quantum number of the orbital angular momentum. Solving the above equation for <math>l</math>, one obtains the equation 
:<math>l\rightarrow l(E) = -n +g(E), \;\; g(E) = -1+i\frac{\pi e^2}{4\pi\epsilon_0h}(2m'/E)^{1/2}.</math>
Considered as a complex function of <math>E</math> this expression describes in the complex <math>l</math>-plane a path which is called a '''Regge trajectory'''. Thus in this consideration  the orbital
momentum can assume complex values.

Regge trajectories can be obtained for many other potentials, in particular also for the [[Yukawa potential]].<ref>Harald J.W. Müller-Kirsten: Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (2012) pp. 395-414</ref><ref>{{cite journal | last=Müller | first=Harald J. W. | title=Regge-Pole in der nichtrelativistischen Potentialstreuung | journal=Annalen der Physik | publisher=Wiley | volume=470 | issue=7–8 | year=1965 | issn=0003-3804 | doi=10.1002/andp.19654700708 | pages=395–411 | bibcode=1965AnP...470..395M | language=de}}</ref><ref>{{cite journal | last1=Müller | first1=H. J. W. | last2=Schilcher | first2=K. | title=High-Energy Scattering for Yukawa Potentials | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=9 | issue=2 | year=1968 | issn=0022-2488 | doi=10.1063/1.1664576 | pages=255–259}}</ref>

Regge trajectories appear as poles of the scattering amplitude or in the related  <math>S</math>-matrix. In the case of the Coulomb potential considered above this <math>S</math>-matrix is given by the following expression as can be checked by reference to any textbook on quantum mechanics:
:<math> 
S = \frac{\Gamma(l-g(E))}{\Gamma(l+g(E))}e^{-i\pi l},
</math>
where <math>\Gamma(x)</math> is the [[gamma function]], a generalization of factorial <math>(x-1)!</math>. This gamma function is a [[meromorphic function]] of its argument with simple poles at <math>x=-n, n=0,1,2,...</math>. Thus the expression for <math>S</math> (the gamma function in the numerator) possesses poles at precisely those points which are given by the above expression for the Regge trajectories; hence the name Regge poles.