q-Laguerre polynomials and fractionized principal quantum number for hydrogen atom

Here and here a semiclassical model based on dark matter and hierarchy of Planck constants is developed for the fractionized principal quantum number n claimed by Mills to have at least the values n=1/k, k=2,3,4,5,6,7,10. This model could explain the claimed fractionization of the principal quantum number n for hydrogen atom in terms of single electron transitions for all cases except n=1/2: the basis reason is that Jones inclusions are characterized by quantum phases q=exp(iπ/n), n> 2. Since quantum deformation of the standard quantum mechanism is involved, this motivates an attempt to understand the claimed fractionization in terms of q-analog of hydrogen atom.

The Laguerre polynomials appearing in the solution of Schrödinger equation for hydrogen atom possess quantum variant, so called q-Laguerre polynomials, and one might hope that they would allow to realize this semiclassical picture at the level of solutions of appropriately modified Schrödinger equation and perhaps also resolve the difficulty associated with n=1/2. Unfortunately, the polynomials correspond to 0<q< 1 rather than complex values of q=exp(iπ/m) on circle and the extrapolation of the formulas for energy eigenvalues gives complex energies.

The most obvious q-modification of Laguerre equation is to replace the ordinary derivative with an average of q-derivatives for q and its conjugate. As a result one obtains a difference equation and one can deduce from the power series expansion of q-Laguerre polynomials easily the energy eigen values. The ground state energy remains unchanged and excited energies receive corrections which however vanish at the limit when m becomes very large. Fractionization in the desired sense is not obtained.

q-Laguerre equation however allows non-polynomial solutions which are square integrable. By the periodicity of the coefficients of the difference equation with respect to the power n in Taylor expansion the solutions can be written as a polynomial of order 2m multiplied by a geometric series. For odd m the geometric series converges and I have not been able to identify any quantization recipe for energy. For even m the geometric series has a pole at certain point, which can be however cancelled if the polynomial coefficient vanishes at the same point. This gives rise to the quantization of energy. It turns out that the fractional principal quantum numbers claimed by Mills correspond very nearly to the zeros of the polynomial with one frustrating exception: n=1/2 producing trouble also in the semiclassical argument. Despite this shortcoming the result forces to take the claims of Mills rather seriously and it might be a good idea for colleagues to take a less arrogant attitude towards experimental findings which do not directely relate to calculations of black hole entropy.

Note added: It turned out that for odd m for which geometric series converges always, allows n=1/2 as a universal solution having a special symmetry implying that solution is product of m:th (rather than 2m:th) order polynomial multiplied with a geometric series of x^m (rather than x^2m). n=1/2 is a universal solution. This is in spirit with what is known about representations of quantum groups and this symmetry removes also the doubling of almost integer states. Besides this one obtains solutions for which n depends on m. This symmetry applies also in case of even values of m studied first numerically.

Note added: The exact spectrum for for the principal quantum number n can be found for both even and odd values of m. The expression for n is simply

n₊= 1/2 + R_n/2,

n_-= 1/2 - R_n/2,

R_n= 2cos(π(n-1)/m)-2cos(πn/m).

This expression holds for all roots for even values of m and and for odd values of m for all but one corresponding to n=(m+1)/2. The remaining zero is of course n=1/2 in this case.

The chapter Dark Nuclear Physics and Condensed Matter contains the detailed calculations. See also the article Could q-Laguerre equation explain the claimed fractionation of the principal quantum number for hydrogen atom?