## Numerical calculations for S4 hydrogen

May 18, 2014 by zulfahmed

The exact integral formula (see 7.391 of Gradshteyn) for us will be:

Let the right hand side above be denoted by . Then we can write for introduced here by

A SUBTLETY

The Dirac operator has no harmonic spinors on and so it is sensible to map to the lowest energy shell of hydrogen. This is not ‘cheating’ and is in fact justified also because the denominator of for is not but instead and the scalar curvature is going to be equal to 12. This is crucial for the Lamb shift issue as well as we will see shortly that there are no degeneracies among energy levels if we consider for example for .

We can proceed now with numerical estimation using python libraries mpmath and math with simple code. We want to map $l

from mpmath import hyp3f2

from math import log, gamma, exp

def Q(r,s,a,b,n):

g1 = gamma(1+r) * gamma(1+s) / gamma(r+s+2)

g2 = (gamma(n+1+a) / gamma(1+n)) / gamma(1+a)

f = hyp3f2( -n, a+b+n+1, r+1, a+1, r+s+2, 1.0)

c = exp(log(2)*(r+s+1))

return c * g1 * g2 * f

def Cnl( n, l ):

return Q(l/2,l/2,l+1,l, n) + 2*Q(l/2+2,l/2+2,l+1,l,n)

print Cnl(3,0)

print Cnl(3,1)

print Cnl(3,2)

print Cnl(3,3)

print Cnl(4,0)

print Cnl(4,1)

print Cnl(4,2)

print Cnl(4,3)

print Cnl(4,4)

The output of this code is the following:

–> n = 3 for Dirac (lowest order shell for hydrogen)

0.271428571428571

1.33333333333333

0.666666666666667

1.5047619047619

–> n=4 for Dirac (n=1 shell for hydrogen)

0.4

2.0

0.857142857142857

2.59393939393939

1.58135198135198

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