Feeds:
Posts

## Numerical experiments seeking exact match to hydrogen spectrum

We are working in the unscaled 4-sphere $S^4(1)$ and we want to understand to what extent the $C_{nl}/(\lambda_n^2-3)$ can represent the relative differences between energy levels in hydrogen.  The figures for the hydrogen energy spectrum we use come from NIST (by entering ‘H’ in this form).  Now cosider the energy difference between 2p1 and 2s1, whose frequencies are 82258.919 and 82258.9544 1/cm.  Relative wavelength differences are directly related to relative energy differences by the $E = h\nu$ formula and we find that this is small at $\delta = 0.43\times 10^{-6}$.

Just for exploration, let us seek $n$ such that $C_{nl}/(\lambda_n^2 - 3) = \delta$ for some $l$.  Since we expect $C_{nl}$ to be O(10^0) bounded, we would want a relatively large $n$ for this problem, such as O(10^3).  Unfortunately our implementation of the formula for $C_{nl}$ produces numerical overflow errors for $n$ much larger than around 170.  So to check the principle that small $\delta$ can be handled, let’s pretend that it is $\delta_0 = 10^{-4}$ so $n=100$ will allow $C_{nl} \times (10^4-3) \sim 1$.  So for example $C_{100 2} = 7.612\times 10^{-2}$ which translates to $C_{100 2}/(10^4-3) \sim 7.6 \times 10^{-6}$ which is in the range of the Lamb shift delta.  This is all assuming that scaling by $h$ will allow us to carry over from calculations on $S^4(1)$ to our model of the real world $S^4(1/h)$.  Thus far we can see that if we are allowed to use a sufficiently high level of the spectrum of $D$ then we can match the actual relative Lamb shift.  But if we do this, then we may lose the clean mapping to the principal energy levels of hydrogen atom and the indexing of the eigenspaces of $D$ because there are larger numbers of possible degeneracies for $l$.

A DIFFERENT IDEA

The empirical hydrogen energy levels are calibrated in such a way if the energy level differences are written in terms of $B(1/n_1^2 - 1/n_2^2)$ for some constant $B$ then using the 2 and 3 orbitals we find $B = 1.09 \times 10^5$.  We can take the Dirac spectrum for $D + F$ where $F = Ze/\sin(t)$ and rescale it by $B$ and then consider whether we have the right order of magnitude for the Lamb shift equivalent for $n=3$ (representing ground state as we explained before since after subracting the scalar curvature term in the denominator).  So now we check to see if with this scaling factor $B$ whether $C_{4,0}/B$ has order of magnitude of the observed redshift.  In this case:  $C_{4,0) = 0.4$ and $C_{4,0}/B = 3.64 \times 10^{-6}$ which is in the same order of magnitude as $\delta = 0.43 \times 10^{-6}$.  In fact, we should really be comparing not $C_{4,0}$ but rather the difference between these, so $(C_{4,2} - C_{4,0})/B = 4.16\times 10^{-6}$.  So we improve the order of magnitude match by this rescaling but still have to understand how to justify this rescaling from the fundamentals.