Experimental measurement of hydrogen atom is obtained from the NIST database: http://physics.nist.gov/cgi-bin/ASD/energy1.pl
The "Level" column below are in electron-Volts. I will show below how these levels can be explained by classical mechanics on a 4-sphere of a fixed radius R ~ 4 x 10^16 meters.
-----------------------------------------------------------------------------------
Configuration | Term | J | Level |
----------------|--------|------|----------------------|---------------------------
| | | |
1s | 2S | 1/2 | 0.0 |
| | | |
2p | 2P* | 1/2 | 10.1988057 |
| | 3/2 | 10.1988511 |
| | | |
2s | 2S | 1/2 | 10.1988101 |
| | | |
3p | 2P* | 1/2 | 12.0874931 |
| | 3/2 | 12.0875066 |
| | | |
3s | 2S | 1/2 | 12.0874944 |
| | | |
3d | 2D | 3/2 | 12.0875065 |
| | 5/2 | 12.0875110 |
| | | |
4p | 2P* | 1/2 | 12.7485319 |
| | 3/2 | 12.7485375 |
| | | |
4s | 2S | 1/2 | 12.7485324 |
| | | |
4d | 2D | 3/2 | 12.7485375 |
| | 5/2 | 12.7485394 |
| | | |
4f | 2F* | 5/2 | 12.7485394 |
| | 7/2 | 12.7485404 |
| | | |
5p | 2P* | 1/2 | 13.0544976 |
| | 3/2 | 13.0545005 |
| | | |
5s | 2S | 1/2 | 13.0544979 |
| | | |
5d | 2D | 3/2 | 13.0545005 |
| | 5/2 | 13.0545015 |
| | | |
5f | 2F* | 5/2 | 13.0545015 |
| | 7/2 | 13.0545019 |
| | | |
5g | 2G | 7/2 | 13.0545019 |
| | 9/2 | 13.0545022 |
| | | |
6p | 2P* | 1/2 | 13.2207009 |
| | 3/2 | 13.2207025 |
| | | |
6s | 2S | 1/2 | 13.2207010 |
| | | |
6d | 2D | 3/2 | 13.2207025 |
| | 5/2 | 13.2207031 |
| | | |
6f | 2F* | 5/2 | 13.2207031 |
| | 7/2 | 13.2207034 |
| | | |
6g | 2G | 7/2 | 13.2207034 |
| | 9/2 | 13.2207035 |
| | | |
6h | 2H* | 9/2 | 13.2207035 |
| | 11/2 | 13.2207037 |
| | | |
7p | 2P* | 1/2 | 13.3209160 |
| | 3/2 | 13.3209171 |
| | | |
7s | 2S | 1/2 | 13.3209161 |
| | | |
7d | 2D | 3/2 | 13.3209171 |
| | 5/2 | 13.3209174 |
| | | |
7f | 2F* | 5/2 | 13.3209174 |
| | 7/2 | 13.3209176 |
| | | |
7g | 2G | 7/2 | 13.3209176 |
| | 9/2 | 13.3209177 |
| | | |
7h | 2H* | 9/2 | 13.3209177 |
| | 11/2 | 13.3209178 |
| | | |
7i | 2I | 11/2 | 13.3209178 |
| | 13/2 | 13.3209179 |
| | | |
8p | 2P* | 1/2 | 13.3859595 |
| | 3/2 | 13.3859602 |
| | | |
8s | 2S | 1/2 | 13.3859595 |
| | | |
8d | 2D | 3/2 | 13.3859602 |
| | 5/2 | 13.3859604 |
| | | |
8f | 2F* | 5/2 | 13.3859604 |
| | 7/2 | 13.3859605 |
| | | |
8g | 2G | 7/2 | 13.3859605 |
| | 9/2 | 13.3859606 |
| | | |
8h | 2H* | 9/2 | 13.3859606 |
| | 11/2 | 13.3859606 |
| | | |
8i | 2I | 11/2 | 13.3859606 |
| | 13/2 | 13.3859607 |
| | | |
8k | 2K* | 13/2 | 13.3859607 |
| | 15/2 | 13.3859607 |
| | | |
9p | 2P* | 1/2 | 13.4305530 |
| | 3/2 | 13.4305535 |
| | | |
9s | 2S | 1/2 | 13.4305530 |
| | | |
9d | 2D | 3/2 | 13.4305535 |
| | 5/2 | 13.4305537 |
| | | |
9f | 2F* | 5/2 | 13.4305537 |
| | 7/2 | 13.4305537 |
| | | |
9g | 2G | 7/2 | 13.4305537 |
| | 9/2 | 13.4305538 |
| | | |
9h | 2H* | 9/2 | 13.4305538 |
| | 11/2 | 13.4305538 |
| | | |
9i | 2I | 11/2 | 13.4305538 |
| | 13/2 | 13.4305538 |
| | | |
9k | 2K* | 13/2 | 13.4305538 |
| | 15/2 | 13.4305539 |
| | | |
9l | 2L | 15/2 | 13.4305539 |
| | 17/2 | 13.4305539 |
| | | |
10p | 2P* | 1/2 | 13.4624504 |
| | 3/2 | 13.4624508 |
| | | |
10s | 2S | 1/2 | 13.4624505 |
| | | |
10d | 2D | 3/2 | 13.4624508 |
| | 5/2 | 13.4624509 |
| | | |
10f | 2F* | 5/2 | 13.4624509 |
| | 7/2 | 13.4624510 |
| | | |
10g | 2G | 7/2 | 13.4624510 |
| | 9/2 | 13.4624510 |
| | | |
10h | 2H* | 9/2 | 13.4624510 |
| | 11/2 | 13.4624511 |
| | | |
10i | 2I | 11/2 | 13.4624511 |
| | 13/2 | 13.4624511 |
| | | |
10k | 2K* | 13/2 | 13.4624511 |
| | 15/2 | 13.4624511 |
| | | |
10l | 2L | 15/2 | 13.4624511 |
| | 17/2 | 13.4624511 |
| | | |
10m | 2M* | 17/2 | 13.4624511 |
| | 19/2 | 13.4624511 |
| | | |
11p | 2P* | 1/2 | 13.4860509 |
| | 3/2 | 13.4860512 |
| | | |
11s | 2S | 1/2 | 13.4860510 |
| | | |
11d | 2D | 3/2 | 13.4860512 |
| | 5/2 | 13.4860513 |
| | | |
11f | 2F* | 5/2 | 13.4860513 |
| | 7/2 | 13.4860513 |
| | | |
11g | 2G | 7/2 | 13.4860513 |
| | 9/2 | 13.4860514 |
| | | |
11h | 2H* | 9/2 | 13.4860514 |
| | 11/2 | 13.4860514 |
| | | |
11i | 2I | 11/2 | 13.4860514 |
| | 13/2 | 13.4860514 |
| | | |
11k | 2K* | 13/2 | 13.4860514 |
| | 15/2 | 13.4860514 |
| | | |
11l | 2L | 15/2 | 13.4860514 |
| | 17/2 | 13.4860514 |
| | | |
11m | 2M* | 17/2 | 13.4860514 |
| | 19/2 | 13.4860514 |
| | | |
11n | 2N | 19/2 | 13.4860514 |
| | 21/2 | 13.4860514 |
| | | |
12p | 2P* | 1/2 | 13.5040010 |
| | 3/2 | 13.5040014 |
| | | |
12s | 2S | 1/2 | 13.5040011 |
| | | |
12o | 2O* | 21/2 | 13.5040014 |
| | 23/2 | 13.5040014 |
| | | |
13p | 2P* | 1/2 | 13.5179704 |
| | 3/2 | 13.5179707 |
| | | |
13s | 2S | 1/2 | 13.5179705 |
| | | |
13q | 2Q | 23/2 | 13.5179707 |
| | 25/2 | 13.5179707 |
| | | |
14p | 2P* | 1/2 | 13.5290547 |
| | 3/2 | 13.5290550 |
| | | |
14s | 2S | 1/2 | 13.5290547 |
| | | |
14r | 2R* | 25/2 | 13.5290550 |
| | 27/2 | 13.5290550 |
| | | |
15p | 2P* | 1/2 | 13.5379970 |
| | 3/2 | 13.5379972 |
| | | |
15s | 2S | 1/2 | 13.5379970 |
| | | |
15t | 2T | 29/2 | 13.5379972 |
| | 27/2 | 13.5379972 |
| | | |
16p | 2P* | 1/2 | 13.5453155 |
| | 3/2 | 13.5453157 |
| | | |
16s | 2S | 1/2 | 13.5453155 |
| | | |
16u | 2U* | 31/2 | 13.5453157 |
| | 29/2 | 13.5453157 |
| | | |
17p | 2P* | 1/2 | 13.5513809 |
| | 3/2 | 13.5513811 |
| | | |
17s | 2S | 1/2 | 13.5513810 |
| | | |
17v | 2V | 31/2 | 13.5513811 |
| | 33/2 | 13.5513811 |
| | | |
18p | 2P* | 1/2 | 13.5564638 |
| | 3/2 | 13.5564640 |
| | | |
18s | 2S | 1/2 | 13.5564639 |
| | | |
18w | 2W* | 35/2 | 13.5564640 |
| | 33/2 | 13.5564640 |
| | | |
19s | 2S | 1/2 | 13.5607655 |
| | | |
19p | 2P* | 1/2 | 13.5607655 |
| | 3/2 | 13.5607655 |
| | | |
19x | 2X | 37/2 | 13.5607656 |
| | 35/2 | 13.5607656 |
| | | |
20s | 2S | 1/2 | 13.5644382 |
| | | |
20p | 2P* | 1/2 | 13.5644382 |
| | 3/2 | 13.5644383 |
| | | |
20y | 2Y* | 37/2 | 13.5644383 |
| | 39/2 | 13.5644383 |
----------------|--------|------|----------------------|---------------------------
| | | |
| Limit | --- | 13.5984340 |
-----------------------------------------------------------------------------------
Christian Bar found a nice way of calculating the eigenvalues of the spinor Dirac operators on space forms of positive curvature. The eigenvalues on n-sphere of radius 1 are simply +/- (n/2 + k), k>0. This implies that the eigenvalues of the Laplace-Beltrami operator are the squares of these: (n/2 + k)^2, with appropriate multiplicities.
For a scaled sphere (one of constant radius R ~ 4 x 10^6 as in the S4 model of the universe), the energy spectrum is simply this spectrum scaled by 1/R^2. Now the hydrogen energy spectrum, by S4 theory should be the pure tones of the hydrogen atom, a smooth function on S^4(R), and thus a multiple of the pure tones of the Universe. (This can be made mathematically more precise).
From the data, if we only look at reciprocal differences of the energy levels for different orbitals, we obtain the following quadratic graph, and question of whether S4 predictions for levels or a quantum mechanical model prediction is a better explanation of the model becomes simply an issue of checking whether a quadratic fit
b +a ( 2 + k )^2
fits as well as
b + a ( 3/2 + k )^2.
Using the standard least square fitting technique on R, there is no difference in the two fits by residual sum of squares. The results are:
For S4 model,
b = -0.0025
a = 6.975 e -8
RSS = 18674
For the quantum mechanical model,
b = -0.002497
a = 6.983 e -8
RSS = 18674
Since the S4 physics model is far simpler to interpret as classical mechanics, I propose that in fact it is a better explanation for the energy spectrum of hydrogen.
For a scaled sphere (one of constant radius R ~ 4 x 10^6 as in the S4 model of the universe), the energy spectrum is simply this spectrum scaled by 1/R^2. Now the hydrogen energy spectrum, by S4 theory should be the pure tones of the hydrogen atom, a smooth function on S^4(R), and thus a multiple of the pure tones of the Universe. (This can be made mathematically more precise).
From the data, if we only look at reciprocal differences of the energy levels for different orbitals, we obtain the following quadratic graph, and question of whether S4 predictions for levels or a quantum mechanical model prediction is a better explanation of the model becomes simply an issue of checking whether a quadratic fit
b +a ( 2 + k )^2
fits as well as
b + a ( 3/2 + k )^2.
Using the standard least square fitting technique on R, there is no difference in the two fits by residual sum of squares. The results are:
For S4 model,
b = -0.0025
a = 6.975 e -8
RSS = 18674
For the quantum mechanical model,
b = -0.002497
a = 6.983 e -8
RSS = 18674
Since the S4 physics model is far simpler to interpret as classical mechanics, I propose that in fact it is a better explanation for the energy spectrum of hydrogen.
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