Not All Integer But Half Has A Place

Continuing from the previous post "Radius Of An Electron", the electron energy \(\psi\) takes discrete up values around the electron.  An underlying assumption was that \(n=1,2,3,4..\), that \(n\) takes up increasing consecutive values with distance \(x\) from the electron's center.

This may not be true.   The table of data from the previous post shows that if \(n=2\) for all values of \(\lambda\)

n	\(\lambda\) (nm)	\(\cfrac { n\lambda  }{ 2\pi  }\)	\(n^{ 2 }ln(\cfrac { 2\pi  }{ n\lambda \times10^{-9} } )\)	\(\cfrac { \lambda  }{ 2\pi  }\) (nm)
2	121.567	38.697	68.270	19.349
2	102.573	32.651	68.950	16.325449626
2	97.254	30.958	69.163	15.478927264
2	94.974	30.232	69.257	15.1160592074
2	93.780	29.852	69.308	14.9260066847
2	93.075	29.628	69.338	14.8137991405
2	92.623	29.484	69.358	14.7418271526
2	92.315	29.386	69.371	14.6928378163
2	92.097	29.316	69.380	14.6581250995
2	91.935	29.265	69.387	14.6323571542
2	91.935	29.265	69.387	14.6323253223
2	91.813	29.226	69.393	14.6128441827

\(n^{ 2 }ln(\cfrac { 2\pi }{ n\lambda \times10^{-9} } )\) varies only in the second last decimal place. As shown in the last column, \(\cfrac { \lambda }{ 2\pi }\) is almost a constant. This suggests that we have a fixed perimeter of \(2\pi x\) onto which different \(\lambda\)s are packed.

 As such,

 \(2\pi x=n\lambda=n\cfrac{c}{f}\)

 \(f=n\cfrac{c}{2\pi x}\) when \(x=a_\psi\)

 \(f=n\cfrac{c}{2\pi a_\psi}\)

Obviously \(f=0\) and \(n=0\) is a valid data point,

n	\(f\)
0	0
1	2466067.5
2	2922728.6
2	3082568.8
2	3156567.3
2	3196759.8
2	3220973.8
2	3236699.1
2	3247491.0
2	3260914.0
2	3260921.1
2	3265268.4

A plot with the regression line is shown below.

The line above is defined by two points and one theoretical point.  Given the gradient of the line,

\(\cfrac{c}{2\pi a_\psi}=1438393.5\)

\(a_\psi=\cfrac{299792458}{2\pi \times1438393.5}=33.2\) nm

From which we have no choice but to admit the existence of a half wave, where \(n=\cfrac{1}{2}\)

 \(x=\cfrac{1}{2}\cfrac{\lambda}{2\pi }\)

And that the lowest frequency point is due to this standing wave half a wave length, and the next series of points are due to the standing wave of one wave length.  From this,

\(a_{\psi\,\frac{1}{2}}=\cfrac{299792458}{2\pi \times2876787.0}=16.6\) nm

With this as a guide we can repeated the analysis in the post "Radius Of An Electron" for better results.