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.