\(2\pi x=i.n\lambda\)
where \(i\) rotates \(\lambda\) to the direction of \(x\), \(n=1,2,3,4...\), \(x\ne0\).
and \(x\) can take on only discrete values dictated by,
\(x=e^{ -\cfrac { Ax+C }{ n^{ 2 } } }\)
\(\lambda\) is also discrete, constrained by,
\(\cfrac { n\lambda }{ 2\pi } =e^{ -\cfrac { A\lambda }{ 2\pi \, n } }.e^{ -\cfrac { C }{ n^{ 2 } } }\)
\( { \lambda } =\cfrac { 2\pi }{ n } e^{ -\cfrac { A \lambda }{ 2\pi \,n } }.e^{ -\cfrac { C }{ n^{ 2 } } }\) --- (*)
A plot of y=2*Pi/n*(e^{-1*x/(2*Pi*n)})*e^{-1/n^2) and y=x is shown below,
This is not a simple plot compared to the case of discrete \(x\). The value of \(C\), on the RHS of the constraint (*) fold the set of curve backwards and the values of \(\lambda\) does not increase monotonously. In the plot above the first value of \(\lambda\) corresponding to \(n=1\) (in red), is in the middle of the spectrum slightly displaced from the value of \(\lambda\) for \(n=3\). An illustrative plots of y=2*Pi/n*(e^{-1*x/(2*Pi*n)})*e^{-C/n^2) for a higher values of \(C\) is given below,
A increasing values of \(C\) unfolds the set of curves and more values of \(\lambda\) increases with increasing \(n\) monotonously. A plot with high value of \(C\) is given below,
With a high value of \(C\), \(\lambda\) increases with increasing \(n\) monotonously, as one would expect initially.
This folding of \(\lambda\) might explain the displaced spectra lines of elements obtained experimentally, where one spectra line seems to be out of place, or the spectra lines looks like the intersection of two series of lines.
We will take a closer look at the frequency of \(\psi\), \(f\).
\(\because f=\cfrac{\lambda}{c}\)
\(f\), frequency is also discrete.
\(\cfrac { c }{ f } =\cfrac { 2\pi }{ n } e^{ -\cfrac { A }{ 2\pi\,n } \cfrac { c }{ f } }.e^{ -\cfrac { C }{ n^{ 2 } } }\)
\( f=\cfrac{nc}{2\pi}e^{ \cfrac { A }{ 2\pi\,n } \cfrac { c }{ f } }.e^{ \cfrac { C }{ n^{ 2 } } }\)
An illustrative plot of y=n/(2*Pi)*(e^{1/(2*Pi*n*x)})*e^{1/n^2), y=x is given below,
The first value of \(f\) for \(n=1\) is displaced. Two other plots of higher \(C\), (\(C=2\),y=n/(2*Pi)*(e^{1/(2*Pi*n*x)})*e^{2/n^2)) and lower \(C\), (\(C=0.1\),y=n/(2*Pi)*(e^{1/(2*Pi*n*x)})*e^{0.1/n^2)) is given below,
The single displaced value of \(f\) corresponding to \(n=1\) increases with increases \(C\). This displaced spectra line (seen as a very close pair of lines) is readily observed in many line spectrum diagram. A further plot of y=n/(2*Pi)*(e^{1/(2*Pi*n*x)})*e^{C/n^2) with \(C=10\) shows two such pairs of close \(f\) solutions.
In this case the solution for \(n=1\) is very high on the plot and cannot be shown with good resolution in the rest of the plot.
Nice, very nice indeed. But what is \(C\)?
