however, draws a sense of déjà vu...
The concepts leading to the expression \(\cfrac { \sqrt [ 3 ]{ n_{ 2 } } -\sqrt [ 3 ]{ n_{ 1 } } }{ \sqrt [ 3 ]{ n_{ 1 }n_{ 2 } } }\) reverses the energy sign of conventional electron energy level transitions; \(E_{1\rightarrow 2}\) is negative and the particle loses energy. \(E_{1\rightarrow 2}\) is emitted
This would make the plot for \(n_1=1\) in the above graph, an absorption line.
For all values of \(n_2\), only when \(n_1=1\) is an absorption line. The plots in black are emission lines against which we see the absorption line without direct illuminations.
Values of the plots in black below \(y=0\) is ignored because \(n_2\ge n_1\).
Rydberg constant is not murdered, instead a new process is given birth.
\(\cfrac { \sqrt [ 3 ]{ n_{ 2 } } -\sqrt [ 3 ]{ n_{ 1 } } }{ \sqrt [ 3 ]{ n_{ 1 }n_{ 2 } } }\) occurs at the same time as \(\cfrac { n_{ 2 }^{ 2 }-n_{ 1 }^{ 2 } }{ (n_{ 1 }n_{ 2 })^{ 2 } }\) due to quantized energy levels in Bohr model theory. The two process has reverse energy signs and hence married with a negative sign. According to Bohr model \(E_{1\rightarrow 2}\) is positive and the particle gains energy and transits to a higher energy level.
We have instead,
\(\Delta E_{2\rightarrow 1}=h.f_{\psi\,c}\left(\cfrac{\sqrt[3]{n_2}-\sqrt[3]{n_1}}{\sqrt[3]{n_1n_2}}-\cfrac { n_{ o2 }^{ 2 }-n_{ o1 }^{ 2 } }{ (n_{o1 }n_{ o2 })^{ 2 } }\right)\)
where a change \(n_1\rightarrow n_2\) is accompanied by a change in orbital energy level \(n_{o1}\rightarrow n_{o2}\).
Only now are the particles in orbits.
Note: \(E_2=E_1+\Delta E_{1\rightarrow 2}\)
