\(2\pi a_{ \psi }=n\lambda\)
\(h_n=2\pi a_{ \psi }mc=mc.n\lambda\)
with \(n=1\).
\(2\pi a_{ \psi }=\lambda_1\)
\(h_1=2\pi a_{ \psi }mc=mc.\lambda_{1}\)
At \(_na_\psi\),
\(h_n=2\pi a_{\psi\,n}.mc=mc.\lambda_n\)
When a particle is excited we have,
\(h_{f}\rightarrow h_{i}\)
and
\(\lambda_{f}\rightarrow\lambda_{i}\)
And when the particle returns from an excited state, \(\lambda_{i}\) is applied to \(h_{f}\) and we formulate,
\(E_{i\rightarrow f}=h_{f}f_{i}=2\pi a_{ \psi\,_f }.mc.f_{i}=\lambda_{f}mc.\cfrac{c}{\lambda_i}\)
which is the energy of the particle at \(h_f\).
Since,
\(2\pi a_{\psi\,n}=2\pi e^{ -\cfrac { Aa_{\psi\,n}+C }{ n^{ 2 } } }= \lambda_{n}\)
\(\cfrac{\lambda_{f}}{\lambda_{i}}= e^{ -\cfrac { Aa_{\psi\,f}+C }{ n^{ 2 }_f }+\cfrac { Aa_{\psi\,i}+C }{ n^{ 2 }_i } }\) --- (*)
We have,
\(E_{i\rightarrow f}=mc^2.e^{ -\cfrac { Aa_{\psi\,f}+C }{ n^{ 2 }_f }+\cfrac { Aa_{\psi\,i}+C }{ n^{ 2 }_i } }=\psi_f+x.F\)
The energy required at \(n_f\) is \(E_{i\rightarrow f}\), the energy in excess is \(x.F\), the work function.
\(E_{\Delta n}=E_{i\rightarrow f}-E_f=mc^2.e^{ -\cfrac { Aa_{\psi\,f}+C }{ n^{ 2 }_f }+\cfrac { Aa_{\psi\,i}+C }{ n^{ 2 }_i } }-mc^2=x.F\)
\(E_{\Delta n}=mc^2\left\{e^{ -\cfrac { Aa_{\psi\,f}+C }{ n^{ 2 }_f }+\cfrac { Aa_{\psi\,i}+C }{ n^{ 2 }_i } }-1\right\}=x.F\) --- (1)
where \(E_{\Delta n}\) is the energy gained by the particle. In this case the, the particle loses energy and \(E_{\Delta n}\) is negative.
In this derivation, the Planck constant \(h\), is formulated for each solution of \(a_\psi\). The resulting \(h_i\) is applied at the appropriate level to obtain \(\psi_i+x.F\), the work function. \(x.F\) is emitted as a photon.
Alternatively,
\(\Delta E=E_{f}-E_{i}=mc^2-mc^2=0\)
What happened?
When we apply \(h_f\) to \(\lambda_i\), we are suggesting that a wave at \(\lambda_i\) is caught at energy level \(h_f\). \(\lambda_i\) adjusts itself to \(\lambda_f\) and there are energy changes in the process. When \(\lambda_i\) is due to a larger \(a_{\psi\,n}\), its amplitude is also smaller (cf. post "H Bar And No Bar"; this amplitude need to be quantified). When \(\lambda_i\) constrict to \(\lambda_f\) at energy state \(n=f\) it does so at smaller amplitude, it loses energy.
This lost in energy appears as the work function \(x.F\) and is emitted as a photon. The material cools. The wave will regain its amplitude as heat is applied. In the diagram above we considered only the fundamental frequency, when \(2\pi a_{\psi\,n}=\lambda_n\).
In the post "Particle Spectrum", we took reference at \(n=1\), where \(h_o=h_1\). The expressions for \(E_{\Delta n}\) and \(\lambda_p\) are valid when the final energy state is \(n=1\).
Intuitively, \(\psi\) has a longer path length at \(n_i\), the difference in energy \(E_{\Delta h}\) due to a change in \(h_n\) as the wave move from \(n_i\) to \(n_f\) is,
\(E_{\Delta h}=h_{f}f_i-h_{i}f_i=\cfrac{c}{\lambda_i}(h_{f}-h_{i})=\cfrac{c}{2\pi a_{\psi\,i}}(h_{f}-h_{i})\)
\(E_{\Delta h}=\cfrac{1}{a_{\psi\,i}}mc^2(a_{\psi\,f}-a_{\psi\,i})\)
\(E_{\Delta h}=mc^2(\cfrac{a_{\psi\,f}}{a_{\psi\,i}}-1)\)
\(E_{\Delta h}=mc^2\left(e^{ -\cfrac { Aa_{\psi\,f}+C }{ n^{ 2 }_f }+\cfrac { Aa_{\psi\,i}+C }{ n^{ 2 }_i } }-1 \right)\)
Which is the same expression as (1). Both derivations are equivalent.
\(\psi\) of a particle is excited to \(n_i\) energy state, \(a_\psi=a_{\psi\,i}\). On its return to a lower \(n_f\) energy state, \(a_\psi=a_{\psi\,f}\) where \(a_{\psi\,f}\lt a_{\psi\,i}\), it loses energy in the form of a photon.
This is consistent with the observation of spectra lines, but this particle is not orbiting around any nucleus.
To be excited is then \(a_{\psi\,i}\rightarrow a_{\psi\,f}\) where \(a_{\psi\,f}\gt a_{\psi\,i}\); to have a higher \(a_\psi\).
Note: \(E_{\Delta h}\) and \(E_{\Delta n}\) are not the difference in energy as the particle move from \(n_i\) to \(n_f\) energy state. But are energy changes as a result of \(h_n\) changes between energy states.
In all cases, for,
\(n_i\gt n_f\), \(f_i\gt f_f\)
which is consistent with the post "Discreetly, Discrete λ And Discrete Frequency, f", unless the frequency profile is folded upwards.
