From the previous post "Amplitude \(A_n\)", it is also possible that a state transition, \(n_f\rightarrow n_i\), occurs when,
\(r=a_{\psi\,i}\)
then,
\(a^2_{\psi\,i}=a^2_{\psi\,f}+A^2_f\)
\(A_f=\sqrt{a^2_{\psi\,i}-a^2_{\psi\,f}}\)
In which case we still have,
\(E_{ \Delta n }\propto A_f\), because \(\psi\) is already energy density.
\(E_{ \Delta n }=E_{ o }\sqrt { a^{ 2 }_{ \psi \, i}-a^{ 2 }_{ \psi \, f} }\)
And we define \(E_o\) as,
\(E_{ o }=-mc^{ 2 }\cfrac { 1 }{ a_{ \psi \, i } } \sqrt { \cfrac { (a_{ \psi \, i }-a_{ \psi \, f }) }{ (a_{ \psi \, i }+a_{ \psi \,f}) } }\)
In this case a state transition occurs when a lower \(\psi\) wave touches the path of an outer wave. This would occur before the perimeters of the paths are equal.
