Touch And Go

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.