\(2a^2_{\psi\,i}-a^2_{\psi\,f}\lt 0\)
\(a_{\psi\,f}\gt \sqrt{2}a_{\psi\,i}\) both \(a_{\psi\,i}\) and \(a_{\psi\,f}\) being positive.
In this case, photon emission based on the equal perimeter criterion does not occur. A state transition is likely to be when \(r=a_i\).
It is tempting to set,
\(\cfrac{1}{2}mc^2=E_{\Delta n}=mc^{ 2 }(\cfrac { a_{ \psi \, f } }{ a_{ \psi \, i } } -1)\)
where \(E_{\Delta n}\) as derived in the post "Amplitude, \(A_n\)" dated 23 Dec 2014, is the energy in \(A_f\). And \(m\) is the mass density of the photon. This simply states that the photon is at light speed along \(A_f\). Then,
\(a_{ \psi \, i }=\cfrac{2}{3}a_{ \psi \, f }\)
Of course,
\(a_{ \psi \, f }\gt a_{ \psi \, i }\)
Does this suggest that unless \(a_{ \psi \, i }\lt\cfrac{2}{3}a_{ \psi \, f }\), we have no emission? That the energy states denoted by \(a_{ \psi \, i }\) and \(a_{ \psi \, f }\) must first be sufficiently far apart for photon emissions to occur with a state transition between them?
Any energy state \(a_{\psi\,f}\) above \(a_{\psi\,i}\), where
\(a_{\psi\,i}\lt a_{\psi\,f}\lt \cfrac{3}{2}a_{\psi\,i}\)
is invisible (with no photon emission) when a state transition from \(a_{\psi\,i}\) occurs. Or does it simply mean that photons can be emitted with velocity less than light speed along \(A_f\)?
What happen to \(A_f\) if not emitted?
