Consider the expression for \(E_{ \Delta h }\),
\(E_{ \Delta h }=\int { F_{ \Delta hi } } da_{ \psi \, i }=mc^{ 2 }\cfrac { 1 }{ a_{ \psi \, i } } (a_{ \psi \, f }-a_{ \psi \, i })\)
Differentiating with respect to \(a_{ \psi \, i }\),
\( \cfrac { \partial \, E_{ \Delta h } }{ \partial \, a_{ \psi \, i } } =F_{ \Delta hi }=-mc^{ 2 }\cfrac { a_{ \psi \, f } }{ a^{ 2 }_{ \psi \, i } } \)
And consider,
\( E_{ \Delta h }=\int { F_{ \Delta hf } } da_{ \psi f }=mc^{ 2 }\cfrac { 1 }{ a_{ \psi \, i } } (a_{ \psi \, f }-a_{ \psi \, i })\)
Differentiating with respect to \(a_{ \psi \, f }\),
\( \cfrac { \partial \, E_{ \Delta h } }{ \partial \, a_{ \psi \, f } } =F_{ \Delta hf }=mc^{ 2 }\cfrac { 1 }{ a_{ \psi \, i } } \)
So, when both \(a_{ \psi \, i } \) and \(a_{ \psi \, f } \) are changing,
\( \cfrac { \partial \, E_{ \Delta h } }{ \partial \, a_{ \psi } } =\cfrac { \partial \, E_{ \Delta h } }{ \partial \, a_{ \psi \, f } } +\cfrac { \partial \, E_{ \Delta h } }{ \partial \, a_{ \psi \, i } } =F_{ \Delta h\, T }=mc^{ 2 }\cfrac { 1 }{ a_{ \psi \, i } } \left( 1-\cfrac { a_{ \psi \, f } }{ a_{ \psi \, i } } \right) \)
And we consider the second derivative,
\( \cfrac { \partial ^{ 2 }E_{ \Delta h } }{ \partial \, a^{ 2 }_{ \psi } } = \cfrac { \partial \, }{ \partial \, a_{ \psi } }\left\{\cfrac { \partial \, E_{ \Delta h } }{ \partial \, a_{ \psi \, f } } +\cfrac { \partial \, E_{ \Delta h } }{ \partial \, a_{ \psi \, i } }\right\}=F^{ ' }_{ \Delta h\,T}\)
\(F^{ ' }_{ \Delta h\,T}=-mc^{ 2 }\cfrac { 1 }{ a^{ 2 }_{ \psi \, i } } +2mc^{ 2 }\cfrac { a_{ \psi \, f } }{ a^{ 3 }_{ \psi \, i } } -mc^{ 2 }\cfrac { 1 }{ a^{ 2 }_{ \psi \, i } } \)
\(F^{ ' }_{ \Delta h\,T}=2mc^{ 2 }\cfrac { 1 }{ a^{ 2 }_{ \psi \, i } } \left( \cfrac { a_{ \psi \, f } }{ a_{ \psi \, i } } -1 \right) =-k\)
And an approximation,
\(F_{ \Delta h\, T }=F^{ ' }_{ \Delta h }.\Delta a_{\psi}\)
If such a system is prompted to oscillate,
\( \omega _{ n }=\sqrt { \cfrac { k }{ m } } =\sqrt { 2 } c\sqrt { \cfrac { a_{ \psi \, f } }{ a^{ 3 }_{ \psi \, i } } } \)
where \( \omega _{ n }\) is the natural frequency of the system. Naturally, such a system is damped as it emits photons, the loss due to damping is,
\(P_{loss}=- \cfrac { \partial \, E_{ \Delta h } }{ \partial \, t }=-m\nu\left(\cfrac{d\,a_{\psi\,i}}{d\,t}\right)^2\)
where \(\nu\) (rads-1) is the damping factor using the model \(f(x,\dot{x})=-kx-m\nu\dot{x}\) and the rate at which \(f(x,\dot{x})\) does work is,
\(P=f.\dot{x}=-kx\dot{x}-m\nu\dot{x}^2=P_{osc}+P_{loss}\)
And consider, \(a_{\psi\,i}\) making a round trip to \(a_{\psi\,f}\) and back in time \(T_d=2\pi\cfrac{1}{\omega_d}\),
\(\cfrac{d\,a_{\psi\,i}}{d\,t}=\cfrac{2(a_{\psi\,i}-a_{\psi\,f})}{T_d}=\cfrac{1}{\pi}(a_{\psi\,i}-a_{\psi\,f})\omega_d\)
where \(\omega_d\) is the damped resonance frequency. We have,
\( \cfrac { \partial \, E_{ \Delta h } }{ \partial \, t }=m\nu\left(\cfrac{d\,a_{\psi\,i}}{d\,t}\right)^2=m\nu\cfrac{1}{\pi^2}(a_{\psi\,i}-a_{\psi\,f})^2\omega^2_d\)
As,
\( \cfrac { \partial \, E_{ \Delta h } }{ \partial \, t } =\cfrac { \partial \, E_{ \Delta h } }{ \partial \, a_{ \psi \, f } }\cfrac{d\,a_{\psi\,f}}{d\,t} +\cfrac { \partial \, E_{ \Delta h } }{ \partial \, a_{ \psi \, i } }\cfrac{d\,a_{\psi\,i}}{d\,t} \)
Assuming \(\cfrac{d\,a_{\psi\,f}}{d\,t}=\cfrac{d\,a_{\psi\,i}}{d\,t}\), that the external driving force affects both equally,
\(mc^{ 2 }\cfrac { 1 }{ a_{ \psi \, i } } \left( 1-\cfrac { a_{ \psi \, f } }{ a_{ \psi \, i } } \right)=m\nu\cfrac{1}{\pi}(a_{\psi\,i}-a_{\psi\,f})\omega_d \)
Since, \(a_\psi\) at \(a_{ \psi \, f }\) loses almost all its energy, the amplitude of the wave at \(a_{ \psi \, f }\), \(A_f\) is small,
\(\nu\approx 1\) rads-1
then,
\(w_d=\pi c^2\cfrac{1}{a^2_{ \psi \, i } }\)
which is a very high frequency, (1034Hz).
At this frequency, the particle will be "resonating" with high output of photons emitted as a result of the energy state transition \(a_{ \psi \, i } \rightarrow a_{ \psi \, f }\).
If this high frequency, \(\omega_d\) can be achieved, we have a photon resonator that might drive that particular particle to wrap speed.
Note: What? Wrong to replace \(\dot{x}\) with average speed?
