More ScFi, Resonator And Wrap Speed

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?