\(F=\psi_{n}-\psi(x)=m\cfrac{d^2x}{dt^2}\)
when we substitute for \(\psi(x)\),
\(\psi(x)=-i{ 2{ mc^{ 2 } } }\,ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }(x-x_z)))+c\) --- (*)
from the post "Not Quite The Same Newtonian Field" dated 23 Nov 2015. Assuming that at \(x=0\), \(\psi(0)=0\),
\(c=i{ 2{ mc^{ 2 } } }\,ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }(x_z)))\)
and so,
\(\psi(x_z)=\psi_{max}=i{ 2{ mc^{ 2 } } }\,ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }(x_z)))\)
Therefore,
\(m\cfrac { d^{ 2 }x }{ dt^{ 2 } } =\psi _{ n }+i2{ mc^{ 2 } }ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_z)))-\psi_{max}\)
\(m\cfrac { d^{ 2 }x }{ dt^{ 2 } } =i2{ mc^{ 2 } }ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_z)))+\psi_{c}\)
where
\(\psi_c=\psi_n-\psi_{max}\)
In cases where,
\(\psi_d=\psi_{max}-\psi_n\)
we have a particle of opposite sign embedded in another particle.
Also,
\(\psi_n=\psi _{ photon }+\psi_{o}\)
\(\psi_n\) is the elevated energy level of the particle from its ground state \(\psi_{o}\) after receiving the photon \(\psi _{ photon }\).
In the case when,
\(\psi _{ photon }=0\), \(\psi_c=\psi_o-\psi_{max}\)
and \(\psi_c\) is a constant. Since,
\(F=m\cfrac { d^{ 2 }x }{ dt^{ 2 } } \)
\(F=i2{ mc^{ 2 } }ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_z)))+\psi_{c}\)
We now consider the effect of exerting an external force on the system, the resulting change in \(x\), the relative displacement of the embedded particle and the containing particle can be obtained from the expression,
\(\cfrac{dF}{dx}=iG \sqrt { 2{ mc^{ 2 } } }.tanh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_z))\)
which is just the force density expression we formulated in the post "Not Exponential, But Hyperbolic And Positive Gravity!" dated 22 Nov 2014. More explicitly,
\(F_{\rho}=e^{ i3\pi /4 }D\sqrt { 2{ mc^{ 2 } } } .tanh\left( \cfrac { { D } }{ \sqrt { 2{ mc^{ 2 } } } }( x-x_o).e^{ i\pi /4 } \right)\)
where we insist that,
\(G=D.e^{ i\pi /4 }\)
is real.
The term \(i\) originates from the expression for \(\psi\), which is a wave in the \(ix\) direction. From the post "Opps Lucky Me" dated 25 May 2015, the Newtonian force \(F_{\small{N}}\)
\(F_{\small{N}}=-\psi\)
and the post "From The Very Big To The Very Small" dated 16 Jul 2015.
\(F_{x}=\int_0^x{F_{\rho}}dx\)
so,
\(F_{\small{N}}=\int{F_{\rho}}dx\)
where \(x=0\) when \(F_{\small{N}}=0\). Combining all these, the Newtonian force \(F_{\small{N}}\) when applied resulting in a deformation \(\Delta x\) is given by,
\(F_{\small{N}}=\cfrac{dF}{dx}|_{x=x_o}.\Delta x=i2{ mc^{ 2 } }ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_z)))|_{x=x_o}.\Delta x\) --- (*)
where \(F_{\small{N}}\) can be the result of an applied electric field, \(E\) or magnetic field \(B\).
When \(\Delta x\) changes \(\theta\) around \(\theta=45^o\), an EMW together with an photon are emitted. The EMW can be amplified and be used as the driving force (positive feedback) and thus generates sustained oscillations.
And we have a problem with \(i\) on the RHS of the expression for \(F_{\small{N}}\), otherwise we have an indication for electrostriction/magnetostriction and piezoelectric.
The problem is \(i\) does make sense,
where the deformation, \(\Delta x\) is in the direction perpendicular to \(F_{\small{N}}\). This is the result of \(\psi\) being perpendicular to the direction of \(x\), from the expression (*). Since the total volume of the material (\(\psi\)), is approximately a constant, there are corresponding deformations in the two perpendicular directions with respect to \(\Delta x\) too.
With the issue of \(i\) aside,
we differentiate between piezoelectricity and its linear dependence on the applied force \(F_{\small{N}}\), and electrostriction and its quadratic dependence on the applied force, base on the same expression (*). Piezoelectricity requires that \(\small{\Delta x}\) changes \(\small{\theta}\) around \(\small{\theta=45^o}\).
where the deformation, \(\Delta x\) is in the direction perpendicular to \(F_{\small{N}}\). This is the result of \(\psi\) being perpendicular to the direction of \(x\), from the expression (*). Since the total volume of the material (\(\psi\)), is approximately a constant, there are corresponding deformations in the two perpendicular directions with respect to \(\Delta x\) too.
With the issue of \(i\) aside,
we differentiate between piezoelectricity and its linear dependence on the applied force \(F_{\small{N}}\), and electrostriction and its quadratic dependence on the applied force, base on the same expression (*). Piezoelectricity requires that \(\small{\Delta x}\) changes \(\small{\theta}\) around \(\small{\theta=45^o}\).
