\(F=-i\sqrt { 2{ mc^{ 2 } } }\,G.tanh\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_z) \right)+C\)
The associated \(\psi\) after integration,
\(\psi= { -i\, 2{ mc^{ 2 } } }ln(cosh(\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_{ z }) \right) ) +Cx+A \)
A sample plot of \(\psi\) with varies values of \(C\) is,
From the diagram, for \(\psi\) to be bounded such that its integral over all positive value of \(\psi\) is finite,
\(0\le C\lt1\)
We see that when \(\psi(x=x_a)=0\), \(x_a\ne 2x_z\) except when \(C=0\). In fact, \(x_a\gt 2x_z\). The far zero of \(\psi\) when \(C\gt0\) is further than when \(C=0\).
For all valid values of \(C\) where \(\psi\) intersects the \(x\) axis again, we have Coulombs' Inverse Square Law.
\(F\propto\cfrac{1}{x^2}\)
If this is the case, we do not have to consider \(\psi\lt0\) at all. \(F\) is due to the interaction of \(\psi\)s, in both derivations of \(F\).
\(|F|=\psi\) and
\(F=\cfrac{1}{4\pi x^2}\int_0^x{F_{\rho}}dx\)
The Newtonian force \(F\), from the post "From The Very Big To The Very Small",
\(F=\cfrac { -i\, 2{ mc^{ 2 } } }{ 4\pi x^{ 2 } } \left[ ln(cosh(\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_{ z }) \right) )+Cx \right] _{ 0 }^{ x_{ a } }\)
\(F=\cfrac { -i\, 2{ mc^{ 2 } } }{ 4\pi x^{ 2 } }\left[ ln(cosh(\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x_a-x_{ z }) \right) )\\-ln(cosh(\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x_{ z }) \right) )+Cx_a\right]\)
If we define,
\(N(x_a)=ln(cosh(\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x_a-x_{ z }) \right)-ln(cosh(\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x_{ z }) \right) ) )\)
And we have,
\(F=\cfrac { -i\, 2{ mc^{ 2 } } }{ 4\pi x^{ 2 } }\left[ N(x_a)+Cx_a\right]\)
the expression for \(F\) splits into two fields. One field due the curve part of \(\psi\), \(N(x_a)\) and another due to the linear part of \(\psi\), \(Cx_a\). A near and a far field, each with its own aggregated constants. Which is which? Others might define far field as \(x\gt x_a\) and near field as \(x\lt x_a\) using the boundary where \(\psi(x_a)=0\) as divide. In which case, \(N(x_a)\) is then a new near field, a nearer field.
Have a nice day.
Have a nice day.
