\(\Delta\,p=\cfrac{2}{c}\int ^{ x_{z} }_{ 0 }{ \psi } d\, x\)
Is \(a_\psi=x_z\)? What does it mean when,
\(2πx=i.nλ\)
as \(\psi\) oscillates at \(f=\cfrac{c}{\lambda}\),
and the variation of \(\psi\) around a particle given by,
\(\psi=-i{ 2{ mc^{ 2 } } }\,ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }(x-x_z)))+c\)
\(c=i{ 2{ mc^{ 2 } } }\,ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }x_z))\)
both of which is from the same wave equation,
\(\cfrac{\partial^2\psi}{\partial\,t^2_c}=ic\cfrac{\partial^2\psi}{\partial\,x\partial\,t_c}\)
where one space dimension has been replaced with a time dimension.
\(a_\psi\ne x_z\). Instead, in a 2D view, \(\psi\) shape like a torus of radius \(a_\psi\) and has extend from \(a_\psi-x_z\) to \(a_\psi+x_z\), the maximum energy density occurs on the perimeter of the circle with radius \(a_\psi\).
In which case we have a bound on \(x_z\),
\(x_z\le a_\psi\)
And,
\(\Delta\,p\le\cfrac{2}{c}\int ^{ a_{\psi} }_{ 0 }{ \psi } d\, x\) --- (*)
The other restriction on \(x_z\) is,
\(m_{ \rho \, p }c^{ 2 }=m_{ \rho }c^{ 2 }-\int _{ 0 }^{ 2x_{ z} }{ \psi } dx=m_{ \rho }c^{ 2 }-2\int _{ 0 }^{ x_{ z} }{ \psi } dx\ge0\)
So,
\(\Delta\,p\le m_{ \rho }c\)
as we set \(x_z=a_{\psi}\) and substitute into (*). \(m_{ \rho }\) is the mass density of the particle fully expressed as mass.
