From the previous post "Photon Momentum", for a mass-less photon,
\(m_{ \rho \, p }c^{ 2 }=m_{ \rho }c^{ 2 }-2\int _{ 0 }^{ x_{ z} }{ \psi } dx=0\)
\(m_{ \rho }c^{ 2 }=2\int _{ 0 }^{ x_{ z} }{ \psi } dx\)
Since \(x_z\le a_\psi\)
\(m_{ \rho }\le \cfrac{2}{c^{ 2 }}\int _{ 0 }^{a_\psi }{ \psi } dx\)
we have an expression for the maximum mass density of a particle, \(m_{\rho\,max}\). We cannot simply substitute the values for \(a_\psi\) discovered previously in the post "Sizing Them Up". However, if the particle with \(a_\psi=15.48\) nm is a photon, then
\(x_z\le 15.48\)
Thus,
\(0\lt m_{ \rho }\le \cfrac{2}{c^{ 2 }}\int _{ 0 }^{15.48}{ \psi } dx\);
a size range restriction upon the mass density of the mother of all particles.
