How Big Is Mama?

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.