Why is Proton mass = 1836 x Electron Mass?

Modern physicists are seeking to understand the deep secrets of the Universe as the totality of everything that exists, by smashing protons at very high velocity into each other at the Large Hadron Collider LHC in Geneva, as if the secrets hide inside protons. 

To assist understanding modern physicist have developed the Standard Model of fundamental particles including the proton and electron as very long-lived particles complemented by a  complete zoo of very short-lived particles supposedly identified by LHC proton collision by mass and charge. 

Does it seem reasonable that the secrets of a very very complex Universe are to be find inside a proton? Is it reasonable that the secrets of Finnegan's Wake can be understood by ripping apart a printout into dots forming the letters and then inspect one of the dots all alike by a microscope?  

Maybe not, but that seems the be strategy. One of the secrets to discover at LHC is why according to measurement the mass of a proton is 1836.152673426(32) times that of an electron?

The idea is that the ratio proton-electron mass of about 1836 must come out from some deeper structure, which maybe can be understood by smashing protons. The Creator must have had a good reason to choose this specific number. What was the reason? 

A natural question is if the Universe would look different with another number than 1836? Now, mass reacts to gravitational force, but the gravitational force is very weak on atomic scales and so mass can be set to zero on atomic scale. Atoms would look the same with a different number than 1836, say any number between 1000-2000 or 100 to 4000. 

But then you say: The electronic kinetic energy in Schrödinger's equation for an H atom takes the form 

• $\frac{\bar h^2}{2m_e}\int\vert\nabla\psi\vert^2dx$

where $\bar h$ is Planck's constant, $m_e$ is the electron mass and $\psi$ is the electron wave function (after normalising $\frac{e^2}{4\pi\epsilon_0}=1$). We see here the presence of $m_e$ and so the physics of an H atom seems to depend on the mass the electron!

But this cannot be true since gravitation is very very weak on atomic scales. In fact, the appearance of $m_e$ in the kinetic energy comes from the ad hoc postulation of the Schrödinger equation with electronic kinetic energy viewed to be analog of the kinetic energy of a macroscopic particle:

• $\frac{p^2}{2m}$   

where $p$ is momentum and $m$ mass with the connection 

• $p=i\nabla\psi$

which however lacks physics as a purely formalistic connection. This is unfortunate and has brought much confusion.

What counts on atomic scales is spatial size or compressibility, but not mass since gravitation is so weak. The size of the proton shows to be about $10^{-5}$ of the size of the electron in an H-atom. More precisely, the factor 

• $\frac{\bar h^2}{2m_e}$   

in Schrödinger's equation describes the size of the electron (as the size of the H-atom), which comes out from a balance between attraction from proton depending on distance and compression of the electron also depending on distance. The ground state of an H atom is formed by a space-filling electron cloud finding minimal energy by seeking to be close to the proton while paying a compression cost. 

But the Schrödinger equation does not carry information about the mass of proton and electron. 

So it seems that the secret of 1836 cannot be uncovered on atomic scales and that 1836 is just any number which is large enough to make a distinction between proton and electron. 

With kinetic energy scaling with $\frac{1}{2m}$ we are led to large $m$ connecting to small size, that is mass density scaling with $m^4$. 

What counts is the compressibility of proton vs electron. This connects to the discussion in previous post of a neutron as an "inverted H-atom" where proton and electron change roles. More to come on this aspect. 

In classical particle physics, particles are point-like without size, which is not physics and so physics was searched in the form a tiny vibration strings of size $10^{-35}$ m or Planck scale, however in 11 spatial dimension and so again physics was missing. In RealQM electrons appear as non-overlapping charge densities without self interaction in 3 space dimensions with real physical meaning.