Neutron as Inverted Hydrogen Atom?

Is this a proton charge density surrounded by an electron charge density. Or is it the other way around? 

The Hydrogen atom consisting of a positively charged proton and a negatively charged electron can in Real Quantum Mechanics RealQM  be mathematically modeled in terms of two spatial charge densities, $\phi (x)$ for the proton $\psi (x)$ for the electron as functions of a Euclidean space coordinate $x$, assuming $\phi$ and $\psi$ have disjoint supports (filling space) meeting at a boundary $\Gamma$ signifying that the proton and the electron do not overlap. 

The ground state of Hydrogen is then characterised as the state of minimal total energy 

• $E(\phi ,\psi ) = PE(\phi ,\psi ) + KE(\phi ,\psi)$

where

• $PE(\phi ,\psi ) = -\int\frac{\phi^2(x)\psi^2(y)}{\vert x-y\vert}dxdy$

 is mutual potential energy, and  

• $KE(\phi ,\psi )=\int\frac{1}{2m}\vert\nabla\phi (x)\vert^2dx+\int\frac{1}{2}\vert\nabla\psi (x)\vert^2dx$

 is the sum of proton and electron compression energies under the normalisation 

• $\int \phi^2(x)dx =1$ and $\int \psi^2(x)dx =1$.

Here $m\approx 1836$ is the ratio of proton to electron mass. Eigenstates of higher energies emerge as stationary points of $E(\phi ,\psi )$. Further, $\Gamma$ is a free boundary included in the minimisation with specific boundary conditions to be decided. 

A proton-electron configuration which agrees with observations is given by a proton charge density of small radius centered at $x=0$ surrounded by an electron charge density of large radius. In the limit with the proton modeled as a constant charge distribution of vanishing radius, this gives the standard Schrödinger equation for the Hydrogen atom with Hamiltonian

• $H = -\frac{1}{2}\Delta -\frac{1}{\vert x\vert}$

in terms of the electron charge distribution $\psi (x)$ alone, with $\psi (x)\sim \exp(-\vert x\vert)$ as ground state.  

Now, a neutron is viewed to also consist of a proton and an electron, and so it is natural to ask if the above model can also describe a neutron? That would correspond to a switch of roles with now the electron at the center surrounded by a proton charge density. The compression energy would now be that of the proton resulting in a change of scale with the neutron radius about $\frac{1}{1836}$ of that of a Hydrogen atom.  

In RealQM the size of an electron, in an atom with electrons organised into shells, increases with distance to the kernel, and so electron size is variable. We may expect the same property of a proton with thus increasing size if harbouring an electron inside in the formation of a neutron. The size of a free proton  is estimated to about $10^{-15}$ meter. We compare with a Hydrogen atom of size $5\times 10^{-11}$  which with the above 1836 scaling, gives a proton size of about $10\times 10^{-15}$ when surrounding an electron in a neutron, about 10 times as big as when free.

These are speculations suggested by RealQM as a classical continuum model in terms of non-overlapping charge densities. RealQM can be seen as a form density functional theory which is different from that pioneered by Walter Kohn and Pierre Hohenberg (Nobel Prize in Physics 1998) formed by averaging in a standard multi-dimensional Schrödinger equation. 

Recall that a free neutron is unstable and decays with mean lifetime of 14 minutes into a proton, an electron and an antineutrino (but not a Hydrogen atom), while neutrons are formed in the fusion process of Hydrogen into Helium in a star like the Sun.