Hydrogen as Two-Density Schrödinger Equation

This is a continuation of the previous post with a Hydrogen Atom modeled according to Real Quantum Mechanics RealQM  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 non-overlapping supports filling space meeting at a common boundary $\Gamma$ with some boundary conditions to be specified, starting for simplicity with zero charge density for both proton and electron. 

We start with the ground state with the proton occupying a fixed sphere of diameter $d$ with the electron filling the exterior volume. We characterise the ground state 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  

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

is the sum of proton and electron compression energies (also referred to as "kinetic energies") under the normalisation 

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

Here the coefficient $d$ sets the size of the proton compared to the electron of unit size and we assume that $d$ is small. 

The proton charge density is given by a spherical harmonic as a "blob" of diameter $d$ centered at $x=0$, while for small $d$ the electron charge distribution is close to the standard Hydrogen ground state with $\psi (x)\sim\exp(-\vert x\vert )$. The total energy comes out as the standard electron compression/potential energy $-\frac{1}{2}$ plus proton compression energy as a constant independent of $d$. Letting $d$ tend to zero and neglecting the proton energy makes the proton into a point source as in the Born-Oppenheimer approximation in terms of only electron density, compare PS below.  

Eigenstates of higher energies emerge as stationary points of $E(\phi ,\psi )$ in a variational setting. 

We see that for small $d$ the two-density model thus reduces to the standard electronic wave function with a constant shift of total energy, which does not affect the spectrum since it corresponds to energy differences. Thus the two-density model for Hydrogen may be seen as a trivial extension of the electronic one-density model, but allowing $\Gamma$ to be a free boundary included in the variational setting may open new views on the interaction between proton and electron. One can then ask if the presence of the electron around the proton affects the proton density, as well as ponder different boundary conditions. 

We note that in RealQM the combined density comes out as the sum of proton and electron densities in 3-dimension physical space, while in standard QM it is the product in 6-dimensional configuration space, which is not physical. 

Altogether, we find that RealQM naturally can be extended beyond electronic interaction. One can then address the question why the proton appears to be so much smaller than the electron in e.g. the Hydrogen atom. It reflects that the proton has a much smaller "resistance to compression" than the electron, which can be accepted as a physical fact asking for deeper analysis.   

Note that it is more natural to connect the compression/kinetic energies to spatial size rather than mass, since the quantum mechanical model concerns electromagnetic interaction without effects of inertia/gravitation. More precisely, the coefficient $\frac{d^2}{2}$ in RealQM corresponds to $\frac{1}{2m}$ in stdQM with $m$ the proton inertial/gravitational mass, which is strange since standard QM primarily concerns electromagnetics. Only in molecular modeling using the Born-Oppenheimer approximation with kernel dynamics treated by classical mechanics, does kernel masses enter. In any case, $d$ appears to scale with $\sqrt{m}$ which with the table value $m=1836$ gives $d\approx 0.02$ to be compared with 1 as atom size. 

Returning to the idea of a neutron as an "inverted Hydrogen atom" with the electron at the center surrounded by a proton of size $d$ will give a large increase of electron compression energy which can be released when the neutron decays observed to be around 1 MeV, which suggests an electron size of $10^{-3}$ which may again suggest a proton size $d\approx 0.02$  

Note that we here speak about "electromagnetic" size, which may be different from a smaller inertial/gravitational size as measured in collision experiments. 

PS1 The article On the hydrogen atom beyond the Born–Oppenheimer approximation considers a two-density model in the spirit of stdQM with a combined wave function as a product of proton and electron densities. Model computations suggest that in RealQM one can assume both proton and electron densities to vanish on the common boundary. 

PS2 The two-density model in the above form contains one parameter $d$ which connects proton mass to electron charge/mass with a direct coupling to the non-dimensional fine structure constant $\alpha\approx\frac{1}{137}$ as expressed here.