Wave Functions: StdQM vs RealQM

Let us compare the wave functions of StdQM and RealQM, for simplicity for $H_2$ or $He$ with two electrons, as a preparation to the questions posed in the ongoing review of first RealQM article reported in the previous post. 

The wave function $\Psi (x_1,x_2)$ of StdQM depends on two 3d spatial variables $x_1$ and $x_2$ covering all of 3d Euclidean space $\Re^3$, altogether $\Re^6$. In computation, dimensionally reduced forms are used, for example as a Slater determinant:

•  $\Psi (x_1,x_2)=\psi_1(x_1)\psi_2(x_2)-\psi_1(x_2)\psi_2(x_1)$         (1)

with $\psi_1$ and $\psi_2$ functions over $\Re^3$, as a linear combination of products, which is anti-symmetric in the sense that 

• $\Psi (x_1,x_2)=-\Psi (x_2,x_1)$.

The wave function $\Phi$ of RealQM has the form 

• $\Phi (x) = \phi_1(x)+\phi_2(x)$                       (2)

where $\phi_1$ and $\phi_2$ are functions over $\Re^3$ with common coordinate $x$ with disjoint supports $\Omega_1$ and $\Omega_2$ covering $\Re^3$ with common boundary $\Gamma$, where $\phi_1$ and $\phi_2$ take on the same value.

We see that $\Psi$ has product form (1) and $\Phi$ has sum form (2). We can make a formal connection by

• $\Psi (x_1,x_2)=\phi_1(x_1)+\phi_2(x_2)-(\phi_1(x_2)+\phi_2(x_1)$.    

We understand that the sum form of RealQM makes clear physical sense with non-overlapping supports (addition of charge densities).  To be compared with the product form preferred in StdQM in the presence of overlapping supports, which has unclear physical meaning (products of probabilities). 

A fundamental difference between StdQM and RealQM is the the effective use in computation of one-electron wave functions with overlapping vs non-overlapping supports. This observation can be added to a comparison of StdQM and RealQM coming up in the review process marked in the previous post. RealQM will meet a 100 year investment in StdQM. Watch out for referee reports.