Comparison RealQM vs StdQM and DFT

Standard Quantum Mechanics StdQM based on Schrödinger's equation SE with standard interpretation of a Hamiltonian acting on wave functions with $3N$ spatial dimensions for a system with $N$ electrons, has only statistical meaning and is computable only for very small $N$, thus can be said to be non-physical and uncomputable.  

Density Functional Theory DFT seeks to reduce StdQM by averaging 3N-dimensional wave functions  into a single electron charge density $\rho (x)$ depending on a 3-dimensional coordinate $x$, and identifying ground states of StdQM with DFT densities satisfying a reduced SE with Hamiltonian only implicitly determined and so has to be approximated. DFT is the main computational method for $N>100$ currently available.

RealQM is based on a different interpretation of the Hamiltonian of SE acting on a wave function $\Psi (x)$ as a sum 

• $\Psi (x) = \sum_{n=1}^N\psi_n(x)$ 

of one-electron wave functions $\psi_n(x)$ with non-overlapping supports depending on a common 3d spatial variable, which meet at a Bernoulli free boundary with continuity and zero normal derivative. The corresponding electron charge density $\rho (x)$ is a sum

• $\rho (x)=\sum_{n=1}^N\psi_n^2(x)$     

of non-overlapping charge densities $\psi_n^2(x)$. 

A fundamental difference between RealQM and DFT is that electron densities in RealQM carry identity by occupying distinct regions in space and so can be numbered, just like pool balls on a pool table, while identity is lost in the common density of DFT (which creates a lot difficulties when having to recreate lost identity to keep physicality).

The zero normal derivative free boundary condition satisfied by meeting wave functions keeps electron identity which is not expressed by continuity alone. 

Recall that wave functions of StdQM have overlapping global supports, which makes identification difficult/impossible, while wave functions in RealQM have non-overlapping local supports, which makes identification direct.

We further recall from recent posts that stability of matter is a direct consequence of the structure of RealQM, but is less obvious in StdQM and DFT.

A basic postulate of StdQM is that electrons carry no identity, that they are indistinguishable, and that is the basic difference with classical physics, which can be viewed to carry identity. So identity vs no-identity can be viewed to be the dividing line between classical mechanics (not including statistical mechanics) and quantum mechanics. 

The dividing line shows that modern physics as microscopic quantum mechanics is different from macroscopic classical mechanics, more precisely so fundamentally different that quantum mechanics is said to be "weird" by the most knowledgable physicists, while saying the same about classical mechanics would simply express ignorance.  

To speak about electrons without any from of identity is according to Leibniz really "weird" since it contradicts his Principle of Identity of Indiscernibles PII.

The fact that RealQM respects PII, while StdQM does not, eliminates the dividing line between microscopic and macroscopic physics and so opens to a unified mechanics on all scales. 

To allow microscopic objects to carry identity allows perception of the microscopic world to be similar to that of the macroscopic world, thus understandable and not only "weird".  

Here is a comparison in condensed form:

• StdQM: explicit Hamiltonian, no electron identity, non-physical, uncomputable, stability non-obvious, "weird".
• DFT: implicit Hamiltonian, no electron identity, physicality?, computable, stability non-obvious, "weird"? 
• RealQM; explicit Hamiltonian, electron identity, physical, computable, stability obvious, understandable not "weird". 

chatGPT says that stdQM violates Leibniz PII and that efforts to change StdQM to "save" identity, like Bohmian mechanics with its "pilot wave", have all failed. Here RealQM comes in...