StdQM/DFT and RealQM as Deterministic Theories

Continued conversation with chatGPT in the previous post comparing StdQM with RealQM, makes clear:

• The StdQM Schrödinger Equation SE for an atom is a deterministic differential equation.
• The eigenvalues of SE as typical deterministic output, represent the spectrum of the atom, with smallest eigenvalue equal to the energy of the ground state of the atom.
• There is no probabilistic element in this picture. SE deterministically predicts the ground state energy of an atom. No game of roulette is involved.

This may seem surprising since it is commonly believed that stdQM involves elements of roulette, to which both Schrödinger and Einstein heavily objected.

StdQM and RealQM thus both fully deterministically predict the ground state of an atom (or spectrum), only in different ways. StdQM works with overlapping electron densities stacked on top of each other in a non-physical multi-dimensional space, while RealQM works with non-overlapping electron densities in ordinary physical 3d space. The essence of Coulomb interaction and electron kinetic energy, is shared. 

RealQM is to be compared with Density Functional Theory DFT, which is StdQM reduced to a common electron density, while RealQM keeps individual electron densities.

I hope this post will add substance to a discussion about StdQM vs RealQM. In particular, it shows RealQM as a more detailed DFT and less detailed StdQM.

How does then the roulette enter StdQM, if not through SE? The common idea is that this is somehow through measurement but then in a way different from random effects on measurements in a classical deterministic setting. This is the measurement problem of StdQM still open after 100 years. In RealQM there is no measurement problem beyond those of classical physics. 

But wait, what about the Schrödinger wave function $\Psi (x,t)$ as solution to a SE describing the evolution of an atomic system over time $t$ from an initial state at $t=0$ to a final state $t=T$ with $\vert\Psi (x,t)\vert^2$ supposed to signify according to Born/StdQM:

• The probability of finding an electron/particle at the point $x$ at time $t$. 

This is a mantra of StdQM you will hear physicists repeat in unison and which chemists have accepted. But the mantra is empty since to determine $\Psi (x,T)$ requires specification of multi-dimensional initial data $\Psi (x,0)$ and computation of the evolution, both impossible to realise. What remains is the eigenvalue problem as a deterministic energy minimisation problem, which does not need any initial data and is deterministic in the same sense as a classical mechanics problem.  

The evolution problem of StdQM represents an uncomputable fiction, while the eigenvalue problem of StdQM is deterministic in a classical sense, but still uncomputable.  

Summary:

• RealQM is a classical deterministic computable theory with potential of predicting both atomic spectra and atomic dynamics since initial data remain 3d.
• DFT is a drastically reduced StdQM theory with potential of predicting atomic spectra, but the reduction carries unresolved problems.