Back to Foundations of Quantum Mechanics

Ground state wave function of Hydrogen (surface plot of 2d section) as minimizer of energy.

Quantum Mechanics QM was formed in the 1920s in a search for a mathematical model of an atom understood to consist of a positively charged kernel surrounded by negatively charged electrons. Attempts by in particular Niels Bohr to form a classical "planetary" model with electrons as particles orbiting a kernel under Coulomb attraction, had miserably failed because orbiting electrons radiate energy and so lose energy in contradiction to observations of stable atoms. 

Schrödinger took on the challenge and in a moment of heavenly inspiration during a vacation trip to the Alps with a companion, came up with a model for the most basic case of the Hydrogen atom with one electron, with the electron being represented by a real-valued wave function $\Psi (x)$ depending on a 3d position variable $x$ with $\Psi ^2(x)$ representing charge density with normalisation 

• $\int \Psi^2 (x) dx = 1$.

The ground state $\Psi (x)$ of Hydrogen would then be represented as the minimiser of the energy

• $\frac{1}{2}\int \vert\nabla\Psi (x)\vert^2dx - \int V(x)\Psi^2(x)dx$, 

where $V(x)=\frac{1}{\vert x\vert}$, as solution of the eigenvalue problem

• $(-\frac{1}{2}\Delta - V)\Psi = E\Psi$ 

with $E$ minimal eigenvalue/energy. Excited states would then correspond to larger eigenvalues/energies. The energies showed close agreement with the observed spectrum of Hydrogen, which rocked Schrödinger to fame. But it came with expectations of generalisation to atoms with more than one electron. A formal mathematical generalisation just adding a new position variable for each new electron presented itself but Schrödinger was not happy with only a formality, since he was asking the model to be Anschaulich or possible visualisable. 

But the physics community with Bohr and Born jumped the formal generalisation giving the multi-dimensional wave function $\Psi$ a non-physical meaning as a possibility rather than physical actuality (as demanded by Schrödinger), which became the Copenhagen Interpretation CI of QM hotly debated in the 1930s as foundation of QM. 

Then the debate ran out of energy until John Bell revived it in 1964 for a short period before oblivion again, until the Nobel Prize in Physics 2022. So we are back to the foundations and CI. What is the meaning of the multidimensional wave function $\Psi$ of the CI? My take is that this $\Psi$ is uncomputable and as such not carrier of any information. This is because the multidimensional wave function for an atom with $N$ electrons depends on $10^{3N}$ variables with a resolution of 10 in each position variable which already for $N=10$ is beyond any thinkable computational power.

Real Quantum Mechanics presents a different generalisation which has physical meaning in the same sense as Schrödinger's equation for the Hydrogen atom. The number of variables is here $10^3N$ to be compared with $10^{3N}$. Computable compared with uncomputable! Time for a fresh restart on the foundations of QM. 

In Real Quantum Mechanics with 3d electron wave functions having disjoint support, locality is manifest, but not in CI because there wave functions have global support.