Recent posts recall that the accepted canon/standard of Quantum Mechanics QM as Schrödinger's equation for an atomic system with $N$ electrons, is a linear partial differential equation in terms of a multidimensional complex-valued wave function $\Psi (x1,x2,...,xN,t)$ depending on $N$ three-dimensional space variables $x1, x2,...,xN$ and a time variable $t$. The equation arises as an ad hoc mathematical generalisation of Schrödinger's equation for a Hydrogen atom with one electron ($N=1$), which can be given a direct physical meaning with the wave function representing an electron density.
The basic question QM concerns the physical meaning of the wave function for $N>1$. The accepted answer coined as the Copenhagen Interpretation CI imprinted into the canon by Born followed by Bohr, is that $\vert\Psi (x1,....,xN,t)\vert^2$ represents a probability density for an electron configuration described by the coordinates $(x1,...,xN,t)$. But a probability density does not describe any state of physics and QM still after 100 years remains a mystery as acknowledged by all prominent physicists.
To a mathematical mind the ad hoc generalisation is direct, and so physicists impressed by mathematics have been forced to accept it as true physics, although mysterious. So instead of searching for a generalisation with physical meaning (see Real Quantum Mechanics), all efforts have gone into loading the multi-dimensional wave function $\Psi (x1,x2,...,xN,t)$ with (some) physical meaning.
To this end physicists have come up with the idea to distinguish between wave functions which are symmetric in the variables $(x1,...,xN)$ (augmented by a two-valued spin variable) supposed to represent boson (photon) and anti-symmetric supposed to represent fermion (such as electron and proton), with any particle being either a boson or fermion. Anti-symmetry was crafted so as to prevent two electrons with the same spin to be at the same place in space-time, expressed as Pauli’s Exclusion Principle.
With this perspective a God creating the world of physics must have been mathematician and not physicist. First He created multidimensional wave functions, which is a very rich Universe, and then He decided that only symmetric or anti-symmetric wave functions would represent actual physics, asking systems of particles to keep track of symmetry or antisymmetry. But a mathematician cannot create physics, only at best model physics in terms of mathematics and computation. God must have been also physicist with richer tools for creation than mathematical symbols on a paper. And then He would not have been satisfied with a purely mathematical Universe designed by mathematicians like the CI of QM...
To sum up: Standard QM for a many-electron system is based on a linear wave equation for a multi-dimensional complex-valued wave function arising from a generalisation of Schrödinger's equation for a Hydrogen atom according to purely mathematical principles. The following questions present themselves as the fundamental conundrums of modern physics:
- Why is the wave equation for a system scalar-valued and not vector-valued?
- Why is the equation linear and thus allowing superposition? Physics is rarely linear.
- What physics is captured by the multi-dimensional wave function? Do electrons play dice?
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