Let us compare the textbook Standard Quantum Mechanics StdQM from 1926 with the recent alternative Real Quantum Mechanics RealQM as concerns the two basic aspects of physical meaning and computability.
Both seek to model a collection of $N$ negatively charged electrons subject to Coulomb interaction with a collection of positively charged nuclei together forming atoms and molecules.
The RealQM model is expressed in terms of a wave function $\psi (x,t)$ depending on a 3d spatial coordinate $x$ and a time coordinate $t$ defined over a subdivision of 3d space into non-overlapping regions acting as supports of one-electron charge densities $\vert\psi (x,t)\vert^2$ meeting with continuity. The corresponding Schrödinger equation has the form of a classical non-linear continuum model for a collection of non-overlapping electron charge densities and thus describes precisely specified real physics. The model is computable in the same sense as classical continuum mechanics with equations such as Maxwell's and Navier's. Computational complexity is (in principle) linear in $N$.
The StdQM model is expressed in terms of a wave function $\Psi (X,t)$ where $X$ is a $3N$ dimensional spatial variable with 3 independent dimensions for each electron. The corresponding Schrödinger equation has non-classical form as a linear equation in multi-dimensions with exponential computational complexity. The physical meaning of $\Psi (X,t)$ has been debated for 100 years without any agreement.
We now compare:
- RealQM: Computable with precise physical meaning.
- StdQM: Uncomputable with unknown physical meaning.
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