Textbook Standard Quantum Mechanics StdQM for an atomic system with $N$ electrons is formulated in terms of a wave function
- $\Psi (x_1,...,x_N)$
depending on $N$ 3d spatial coordinates $x_1,...,x_N$ ranging over $N$ copies of the same physical 3d Euclidean space $\Re^3$, altogether forming a $3N$ dimensional Euclidean space $\Re^{3N}=\Re^3\times\Re^3\times....\times\Re^3$ referred to as configuration space. The wave function satisfies Schrödinger's Equation SE, which is a linear partial differential equation over configuration space.
SE was first formulated by Schrödinger for the H atom with $N=1$ in 1926, in which case configuration space $\Re^{3N}$ is identified with physical 3d Euclidean space $\Re^3$ and $\Psi^2(x)$ with $x\in\Re^3$ represents electron charge density of clear physical meaning.
But in the generalisation to $N>1$ in StdQM, which quickly followed with assistance of Bohr-Born-Heisenberg, the meaning of $\Psi (x_1,...,x_N)$ defined over configuration space is not clear and in fact has been the subject of intense debate for 100 years without ever any consensus being reached. Basic questions are:
- What is the physical meaning of SE?
- What is the physical meaning of $\Psi (x_1,...x_N)$?
- What is the meaning of the coordinates $x_1,....,x_N$?
- What is the physical meaning of configuration and configuration space?
- Does $x_i\in\Re^3$ somehow represent electron $i$ with $i=1,...,N$?
- If so, does $x_i$ represent physical presence?
- What does labelling of identical electrons mean?
- What does imposed anti-symmetry of $\Psi (x_1,...x_N)$ mean?
- What does exchange and correlation of identical electrons mean?
- $\Psi^2(x_1,...,x_N)$ represents "all there is to know" about an atomic system.
- "All there is to know" does not include actual states of the system, but is reduced to "outcomes of experiments".
- $\Psi^2(x_1,...,x_N)$ is a "probability density of electron configurations".

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