What Is an Electron? RealQM vs StdQM.

Electrons are described by wave functions as solutions to Schrödinger's Equation SE as the basic mathematical model of the quantum mechanics of atoms and molecules which comes in two forms: Standard Quantum Mechanics StdQM and Real Quantum Mechanics RealQM:

• Wave functions of StdQM for a system with $N$ electrons have global supports and depend on $N$ 3d spatial coordinates, altogether $3N$ spatial coordinates. Wave functions are computable only for very small $N$ since computational complexity grows exponentially with $N$.
• Wave functions of RealQM are sums of one-electron wave functions with non-overlapping supports depending on the same 3d spatial coordinate, and meet with continuity at a free boundary between supports. Wave functions are computable for all $N$, since computational complexity grows linearly with $N$. 

We now consider in more detail electrons according to RealQM. Each electron is described as a charge density $\psi (x)^2$ of a real-valued wave function $\psi (x)$ with support (non-zero value) in a certain region $\Omega$ in space with boundary $\Gamma$. The electron charge density does not have to vanish on $\Gamma$. Electrons sharing a common boundary piece meet with continuity as a free boundary condition. This allows the kinetic energy of an electron measured by $\vert\nabla\psi (x)\vert^2$ to be small even if the size of $\Omega$ is very small, which is not possible if $\psi (x)$ is forced to vanish on $\Gamma$. In other words, in the presence of other electrons, an electron can have small support and small kinetic energy, which means that it can circumvent the dictate of Heisenberg's Uncertainty Principle.

RealQM describes protons in the same way as electrons, with only a shift of sign of the charge.  

RealQM allows electrons to appear in two different forms in atoms and atomic nuclei as: 

• Charge densities with large support in atoms around a nucleus of vanishing size, meeting the nucleus freely (RealAtom). 
• Charge densities with very small support in kernels of nuclei surrounded by protons meeting proton charge densities with continuity, making sense since the size of the kernel is not small compared to the nucleus (RealNucleus). 

RealQM thus offers a complete model of an atom + nucleus in terms of: 

• A Schrödinger equation for a collection of positive and negative charge densities with non-overlapping supports interacting by Coulomb potentials, 
• The large difference in mass between proton and electrons allows the electron to serve a double role with presence both outside and inside the nucleus.  

RealQM is non-relativistic with motivation from the fact that there is no charge density motion in neither atom nor nucleus at all, and then certainly not at any relativistic speed. This is fundamentally different in StdQM where inner core electrons of heavy elements like Gold are claimed to move at half the speed of light by a purely formal argument connecting kinetic energy to velocity. Electrons can in giant particle accelerators be accelerated to relativistic speeds with massive input of energy, but its is difficult to fathom that the same thing happens in an atom of Gold...

Summary:

• RealQM describes a collection of $N$ electrons as a charge densities with non-overlapping local supports in 3 space dimensions meeting with continuity and interacting by Coulomb potentials. RealQM is computable for all $N$.
• StdQM the collection as overlapping charge densities with global support in $3N$ space dimensions. StdQM is computable only for very small $N$. 
• RealNucleus describes a nucleus as a collection of non-overlapping electron and proton charge densities meeting with continuity and interacting by Coulomb potentials.
• The Standard Model of StdQM describes a nucleus as a collection of protons and neutrons interacting by a residual of a strong force.