The Kinetic Energy of StdQM and RealQM

The Schrödinger Equation SE as the basic mathematical model of Standard Quantum Mechanics StdQM in terms of (here for simplicity) a one-electronic wave function $\Psi (x)$ depending on a 3d spatial coordinate $x$, gives rise to a contribution to total energy named kinetic energy of the form 

• $E_{kin}(\Psi )=\frac{h^2}{2m}\int\vert\nabla\Psi (x)\vert^2 dx$ with $\int\Psi^2(x)dx=1$, 

where $h$ is Planck's constant and $m$ the mass of the electron. If $\Psi (x)$ is globally defined with $\Psi (x)$ tending to zero for $\vert x\vert$ tending to infinity, the coefficient $\frac{h^2}{m}$ determines  the size of the electron to scale with the $\frac{h}{\sqrt{m}}$ with thus a larger size for smaller $m$. 

Viewed the other way around, $E_{kin}(\Psi )$ scales with the factor $D^{-2}$ with the size $D$ of the electron, which means that electron concentration comes with large kinetic energy. 

In RealQM as an alternative to StdQM an electronic wave function has local support over a domain in space and is not restricted to vanish on the boundary of the domain. This allows the kinetic energy to stay bounded with decreasing size of the electron. This is the secret of covalent bonding as shown here and allows RealQM to extend to a model of an atomic nucleus as shown here. 

The salient feature of RealQM is that electrons have wave functions with non-overlapping support representing non-overlapping unit charge densities which meet a free boundary with continuity. 

Covalent bonding is thus realised in RealQM by allowing electrons to meet between kernels without increase of kinetic energy. 

A RealQM model without need of a strong force of a nucleus can thus be built as an electron density of very small size surrounded by protons of larger size. Electrons thus appear in two sizes, large for an atom and small for a nucleus. 

Covalent bonding is not well explained within StdQM, and is still after 100 years subject to debate without conclusion. The Standard Model of a nucleus built by quarks and gluons/strong force is very complicated.