Stability of Bulk Matter by RealQM

This is a condensation of the preceding sequence of posts with a clear and simple message.

Stability of bulk matter as a collection of a large number $N$ of atoms, is on a basis of Standard Quantum Mechanics StdQM considered to be very difficult to prove mathematically, as evidenced in the work by Dyson-Lenard and Lieb-Thirring. Stability is expressed by a lower bound on total energy scaling with $N$, making total energy an extensive quantity. 

The difficulty is that electrons in StdQM have global support and so in principle can interact with many kernels to give a lower bound scaling with a power of $N$ possibly bigger than 1 with then total energy tending to minus infinity.

On the other hand with RealQM as an alternative to StdQM, stability of bulk matter directly follows from atomic stability which is a simple consequence of a the Hardy inequality: (see this post):

• $\int\frac{\psi^2(x)}{\vert x\vert}dx\le (\int\psi^2(x)dx)^{\frac{1}{2}}(\int\vert\nabla\psi (x)\vert^2dx)^{\frac{1}{2}}$ for all $\psi\in H^1(R^3)$, 

used by Kato in his analysis of Schrödinger's equation as the foundation of StdQM. This easy-to-prove inequality gives a bound on potential energy in terms of kinetic energy proving stability. 

The wave function $\Psi (x)$ of RealQM for an atomic system is a sum

• $\Psi (x)= \sum_{n=1}^N\Psi_n(x)$

of atomic electronic wave functions $\Psi_n(x)$ depending on common 3d variable $x$ with non-overlapping supports of electrons meeting at a Bernoulli free boundary with continuity and zero normal derivative. This means  that $\Psi\in H^1(R^3)$ and so satisfies the Hardy inequality bounding total potential energy by kinetic energy with extensivity directly following from the fact that the $\Psi_n(x)$ are sums of electronic wave functions with non-overlapping support and so then also $\Psi (x)$.

The structure of RealQM thus gives a lower bound on the total energy $E$ of a system of $N$ atoms each with kernel charge $Z$ of the form 

• $E\ge -CNZ^2$

with $C$ an absolute constant expressing stability of the 2nd kind according to Dyson-Lenard-Lieb-Thirring. 

RealQM is a deterministic continuum model in 3d based on Coulomb physics combined with a form of electronic "kinetic energy" measured by $\vert\nabla\psi (x)\vert^2$ with Pauli Exclusion Principle safely built in by non-overlap of electronic wave functions. 

RealQM is a physical computable model to be compared with StdQM which is non-physical and uncomputable. The fact that stability of matter is safely built into the structure of RealQM, but much less so in StdQM, can be seen as a major additional advantage of RealQM.