Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter

Abstract

We first prove that $\Sigma \mathrm{}\left|e\right(V\left)\right|$, the sum of the negative energies of a single particle in a potential $V$, is bounded above by $\left(\frac{4}{15\pi }\right)\int {\mathrm{}\left|V\right|\mathrm{}}^{\frac{5}{2}}$. This, in turn, implies a lower bound for the kinetic energy of $N$ fermions of the form $\frac{3}{5}{\left(\frac{3\pi }{4}\right)}^{\frac{2}{3}}\int {\rho }^{\frac{5}{3}}$, where $\rho \mathrm{}\left(x\right)\mathrm{}$ is the one-particle density. From this, using the no-binding theorem of Thomas-Fermi theory, we present a short proof of the stability of matter with a reasonable constant for the bound.