An uncertainty relation for quantum systems with an arbitrarily large number of particles and the expansion rate of matter in the bulk

For any quantum system with an arbitrarily large number $N$ of particles for which the lower end of the spectrum has a nonzero width and is bounded below by $N$, we rigorously derive an uncertainty relation for the product of $N$ and the survival time $T$, a measure of the system resistance to change. We then use the derived inequality to investigate the highly nontrivial problem of the expansion rate of bulk matter as a function of the number $N$ of electrons for large $N$. We approach the application to this problem by noting that resistance to the increase of the expansion rate can be quantum mechanically defined in terms of its survival time against such an increase. We show that a sufficient condition for matter to have a nonvanishing survival time against expansion rate increase is that the lower end of the spectrum is the lower end of an energy width for which applying the derived uncertainty relation becomes obvious. In turn, in particular, we show that if the expansion rate increases with its “size” of radius $R$, then the survival time decreases not faster than $1/R^3$ for large $R$. For completeness and consistency of the analysis, we also consider the formal zero-width limit. Because the derived uncertainty relation is general, we expect it also to have other applications.