Density of real numbers and density function

Question

In Quantum mechanics, given a certain material, it is possible to write the density of energy states $\rho (E)$ as a function of $E$. That is: let's consider all the real values contained in the interval $[E,E + dE]$ and $\rho (E)dE$ of them are allowed energy values for the electrons of the atoms.

But how much real numbers are actually contained between the number $E$ and the number $E + dE$, where $dE$ is an infinitesimal? I would say one, at most, but this is clearly not true, because typically $\rho(E) \sim \sqrt{E}$ and so between $16$ and $16 + dE$ we have $\sim 4dE$ states. The number of states should be always less then (or equal to) the number of real numbers in a certain interval.

In other words, maybe with a little abuse of terminology, what is the density function of the real numbers?

Answer 1

There are as many real numbers in any interval of the real line as there are in the entire real line.

Quantum mechanics doesn't play nice with real analysis, as there are minimal distances in quantum mechanics, while that is not a problem in mathematics.

Answer 2

If you consider a hyperfinite partition of a typical interval say $[16,17]$ into $N$ infinitesimal subintervals where $N$ is infinite, then the length of each subinterval is an infinitesimal $\alpha=\frac{1}{N}$ and if you take your $dE$ to be say $100\alpha$ then there will be about $25$ partition points between $16$ and $16+dE$ in your case $\rho(E)=\sqrt{E}$. The real number system does not contain infinitesimals so to formalize your argument you need to consider an extension that does. See Elementary Calculus for details.