Number of particles in a given energy interval

While reading statistical mechanics from beiser,I encountered statements like *If $$n(E)$$ is the number of particles with energy $$E$$,then the number of particles with energy between $$E$$ and $$E+ dE$$ is $$n(E)dE$$.

Why is it true? I believe the reason for taking small interval $$dE$$ is for approximating that all energy levels from $$E$$ to $$E+dE$$ have same number of particles $$n(E)$$. So the product says there are $$dE$$ energy levels with each having particles $$n(E)$$,so the total is $$n(E)dE$$. But how is it possible that $$dE$$ is the number of energy levels between $$E$$ and $$E+dE$$? $$dE$$ just means difference in energy and not any number.I am really confused why the difference in magnitude of energy is being treated as number of energy levels which is hindering me from understanding further concepts.

• Are you sure you have the right definition of $n$? It looks like it's a density, not number. Also, saying there are "$\text dE$" energy levels doesn't make sense Commented Jul 26 at 0:26

There's probably a mistake in your reference. Likely $$n(E)$$ is not the number of particles with energy $$E$$, but rather the number of particles per energy. This way $$n(E) dE$$ will have the correct units of [number of particles].

all energy levels from $$E$$ to $$E+dE$$ have same number of particles $$n(E)$$.
A few potential misunderstandings may hamper full mastery of the concept of density of states in energy ($$n(E)$$).
The most important thing to keep in mind is that $$n(E) dE$$ represents the number of states in the interval between the energy $$E$$ and $$E+dE$$ per unit volume. The number of states is usually not the same as the number of energy levels due to the possibility of degeneracy (more than one state at the same energy).
In a one-particle approximation for a fermionic system of particles with spin $$1/2$$, the number of states can be further connected to the number of particles, assuming that all the states in the energy interval of width $$dE$$ are occupied. In such a case, assuming that the states counted by $$n(E)$$ are spatial states, we must consider that each state can accommodate two fermions. Under such additional hypotheses, $$n(E)dE$$ can be connected to the number of particles per unit volume.
In summary, it is not $$dE$$ that is proportional to the number of states but $$n(E)dE$$. The reason for a function $$n(E)$$ and not a constant is the presence of the degeneracy of the energy levels. Such a degeneracy depends on the dimensionality of the space and the interaction.