Integrating Equation 37.5 over all energies gives the total number of states per unit volume in a metal. Therefore, integrating from \(E=0\) to \(E=E_{\mathrm{F}}\) - that is, over the occupied states only-gives the number of conduction electrons per unit volume. Carry out this integration to show that the electron number density is given by $$n=\left(\frac{2^{9 / 2} \pi m^{3 / 2}}{3 h^{2}}\right) E_{\mathrm{F}}^{3 / 2}$$

Semiconductor materials play a critical role in modern electronic devices, being the foundation for chips and transistors. Unlike metals, which conduct electricity freely, and insulators, which do not conduct at all, semiconductors have a conductivity that lies between the two extremes. This property is governed by their unique electronic band structure.
At absolute zero temperature, semiconductors' valence bands are fully occupied while their conduction bands are empty. However, with increasing temperature, or through 'doping' with impurities, charge carriers (electrons and holes) can be excited into the conduction band, facilitating conductivity.
These charge carriers are influenced by quantum mechanics, and the number density of electrons is a fundamental characteristic of semiconductors that reflects the number of conduction electrons per unit volume. This concept is directly connected to the exercise, where electron number density is computed through integration, reflecting the quantum mechanical nature of these materials.

A core concept in understanding electronic properties of materials is density of states (DOS), which is a measure of the number of states that are available for electrons to occupy, at each energy level, within a material. It plays a significant role in determining the electrical and thermal properties of semiconductors.
For a three-dimensional system like a semiconductor, the density of states for electrons is proportional to the square root of the energy. This relationship forms the basis of the integral in the exercise. The integration of the DOS up to the Fermi energy gives us the total number of electrons per unit volume that contribute to conduction, which is of paramount importance for understanding the functionality of the semiconductor devices.

The Fermi energy (\(E_{\text{F}}\)) is a fundamental quantity in the study of solid-state physics, typically associated with conductors and semiconductors. It is the energy level at which the probability of an electron being present at absolute zero temperature is 50%. For all practical temperatures in a metal, it is essentially the highest occupied energy level.
In semiconductors, the Fermi level is crucial for determining the distribution of electrons in the conduction and valence bands. It helps in defining the position of the energy level where the probability of finding an electron drops from near certainty to near zero. The solution to the exercise involves integrating up to the Fermi energy to find the total number of electrons per unit volume, indicating how the Fermi energy serves as an upper limit for occupied electron states at absolute zero.

Quantum mechanics is the underlying theory that explains the behavior of particles on the atomic and subatomic scales. It is the framework within which semiconductor physics and concepts such as density of states and Fermi energy can be understood.
Essential to quantum mechanics are the quantized energy levels and wave-particle duality, where entities like electrons exhibit both wave-like and particle-like properties. The quantization leads to discrete energy states in atoms and solids, influencing how carriers behave in a semiconductor. In the context of the exercise, using the density of states and integrating to the Fermi energy, we utilize principles of quantum mechanics to arrive at the electron number density, which is determined by the allowed energy states and their occupation by electrons.