For an electron in a three-dimensional metal, show that the average energy is given by $$\bar{E}=\frac{1}{N} \int_{0}^{E_{F}} E g(E) d E=\frac{3}{5} E_{F}$$

The density of states function is a fundamental concept in quantum mechanics and solid state physics. It describes how many electron states are available at a given energy level within a material. For electrons in a three-dimensional metal, the density of states function, denoted as \(g(E)\), increases with the square root of energy, indicating that higher energy levels have more available states.

This function relies on several physical constants and properties: the volume \(V\) of the metal, the mass \(m\) of an electron, and the reduced Planck constant \(\hbar\). In mathematical terms, for a three-dimensional metal, it's given by the equation \(g(E) = \frac{V}{2\pi^2} \left(\frac{2m}{\hbar^2}\right)^{\frac{3}{2}} E^{\frac{1}{2}}\).

To calculate the average energy of electrons, you integrate the product of energy \(E\) and the density of states function across all energies up to the Fermi energy. This calculation clearly considers how the availability of states at different energy levels contributes to the average energy of the electron.

Fermi energy, denoted as \(E_F\), is a vital concept in the study of metals and semiconductors. It represents the highest energy level occupied by electrons at absolute zero temperature. Essentially, it is the energy cut-off point beyond which no electron states are filled.

In calculations of the average energy for electrons in metals, the Fermi energy serves as the upper limit of the integral that determines the average energy. As demonstrated in the given exercise, the average energy \(\bar{E}\) is proportionally less than the Fermi energy, specifically, it is \(\frac{3}{5}E_F\). This shows that while electrons can occupy states up to the Fermi energy, their average energy is lower due to the way states are filled according to quantum mechanical principles.

Quantum mechanics plays a critical role in understanding solid-state physics, which deals with the physical properties of solids such as metals. Quantum mechanics provides a framework for describing the behavior of electrons in a solid, which is necessary to understand various phenomena such as electrical conductivity and heat capacity.

The principles of quantum mechanics dictate that electrons occupy discrete energy states within a material. The application of quantum mechanics in the exercise dealing with the average energy of electrons involves using the concept of the density of states function. This application emphasizes how the discrete nature of electron states, which is a quantum mechanical concept, affects macroscopic properties, like the average energy of electrons in a metal.

Electrons in metals behave differently from those in other states of matter. Their movement and energy distribution are key to the electrical and thermal properties of the metal. Under the framework of quantum mechanics, the behavior of electrons in metals is described using models like the free electron gas model.

This model assumes that electrons move freely within the metal, similar to a gas, and their behavior can be described using the principles of quantum mechanics without considering interactions between electrons or with the lattice ions. From the exercise, we see that the average energy of electrons is a fraction of the Fermi energy, illustrating that while electrons have the freedom to occupy states up to \(E_F\), their distribution across these states is unequal and follows quantum rules, leading to the specific average energy found.