$$ \rho(k) d E_{k}=\left(\frac{L}{2 \pi}\right)^{3} k^{2} d k \sin \theta d \theta d \phi $$ Since \(E_{k}=\hbar^{2} k^{2} / 2 \mu, d E_{k} / d k=\hbar^{2} k / \mu\), and we obtain for \(\rho(k)\) $$ \rho(k)=\frac{\mu L^{3}}{8 \pi^{3} h^{2}} k \sin \theta d \theta d \phi $$

The wave vector, denoted as \(k\), is a fundamental concept in quantum mechanics and wave physics. It describes the number of wave cycles per unit distance and relates to the wavelength \( \lambda \) through the equation \(k = \frac{2\pi}{\lambda}\). In three-dimensional space, the wave vector has components \( (k_x, k_y, k_z) \) and is often expressed in spherical coordinates for ease of integration.

Understanding the wave vector is crucial because it connects the spatial properties of a wave with its energy and momentum. For instance, the energy of a free particle is related to its wave vector by the expression \(E_k = \frac{\hbar^2 k^2}{2 \mu}\), where \(\hbar\) is the reduced Planck constant, and \(\mu\) is the particle's mass.

In many problems involving quantum states, we use the wave vector to describe the state of a particle in a potential-free region. The wave functions, which describe the probability amplitude of finding a particle, are then expressed as plane waves: \( \psi(x) = A e^{i k x} \). This formulation makes it easier to understand and calculate various physical properties, including the density of states.

Energy levels in quantum mechanics are the allowed energies that a particle or a system of particles can have. These levels are quantized, meaning particles can only occupy certain discrete energy values. The energy levels are typically derived from the Schrödinger equation.

In the context of a free particle, where no potential is acting on the particle, the energy \(E_k\) is related to the wave vector \(k\) by the formula \(E_k = \frac{\hbar^2 k^2}{2 \mu}\). This relationship shows that the energy levels are continuous and dependent on the square of the wave vector.

When dealing with multiple dimensions, the energy levels get more complex because they depend on the square sum of all the wave vector components. However, the fundamental principle remains that each permissible \(k\) value corresponds to a specific energy level. This understanding allows us to explore other properties, such as the density of states, which gives insight into how many quantum states are available at a given energy.

The density of states (DOS) is a crucial concept in quantum mechanics and solid-state physics. It describes the number of available quantum states at each energy level for a system of particles. The DOS function, often denoted as \( \rho(E) \), provides critical information about the distribution of states in energy space.

Mathematically, the density of states for wave vector \(k\) is given by the equation:
\( \rho(k) = \frac{\mu L^3 k}{8 \pi^3 \hbar^2} \sin \theta d \theta d \phi \).
Here, \(\mu\) is the particle's mass, \(L\) is the dimension of the system, \( \hbar \) is the reduced Planck constant, and \(\theta \) and \(\phi\) are spherical coordinate angles.

This expression results from isolating \(\rho(k)\) in the equation that relates it to the energy and wave vector. To better understand it, recall the step-by-step derived formula where we replaced \(dE_k\) and isolated \(\rho(k)\). The final expression indicates how the density of states varies with the wave vector \(k\) and incorporates factors like the system size and spatial dimensions. Understanding the DOS helps in analyzing the electronic and optical properties of materials, as it determines how particles fill up the energy levels in a given system and plays a vital role in phenomena like conductivity and heat capacity.