16.1 Quantum states and density of states

Quantum States and Energy Levels

Quantum states and energy levels form the foundation of quantum statistical mechanics. Every quantum system can only exist in specific, discrete states, and the way those states are distributed in energy determines the thermodynamic behavior of the system. The density of states then tells you how many of those states are available at each energy, which is what connects the quantum picture to macroscopic quantities like heat capacity, entropy, and particle number.

Quantum states and energy levels

A quantum state is a complete description of a quantum system at a given moment, specified by a unique set of quantum numbers. For an electron in an atom, these are the principal ($n$), angular momentum ($l$), magnetic ($m_l$), and spin ($m_s$) quantum numbers. Each distinct combination corresponds to a specific energy level determined by the system's Hamiltonian.

Energy in quantum systems is quantized: only certain discrete energy values are allowed. Which values those are depends on the potential energy landscape and the boundary conditions. A particle in a 1D box of length $L$, for example, has energy levels:

$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$

where $n = 1, 2, 3, \ldots$ This is one of the simplest cases, but the same principle applies to atoms (Coulomb potential), molecules, and solids.

Wave functions provide the mathematical description of each quantum state. The wave function $\psi(\mathbf{r})$ is a solution to the Schrödinger equation, and its squared modulus $|\psi(\mathbf{r})|^2$ gives the probability density of finding the particle at position $\mathbf{r}$.

Density of States

Density of states calculations

The density of states (DOS) counts how many quantum states exist per unit energy interval. It's the bridge between individual quantum levels and statistical mechanics: once you know the DOS and the relevant distribution function (Fermi-Dirac, Bose-Einstein, or Maxwell-Boltzmann), you can compute thermodynamic averages.

The general definition is:

$D(E) = \frac{dN}{dE}$

• $D(E)$: number of states per unit energy at energy $E$
• $N$: total number of states with energy up to $E$

To calculate the DOS analytically, you typically:

1. Solve for the allowed wavevectors $\mathbf{k}$ using boundary conditions (usually periodic or hard-wall).
2. Determine the dispersion relation $E(\mathbf{k})$ that maps wavevectors to energies.
3. Count the number of $\mathbf{k}$-states in a shell of energy between $E$ and $E + dE$ in $k$-space.
4. Convert that count into $D(E)$ by dividing by $dE$.

For a free particle in 3D with $E = \hbar^2 k^2 / 2m$, this procedure gives the well-known result $D(E) \propto E^{1/2}$. For more complex or disordered systems, numerical methods (such as the recursive Green's function technique) are used instead.

Density of states across dimensions

The dimensionality of a system has a dramatic effect on the functional form of the DOS. For free (or nearly free) particles:

• 3D systems: $D(E) \propto E^{1/2}$ The DOS increases with energy. This applies to bulk semiconductors (silicon, germanium) and metals (copper, aluminum). Specifically, for a free electron gas in 3D:

  $D(E) = \frac{V}{2\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} E^{1/2}$

• 2D systems: $D(E) \propto E^{0}$ (constant) The DOS is a step function, independent of energy within each subband. Examples include semiconductor quantum wells (GaAs/AlGaAs heterostructures) and graphene (though graphene's linear dispersion gives a DOS that goes as $|E|$, not a constant, so it's an important exception).

• 1D systems: $D(E) \propto E^{-1/2}$ The DOS diverges as energy approaches the bottom of a subband. These van Hove singularities appear in quantum wires and carbon nanotubes.

• 0D systems (quantum dots): The DOS becomes a series of delta functions, since the energy levels are fully discrete with no continuous bands at all.

The trend to remember: as you reduce dimensionality, the DOS becomes increasingly singular near band edges.

Applications of density of states

The total number of states in an energy window is found by integrating the DOS:

$N = \int_{E_1}^{E_2} D(E) \, dE$

This integral, combined with the appropriate quantum distribution function, gives you physically measurable quantities:

1. Carrier concentrations in semiconductors: Multiplying $D(E)$ by the Fermi-Dirac distribution $f(E)$ and integrating yields the electron (or hole) density. This is how you predict conductivity and doping behavior.
2. Thermodynamic properties: The electronic contribution to heat capacity in metals, for instance, depends directly on $D(E_F)$, the DOS evaluated at the Fermi energy.
3. Optical properties: Absorption and emission spectra depend on the joint density of states between valence and conduction bands.
4. Quantum device design: Engineering the DOS through confinement (quantum wells, wires, dots) lets you tailor properties like laser threshold current, LED emission wavelength, and thermoelectric efficiency.

The DOS is, in short, the central quantity that connects the microscopic quantum description of a system to its macroscopic, measurable behavior.