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{{DISPLAYTITLE:Quantum Density of states}}
{{Quantum book backlink|Wavefunctions and modes}}
'''Quantum density of states''' describes how many quantum states are available within a given energy interval. It is commonly written as <math>g(E)</math>, where <math>g(E)\,dE</math> gives the number of states between <math>E</math> and <math>E+dE</math>.<ref>[https://www.britannica.com/science/band-theory Band theory – Britannica]</ref>

[[File:Quantum_density_of_states.svg|thumb|400px|Density of states showing how the number of available quantum states varies with energy in a quantum system.]]

== Definition ==

The density of states is a counting function in energy space. It becomes useful when individual quantum levels are so closely spaced that the spectrum can be treated as effectively continuous.<ref name="FERMI">[https://www.britannica.com/science/Fermi-surface Fermi surface – Britannica]</ref>

== Origin from quantization ==

In confined systems, boundary conditions restrict wavefunctions to discrete standing-wave solutions. As the size of the system increases, these discrete levels become densely packed, and a continuous density-of-states description becomes appropriate.<ref>[https://openstax.org/books/university-physics-volume-3/pages/7-4-the-quantum-particle-in-a-box The Quantum Particle in a Box – OpenStax]</ref>

== Free-particle and solid-state picture ==

In the free-electron model, electrons are treated as particles in a three-dimensional box. Counting the allowed quantum states in momentum space leads to an energy-dependent density of states.<ref>[https://openstax.org/books/university-physics-volume-3/pages/9-4-free-electron-model-of-metals Free Electron Model of Metals – OpenStax]</ref>

In solids, the available quantum states are organized into bands, and the density of states helps determine how electrons populate those bands.<ref>[https://www.britannica.com/science/band-theory Band theory – Britannica]</ref>

== Dependence on dimensionality ==

The density of states depends strongly on the dimensionality of the system:

* in one dimension, <math>g(E)</math> decreases with energy
* in two dimensions, <math>g(E)</math> is constant for an ideal free-particle system
* in three dimensions, <math>g(E)</math> increases with <math>\sqrt{E}</math>

These differences are important in nanoscale systems such as quantum wells, wires, and dots.<ref>[https://openstax.org/books/university-physics-volume-3/pages/9-4-free-electron-model-of-metals Free Electron Model of Metals – OpenStax]</ref>

== Physical interpretation ==

The density of states tells how many quantum states are available at a given energy, but not whether they are occupied. Actual populations are determined only when the density of states is combined with a statistical distribution.<ref name="FERMI" />

== Applications ==

Density of states is fundamental in:

* solid-state physics
* semiconductor theory
* nanostructures and quantum wells
* statistical mechanics

It helps determine electrical, thermal, optical, and transport properties of materials.<ref>[https://www.britannica.com/science/hole-solid-state-physics Hole – Britannica]</ref>

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}
{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Density of states|1}}

Physics:Quantum Density of states - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

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