Density of States - Statistical Physics - University Physics

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The density of states is a concept that's very weird, and in all honesty after learning it many times in my degree I still don't think I have a perfect understanding. In this video I try and explain it as best I can, and derive what the DOS is for a three dimensional system.

In this video we draw inspiration from the one dimension quantum well (or "Particle in a box"). What was found was once the particle was confined, it could only take discrete energy states. The other observation was that there were no degenerate energy states for this system (that is to say, one or more states with the same energy), however in most systems there are many degenerate energy levels.

The density of states quantifies this degeneracy in terms of a smooth analytical function, ρ(E). This function determines what energies each particle in the system has, and how states are filled by particles. In this system we assume the system is isotropic and hence perform an integral in spherical polar coordinates in k-space.

A good exercise to practice is deriving what the DOS is for different system dimensionalities, along with different dispersion relation (i.e photons in 2D, confined systems in 3D, etc). For isotropic systems, instead of a sphere in 3D it will be an integral over a circle in 2D and a line in 1D due to the different symmetries in different dimensions.

If anyone has any questions, let me know in the comments and I'll try to answer as best and as quickly as I can!