- #1

VortexLattice

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I was deriving the free electron gas for practice in 1, 2, and 3 dimensions, and I started wondering why they have different dependencies on energies and what that means. I got:

1D: ##g(E) = \frac{1}{\pi\hbar} \sqrt{\frac{2m}{E}}##

2D: ##g(E) = \frac{m}{\pi\hbar^2}##

3D: ##g(E) = \frac{1}{2\pi^2} (\frac{2m}{\hbar^2})^{3/2} \sqrt{E}##

Why does this happen, though? I mean, I did all the math so I see where it comes from, but I don't have a good intuitive reason... It's very strange to me that in 3D, it has a 'reasonable' dependence on the energy, but then it becomes constant when confined to 2D, and then gets energy dependence again when confined to 1D.

Also, what does it mean that in the 1D case, the density of states diverges at ##E = 0##? Does this mean that at low energies, there are (approaching) infinite states for electrons to fill? That starts to contradict what I know about the density of states.

Thank you!