It's known that the Density of States in 2D is given by, [tex] g_2(E)dE = \frac{a^2m}{\pi\hbar^2}dE[/tex] The density of states in 1D and 3D are as follows, [tex] g_1(E)dE = \left(\frac{a}{\pi}\sqrt{\frac{2m}{\hbar^2}}\right)\frac{1}{\sqrt{E}}dE[/tex] [tex] g_3(E)dE = \frac{a^3}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{\frac{3}{2}}\sqrt{E}dE[/tex] It's clear that the 1D and 3D Density of States are dependent on energy but it seems for the 2D case the energy density is constant. I was wondering why this was the case?
this is for a free electron. Not necessarily for every system. I don't have a physical intuitive explanation. It just arises from the math of the system.
I think I have a mathematical explanation that is somewhat intuitive. If one accepts that the energy of the free electron is given by: [itex] E=\frac{\hbar^2 k^2}{2m} [/itex] and that the "density" of states, when imagined as the density of points in reciprocal space, in 2D will be an areal density (area rather than volume or length for 1D), then you can see that those [itex]k[/itex] are going to cancel. All it means is that as you imagine going further out, from the origin, in your reciprocal space, the energy increases quadratically. Remember that we discretise the reciprocal space dependent on the number of electrons in the system. In all dimensions then, the number of points contained in a region (a length, area or volume) bounded by a k-point is determined by our deliberate choice to discretise reciprocal space, so as to give each point an equal fraction of the total space. It is only in the 2 D case that the number of points contained in a region increases quadratically with the size of your region (area of circle is proportional to radius squared). In 1 D and 3 D the increase is linear, and cubic, thus increasing slower and faster than the energy of state k at the boundary, respectively. Conclusion: our choice of even length/area/volume per k-point and the physical reality that the energy is proportional to k^2 gives the result. The first part (the choice) is necessary for an easy derivation of the density of states, and so the actual answer is the nature of the energy equation.