- #1
JohnnyGui
- 802
- 51
- TL;DR
- Why exactly is the number of quantum states equal to the volume in n-space if it is actually about the number of lattice points within that volume instead?
I'm trying to understand the detailed concept of why the density of states formula is accurate enough to calculate the number of quantum states of an energy level, including degeneracy, within a small energy interval of ##dE##.
The discrete energie levels are calculated by
$$E = \frac{h^2 \cdot (n_x^2+n_y^2+n_z^2)}{8mL^2}$$
Where the 3 dimensions of ##n## are integer values. The number of quantum states between ##E \geq E + dE## is deduced by calculating the volume of an 8th of a shell in n-dimensions with thickness ##d\bigg(\sqrt{n_x^2+n_y^2+n_z^2}\bigg)## (which is a piece of the n-sphere's radius).
$$N_{E_k} = \frac{1}{8} \cdot 4\pi (n_x^2+n_y^2+n_z^2) \cdot d\bigg(\sqrt{n_x^2+n_y^2+n_z^2}\bigg)$$
I have some questions about its accuracy but it's best to start off with these 2 questions first
1. The n-values are integer which means that the true number of quantum states is equal to the number of lattice points of the n-grid within a certain n-volume. However, when calculating the volume instead, you're associating 1 unit volume to 1 quantum state while in fact, 1 unit of volume can have more lattice points (i.e. one n-cube has 8 corners).
This Wiki states for a circle that for large n-values, the average number of lattice points per unit volume goes down to 1, which explains why the number of lattice points would be qual to the circle area. Can this reasoning be extrapolated to volume?
2. Does the thickness ##d\bigg(\sqrt{n_x^2+n_y^2+n_z^2}\bigg)## actually stay constant or does it change depending on the value of ##\sqrt{(n_x^2+n_y^2+n_z^2)}## as the radius at which you're calculating the n-shell volume?
The discrete energie levels are calculated by
$$E = \frac{h^2 \cdot (n_x^2+n_y^2+n_z^2)}{8mL^2}$$
Where the 3 dimensions of ##n## are integer values. The number of quantum states between ##E \geq E + dE## is deduced by calculating the volume of an 8th of a shell in n-dimensions with thickness ##d\bigg(\sqrt{n_x^2+n_y^2+n_z^2}\bigg)## (which is a piece of the n-sphere's radius).
$$N_{E_k} = \frac{1}{8} \cdot 4\pi (n_x^2+n_y^2+n_z^2) \cdot d\bigg(\sqrt{n_x^2+n_y^2+n_z^2}\bigg)$$
I have some questions about its accuracy but it's best to start off with these 2 questions first
1. The n-values are integer which means that the true number of quantum states is equal to the number of lattice points of the n-grid within a certain n-volume. However, when calculating the volume instead, you're associating 1 unit volume to 1 quantum state while in fact, 1 unit of volume can have more lattice points (i.e. one n-cube has 8 corners).
This Wiki states for a circle that for large n-values, the average number of lattice points per unit volume goes down to 1, which explains why the number of lattice points would be qual to the circle area. Can this reasoning be extrapolated to volume?
2. Does the thickness ##d\bigg(\sqrt{n_x^2+n_y^2+n_z^2}\bigg)## actually stay constant or does it change depending on the value of ##\sqrt{(n_x^2+n_y^2+n_z^2)}## as the radius at which you're calculating the n-shell volume?