Accuracy of the Density of States

In summary, the conversation discusses the concept of the density of states formula, which is used to calculate the number of quantum states within a small energy interval. The formula involves calculating the volume of an 8th of a shell in n-dimensions with a certain thickness, and the conversation raises questions about its accuracy and the relationship between volume and the number of lattice points. The conclusion is that the number of lattice points enclosed is proportional to the volume enclosed, explaining why the formula accurately calculates the number of quantum states.
  • #1
JohnnyGui
796
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TL;DR Summary
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?
 
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  • #2
JohnnyGui said:
the density of states formula

What density of states formula? And where are you getting it from? Neither reference you gave gives one.
 
  • #3
JohnnyGui said:
Summary:: 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?
Because the number of lattice points enclosed is proportional to the volume enclosed.

Why do keep asking the same question?
 
  • #4
Thread locked, duplicate question.
 

Related to Accuracy of the Density of States

1. What is the density of states and why is it important?

The density of states is a concept in physics that describes the number of energy states per unit volume available to a particle or system. It is an important quantity in understanding the behavior of materials and their electronic properties.

2. How is the density of states calculated?

The density of states is typically calculated using mathematical models and equations that take into account the energy levels and spatial distribution of particles in a system. It can also be experimentally measured using techniques such as spectroscopy.

3. What factors affect the accuracy of the density of states?

The accuracy of the density of states can be affected by various factors such as the quality of the data used in calculations, the assumptions and approximations made in the mathematical models, and the complexity of the system being studied.

4. How is the accuracy of the density of states verified?

The accuracy of the density of states can be verified through comparison with experimental data or by using different mathematical models and techniques to calculate it. Additionally, the results can be cross-checked with other related properties to ensure consistency.

5. Can the accuracy of the density of states be improved?

Yes, the accuracy of the density of states can be improved by using more sophisticated mathematical models, incorporating more precise experimental data, and considering a wider range of factors that may affect the system. Continuous refinement and validation of the calculations can also lead to improved accuracy.

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