About weak potential (solid state)

[armandowww]

In solid state physics, a proof of bearing energy gaps in one-dimesional dispersion relations is given by a quantum-mechanical perturbative approach applied to the level belonging to two bands at once, that is twice degenerate. For example for k=pi/a there's one of these levels between n=0 band and n=-1 band.
The method is able to make visible the reason for which lower level is going lower again and upper, upper itself.
I read somewhere that passing to tridimensional problem is not clever, not for a computational subject, but for the intrinsic long range property of electron-electron and electron-ion interactions. They go like 1/x.
Now, here's my question: Should I deal with Fourier trasforms to reach some kind of a comprehension of this inability? They should go like 1/k^2, too slow to compensate in k-space (i.e. N-space) the fast growth of sites number: the volume of infinitesimal spherical shell N^2dN. If so, where's the cause?
Thanks for your paid attention.

 

[BigJDubs]

The inability to pass to a three-dimensional problem is due to the fact that the long-range nature of electron-electron and electron-ion interactions make it difficult to accurately calculate the energy gaps in a three-dimensional system. Even with perturbative methods, the energy gaps cannot be accurately calculated for three-dimensional systems due to the fact that the interactions become less important as the distance between particles increases. The interactions also become less important as the wave vector magnitude increases, meaning that the energy gaps cannot accurately be calculated using perturbative methods for higher wave vector magnitudes. Therefore, when dealing with the energy gaps in three-dimensional systems, Fourier transforms may be used to effectively calculate the energy gaps at lower wave vector magnitudes.

 

[charng]

I would like to provide a response to the content you have shared about the weak potential in solid state physics. It seems that you are discussing a specific method used to study the energy gaps in one-dimensional dispersion relations, where the levels belonging to two bands are considered to be twice degenerate. This method is able to explain why the lower level goes lower and the upper level goes higher.

You also mention that this method may not be suitable for studying three-dimensional problems due to the long-range nature of interactions between electrons and ions. These interactions follow a 1/x relationship, which is much slower than the 1/k^2 relationship observed in Fourier transforms. This could potentially lead to difficulties in understanding the underlying cause.

In order to fully comprehend this inability, it may be necessary to use Fourier transforms to gain a better understanding of the situation. However, it is important to note that Fourier transforms alone may not be sufficient to explain the cause of this limitation. Further research and analysis may be required to fully understand the underlying factors at play.

Thank you for bringing this topic to my attention and I appreciate your interest in this area of study. it is important to continually question and explore the limitations of our current methods and strive to find new and improved ways of understanding the natural world.