Independent and interacting electron models

In summary, the reason we can treat electrons as independent and non-interacting in solid state physics is because of Landau's adiabatic switching on procedure, where the eigenstates of a non-interacting system evolve into the eigenstates of an interacting system. In this process, the system is described in terms of quasiparticles which have the same quantum numbers as the non-interacting electrons. However, this is not always the case and can depend on certain assumptions. This is known as a "normal" Fermi liquid.
  • #1
Nusc
760
2
Why is it that we can treat electrons as independent and non-interacting in solid state physics?

What is the difference between mean-field theory and density functional theory?
In lame terms please.
 
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  • #2
Okay forget lame terms.
 
  • #3
Nusc said:
Why is it that we can treat electrons as independent and non-interacting in solid state physics?

Suppose you're considering a system of noninteracting fermions (e.g., electrons with no coulomb interactions), called a Fermi Gas. Since this system is non-interacting, it's possible to solve for the energy spectrum of this system. In a real system, of course, the electrons are interacting, so imagine that you started with this Fermi gas system and slowly turned on the coulomb interaction. In doing so, the eigenstates of the Fermi Gas system (that is, the various characteristic energy states of the system) will evolve adiabatically into the eigenstates of the interacting system. So, there is a mapping from the eigenstates of the non-interacting system to the eigenstates of the interacting system. Strictly speaking, then, electrons themselves aren't actually treated as independent and non-interacting - the system is actually described in terms of "quasiparticles", elementary excitations of the system with the same quantum numbers as the non-interacting electrons, which may be treated as non-interacting. The quasiparticle description is obtained through this adiabatic evolution of eigenstates.

This process is called "Landau's adiabatic switching on procedure". There are some caveats I didn't mention, mainly because I don't remember them all but also because they wouldn't be too enlightening as part of a basic description. If I recall correctly this switching-on procedure isn't strictly necessary, but is the fastest way to get to the quasiparticle description of the system.
 
  • #4
Mute said:
Suppose you're considering a system of noninteracting fermions (e.g., electrons with no coulomb interactions), called a Fermi Gas. Since this system is non-interacting, it's possible to solve for the energy spectrum of this system. In a real system, of course, the electrons are interacting, so imagine that you started with this Fermi gas system and slowly turned on the coulomb interaction. In doing so, the eigenstates of the Fermi Gas system (that is, the various characteristic energy states of the system) will evolve adiabatically into the eigenstates of the interacting system.

Sometimes this is the case, and sometime it is not. this is actually an assumption, which does not always turn out to be true. If the assumption is true then we say we are dealing with a "normal" Fermi liquid.

So, there is a mapping from the eigenstates of the non-interacting system to the eigenstates of the interacting system. Strictly speaking, then, electrons themselves aren't actually treated as independent and non-interacting - the system is actually described in terms of "quasiparticles", elementary excitations of the system with the same quantum numbers as the non-interacting electrons, which may be treated as non-interacting. The quasiparticle description is obtained through this adiabatic evolution of eigenstates.

This process is called "Landau's adiabatic switching on procedure". There are some caveats I didn't mention, mainly because I don't remember them all but also because they wouldn't be too enlightening as part of a basic description. If I recall correctly this switching-on procedure isn't strictly necessary, but is the fastest way to get to the quasiparticle description of the system.
 
  • #5
olgranpappy said:
Sometimes this is the case, and sometime it is not. this is actually an assumption, which does not always turn out to be true. If the assumption is true then we say we are dealing with a "normal" Fermi liquid.

Yes, this is one of the caveats I didn't mention, since I don't know much about non-normal Fermi Liquids.
 

1. What are independent electron models?

Independent electron models are theoretical models used in condensed matter physics to describe the behavior of electrons in a material. These models assume that each electron in the system behaves independently of the others, meaning that their interactions with each other and with the lattice of the material are negligible.

2. How do independent electron models differ from interacting electron models?

Unlike independent electron models, interacting electron models take into account the interactions between electrons and the effects of the lattice in a material. This means that they can provide a more accurate description of the electronic properties of a material, but they are also more complex and difficult to solve mathematically.

3. What are some examples of materials that can be described by independent electron models?

Independent electron models are commonly used to describe the behavior of electrons in metals, where the electrons are relatively free to move and interact with each other. They are also used to study semiconductors and insulators, but their accuracy in these materials is limited.

4. How do independent electron models contribute to our understanding of materials?

Although independent electron models may not provide a completely accurate picture of the electronic properties of a material, they are still useful in helping us understand the behavior of electrons in different types of materials. They provide a baseline for comparison with more complex models and can help identify important factors that may be neglected in independent electron models.

5. What are the limitations of independent electron models?

Independent electron models have several limitations, including the assumption that electrons behave independently, which is not always the case. They also do not take into account factors such as electron-electron interactions, which can be significant in certain materials. Additionally, these models do not accurately describe the behavior of electrons at very low temperatures or in highly correlated systems.

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