Independent and interacting electron models

[Nusc]

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.

 

[Nusc]

Okay forget lame terms.

 

[Mute]

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.

 

[olgranpappy]

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.

 

[Mute]

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.