Energy gaps for quasi-free electrons in a 2D lattice

[Adele]

Homework Statement

Solid state

Relevant Equations

V(r) = exp{-|r|/b}

Hi!
Situation: quasi-free electron in a 2D lattice, considering atomic potential V(r) = exp{-|r|/b} (r is the distance from the atom)
I'm trying to compute the first five energy gaps at point (10),
firstly I don't understand the meaning of calculated 5 energy gaps at one point and usually we use E= 2|Vg| where Vg s are the Fourier components but here... how?Thanks for the attention and if you have some ideas

 

[Nate810]

I will be glad to hear them!The energy gaps you are referring to are the gaps between the energy levels of the quasi-free electron in the 2D lattice. They are typically referred to as the quantum size effect (QSE) gaps or the energy level spacings of the electron in the lattice. To calculate these energy gaps, you first need to calculate the wavefunction and energy of the electron in the lattice. This can be done by solving the Schrödinger equation for the electron, with the atomic potential V(r) = exp{-|r|/b} taken into account. This will give you the ground state energy E0 and the wavefunction Ψ0. You can then calculate the energy of the first excited state E1 using perturbation theory. This will give you the gap between the ground state and the first excited state, which is the first energy gap. Similarly, you can calculate the energy of the higher excited states and the corresponding energy gaps.Once you have calculated the energy gaps, you can use them to estimate the point (10) potential (Vg). This is done by taking the absolute value of the gap between the ground state and the first excited state (|E1-E0|), and then multiplying it by two: Vg = 2|E1-E0|. This is an approximation, but it should give you a good estimate of the potential at point (10).