Central Equation Derivation: NxN Matrix?

[semc]

I went through the derivation for the central equation

$$(\lambda_k - \epsilon)C_k + \Sigma_G U_G C_{k-G} = 0$$

and everywhere I look everybody just says this is a NxN matrix. I don't see how this is true. Isn't this just a one line equation with summation across all reciprocal lattice? I don't understand why at one value of k we get NxN matrix. Any help is greatly appreciated!

 

[Lord Jestocost]

Maybe, this will be of help:
[PDF]Lecture 12 - MIT OpenCourseWare

 

[semc]

Yeah this is what I found but the central equation is
$$ \frac{\hbar^2k^2}{2m} C_k + \sum_{G}^{} V_GC_{k-G}=EC_k$$
so shouldn't I just have
$$ \frac{\hbar^2k^2}{2m} C_k + V_0C_{k-g} + V_0C_{k+g}=EC_k$$
for G=ng and $$ V=V_0e^{igx} +V_0e^{-igx} ?$$
How do you get the other equations
$$ \frac{\hbar^2(k-g)^2}{2m} C_{k-g} + V_0C_{k-2g} + V_0C_{k}=EC_{k-g}$$

 

[Lord Jestocost]

The "central equation" is a set of equations. For a fixed k in the first Brillouin zone, this set of equations for all reciprocal lattice vectors G couples those coefficients C_k , C_k−G , C_k−G' , C_k−G'' ,... whose wave vector differ from k by a reciprocal vector. Equation (107) in [1] which follows the presentation in the textbook “Solid State Physics” by Neil W. Ashcroft and N. David Mermin should illustrate the meaning of the "central equation". Have a look at [2], too.

[1] [PDF]Introduction to Solid State Physics
[2] [PDF]7.6 The Schrödinger equation of electron in a periodic potential

 

[semc]

Well I read the book by ashcroft and mermin and [2] before. Everybody just says that the C(k) couples to C(k+G). Are they referring to the coupling via V₀? I really don't see where the other equations come from

 

[semc]

I guess I am asking why is C(k) a vector? Isn't C(k) just the Fourier coefficient of the wavefunction?

 

[Lord Jestocost]

I guess I am asking why is C(k) a vector?

C(k) isn't a vector. You simply write the set of equations which have to be solved in a matrix form.