Kittel, Chapter 7, Central Equation

[DrBrainDead]

Homework Statement

Kittel states at page 172, that the central equation for the Fourier components of waves in a periodic lattice is given by:
($$\lambda$$_k-E)*C(k)+$$\sum$$U_G*C(k-G)=0

From this he goes on to say "Once we determine the C's from (27), the wavefunctions (25) is given as:

$$\psi$$_k(x)=$$\sum$$C(k-G)*exp(i(k-G)x)

I'm completely in the blank about how he got the C's isolated and got the above equation for the wavefunction..any help?

Homework Equations

Equation 27:
($$\lambda$$_k-E)*C(k)+$$\sum$$U_G*C(k-G)=0

Equation 25:
$$\psi$$(x)=$$\sum$$C(k)*exp(i(k)x)

 

[nasu]

He did not actually get the coefficients.
He even tells you that solving the equation is a very difficult task in general.
He tells what can be done after the equation is solved.

About how he gets the wave-function: he assumes that it can be written in the form (25), as a combination of plane waves, with various coefficient. (This form is like a Fourier expansion)
To find the coefficients, you plug in the function in Schrödinger eq (the one with Bloch functions) and you get some equations for the coefficients.

This is actually a quite general method of solving differential equations, by using Fourier series. Is not something specific for solid state or Bloch functions.