# 2D problem of nearly free electron model

Tags:
1. May 16, 2015

### unscientific

1. The problem statement, all variables and given/known data

(a) Find energies of states at $(\frac{\pi}{a},0)$.
(b) Find secular equation

2. Relevant equations

3. The attempt at a solution

Part(a)

In 1D, the secular equation for energy is:
$$E = \epsilon_0 \pm \left| V(x,y) \right|$$

When represented in complex notation, the potential becomes
$$V(x,y) = V_{10} \left[ e^{i\frac{2\pi x}{a}} + e^{-i\frac{2\pi x}{a}} + e^{i\frac{2\pi y}{a}} + e^{-i\frac{2\pi y}{a}} \right] + V_{11} \left[ e^{i\frac{2\pi x}{a}} + e^{-i\frac{2\pi x}{a}} \right] \left[ e^{i\frac{2\pi y}{a}} + e^{-i\frac{2\pi y}{a}} \right]$$

$$E = \epsilon_0 \pm \sqrt{V_{10}^2 + V_{11}^2 }$$

Part(b)
I know the central equation is given by
$$\left(\epsilon_0 - E \right) C_{(k)} + \sum\limits_{G} U_G ~ C_{(k-G)} = 0$$

How do I find the 4x4 matrix?

2. May 21, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. May 23, 2015

### unscientific

bumpp

4. May 31, 2015

bumpp