能带拓扑性质

考虑一维简单晶格, 晶格常数为 $a$. 则哈密顿量为

$$ \hat{H} = \frac{\hat{p}^{2}}{2m} + U(x), (U(x + na) = U(x)) $$

我们可以将波函数展开为单原子波函数的线性组合:

$$ \hat{H}_{0} = \frac{\hat{p}^{2}}{2m} + V(x), \hat{H}_{0}\phi(x) = E_{0}\phi(x)\\ \psi(x) = \sum_{n}a_{n}\phi_{n}(a_{n} = \frac{1}{\sqrt{N}}e^{ikx_{n}}) $$

引入

$$ \Delta U(x) = U(x) - V(x) $$

则原本的薛定谔方程变为

$$ \hat{H}\psi = (\hat{H}_{0} + \Delta U)\psi = E\psi $$

则

$$ \sum_{n}\langle\phi_{m}|\Delta U(x-x_{0})a_{n}|\phi_{n}\rangle = (E - E_{0})a_{m} $$