Energy Levels in a Crystalline Solid — Question 10

Question 10

An avoided crossing exchanges orbital character while maintaining a finite gap

Two crystal orbitals have bare energies $E_0+\delta$ and $E_0-\delta$, coupled by real t. Use $H=\begin{pmatrix}E_0+\delta&t\\t&E_0-\delta\end{pmatrix}$.

1. Derive the eigenenergies and minimum separation as $\delta$ varies.

2. For $\delta=3t$ and $t>0$, find the splitting and compare it with the uncoupled separation.

Question 10 - Solution

An avoided crossing exchanges orbital character while maintaining a finite gap

(a) Method and calculation

$$\det(H-EI)=(E_0-E)^2-\delta^2-t^2=0.$$$$\boxed{E_\pm=E_0\pm\sqrt{\delta^2+t^2},\quad\Delta E_{\min}=2|t|}.$$

(b) Interpretation and further calculation

$$\delta=3t:\quad\boxed{\Delta E=2\sqrt{10}\,t=6.325t}.$$$$\Delta E_{t=0}=2|\delta|=6t\Rightarrow\boxed{\Delta E/(2|\delta|)=\sqrt{10}/3=1.054}.$$

Physical check

The minimum gap is nonzero for nonzero coupling. Far from the crossing, eigenstates approach their bare-orbital character; near the crossing they mix strongly.