fix hopping sign in 8.2.2 solutions

docs/8_many_atoms_solutions.md

@@ -73,8 +73,8 @@ The band gap is $\Delta\omega = \omega_+-\omega_-$
 ### Exercise 2: Analyzing the LCAO chain with alternating hoppings
 
 1. The unit cell contains exactly two atoms. By looking at the figure, we can immediately write down the Schrodinger equation relating the probability amplitudes of the atomic orbitals:
-$$ E \phi_n = \varepsilon_0 \phi_n - t_1 \psi_n - t_2 \psi_{n-1} $$
-$$ E \psi_n = \varepsilon_0 \psi_n - t_1 \phi_n - t_2 \phi_{n+1} $$
+$$ E \phi_n = \varepsilon_0 \phi_n + t_1 \psi_n + t_2 \psi_{n-1} $$
+$$ E \psi_n = \varepsilon_0 \psi_n + t_1 \phi_n + t_2 \phi_{n+1} $$
 
 2. This follows directly from substituting the Ansatz