Update tunneling section & adiabatic part

src/adiabatic_proof.md

@@ -189,11 +189,11 @@ c_{n\neq m}(T) &\approx \int _0^T \frac{\overline{\langle \psi_n | \dot H |\psi_
 $$
 In the last line we made use of the fact that the last term of the previous line is oscillating. Since we are only interested in upper bounds, we can approximate it with a constant 1. To get our adiabaticity criterion, we now remember that we need to have $|c_{n\neq m}(T)| \ll 1$, so that
 $$
- \frac{\hbar \overline{\langle \psi_m | \dot H |\psi_n\rangle} }{\overline{E_n - E_m}^2} << 1
+ \frac{\hbar \overline{\langle \psi_m | \dot H |\psi_n\rangle} }{\overline{E_n - E_m}^2} \ll 1
 $$
 or in other words
 $$
- \frac{\overline{\langle \psi_m | \dot H |\psi_n\rangle} }{\overline{E_n - E_m}} << \frac{\overline{E_n - E_m}}{\hbar}\,.
+ \frac{\overline{\langle \psi_m | \dot H |\psi_n\rangle} }{\overline{E_n - E_m}} \ll \frac{\overline{E_n - E_m}}{\hbar}\,.
 $$
 This is a quantitative estimate for the adiabaticity criterion.