MSW effect ($2E\sqrt{2}{G}_F{N}_e=\Delta m^2\cos 2\theta$)

Matter changes the effective mixing angle and level splitting of propagating neutrinos. The electron-flavour charged current potential is $\sqrt{2}{G}_F{N}_e$ relative to the other active flavours. In the two-flavour Hamiltonian, equality of its two diagonal entries gives the displayed resonance condition. Adiabatic evolution through a varying density can therefore cause a large flavour change even when the local vacuum mixing angle is smaller. The equality alone does not guarantee adiabatic propagation.