We discuss the emergence of logarithmic Sobolev inequalities from energy/entropy inequalities and then derive from them the existence and uniqueness of the ground state of Hamiltonians as well the spectral gap. The method is an infinitesimal extension of the one introduced by Len Gross in case the ground state is a probability or a trace and is based on the monotonicity of the relative entropy. Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page