 Solution

Set $u=A\psi$, $v=B\psi$ and $z=⟨u,v⟩$. The self-adjointness of $A,B$ impliesThe Cauchy-Schwarz inequality followed by $|z{|}^2\ge (\mathrm{Im}z{)}^2$ givesBoth inequalities must be equalities. The exact conditions are $A\psi$ and $B\psi$ are linearly dependent, and $\mathrm{Re}⟨A\psi ,B\psi ⟩=0$. Equivalently, equality holds if either vector is zero; when both are nonzero it holds precisely whenThe zero-vector cases matter: they saturate the inequality but cannot all be expressed by solving for $B\psi$ as a multiple of $A\psi$. The real-part condition can also be written ${\mathbb{E}}_{\psi }(AB+BA)=0$, using the anticommutator. This is the uncentered second-moment commutator inequality.