= Variational costate gradient for Hamiltonian quantum control
{title2=$\frac{\delta J}{\delta f(t)}=\frac2\hbar\operatorname{Im}\langle\chi(t)|H_1|\psi(t)\rangle-\lambda f(t)$}
For $J=\langle\psi(T)|A|\psi(T)\rangle-(\lambda/2)\int_0^Tf^2dt$ and $i\hbar\dot\psi=(H_0+fH_1)\psi$, enforce the state equation with $-2\operatorname{Re}\int\langle\chi|\dot\psi+iH\psi/\hbar\rangle dt$. Variation gives the same <Schrödinger equation> for the <costate>, propagated backward from $\chi(T)=A\psi(T)$. Control variation gives the displayed real <gradient>. Forward state propagation, backward <costate> propagation, and a constrained <gradient> update form <direct-adjoint looping>. These are necessary stationarity equations, not a guarantee of a global optimum.
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