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Let's say I have a Hamiltonian operator $H(x_a..)$, where $x_a,x_b$ are some parameters of my Hamiltonian.
I can calculate the quantum canonical ensemble expectation value of $\frac{\partial H}{\partial x_a}$ (this is also an operator)
$$\left\langle \frac{\partial H}{\partial x_a} \right\rangle=...