Calculus and perturbing expectation values

Consider the following quantity: $$f_O(|\psi\rangle) = \langle \psi | O | \psi \rangle$$

How would we study a perturbation on $$|\psi\rangle$$, given that it has to be a valid quantum state? What mathematical tools underlie this quantity?

For example, we might be interested in understanding whether there are specific $$|\psi\rangle$$ with high sensitivity to disruption.

• you want to preserve normalization, thus a perturbation $\delta\psi$ needs to satisfy $\langle\delta\psi|\psi\rangle=0$. For the expval you just get, assuming Hermitianity, $2\langle\delta\psi|O|\psi\rangle$, so eg eigenstates of $O$ are "insensitive to perturbations" in the first order. Is this what you're asking?
– glS
Commented Apr 4, 2023 at 19:33
• @glS why is it that a perturbation must satisfy orthogonality? Also, I believe I'd be interested more in the sensitivity of f_O given inputs |\psi> vs |\psi'> Commented Apr 4, 2023 at 23:27
• because you need to preserve the normalisation $\langle\psi|\psi\rangle=1$. It's like saying that tangent vectors in a hypersphere attached to $v$ are orthogonal to $v$. I don't really understand the other question. You defined $f_O$ as having a single input. What do you mean with sensitivity "given inputs $|\psi\rangle$ vs $|\psi'\rangle$"?
– glS
Commented Apr 5, 2023 at 0:00