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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.

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    $\begingroup$ 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? $\endgroup$
    – glS
    Commented Apr 4, 2023 at 19:33
  • $\begingroup$ @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'> $\endgroup$
    – C. Kang
    Commented Apr 4, 2023 at 23:27
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    $\begingroup$ 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 $|\psi'\rangle$"? $\endgroup$
    – glS
    Commented Apr 5, 2023 at 0:00

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