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Physics

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    $\begingroup$ Expand $|\psi_k\rangle$ in eigenstates of $\hat f$ and recall first order perturbation theory. $\endgroup$ Commented Jul 7, 2020 at 13:29
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    $\begingroup$ @CosmasZachos I can see where the expansion may help, but not how perturbation theory helps.. $\endgroup$ Commented Jul 7, 2020 at 14:05
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    $\begingroup$ The derivative around a point is essentially a first order expansion in the excursion around that point, no? $\endgroup$ Commented Jul 7, 2020 at 14:09
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    $\begingroup$ Hint: before meaningless abstraction, experiment with $\hat f= \sigma_3 + \epsilon \sigma_1$ in a couple of states. Try $|\psi\rangle = (\cos\theta-\epsilon \sin\theta, \sin\theta+ \epsilon \cos\theta)^T$. Does your conjecture hold? $\endgroup$ Commented Jul 7, 2020 at 15:07
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    $\begingroup$ Your second equation is unexceptional and trivial. Your first is a conjecture. Try any and all wave functions. I gave you a simple one. $\endgroup$ Commented Jul 7, 2020 at 15:37