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My question is somewhat related to this one. I want to know if

$$ \frac{d}{dk}\left\langle \hat{f}_k \right\rangle_{\psi_k} = \left\langle \frac{d}{dk} \hat{f}_k \right\rangle_{\psi_k} $$

holds for some quantum-number like parameter $k$ and any operator $\hat{f}_k$. I am using the letter $k$ because it might be some variable similar to the crystal momentum in solid state physics. Note that the states $\left|\psi_k\right\rangle$ "depend" on this quantum number. It can be assumed to be continuous to allow for the differentiation to be well defined. It is not an "external parameter", i.e. something about the system that can be changed. Does that change whether or not this holds?

This can of course be rewritten as

$$ \frac{d}{dk}\left\langle \hat{f}_k \right\rangle_{\psi_k} = \left\langle \frac{d}{dk} \hat{f}_k \right\rangle_{\psi_k} + \frac{d\left\langle\psi_k\right|}{dk} \hat{f}_k \left|\psi_k\right\rangle + \left\langle \psi_k\right| \hat{f}_k \frac{d\left| \psi_k\right\rangle}{dk}$$

But I don't know how to continue from there. Is there a way to show this?

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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

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Here is a hint to assist your intuition, before you launch into abstractions and generalizations.

A derivative is $(\hat f_\epsilon -\hat f_0)/\epsilon$ to lowest order in ε.

Take $$ \hat f_\epsilon = \sigma_3 + 3\epsilon \sigma_1 ~. $$

Consider the state $$ |\psi_\epsilon\rangle= (\cos 𝜃−𝜖\sin 𝜃,\sin 𝜃+𝜖\cos 𝜃 )^T, $$ normalized to lowest order in ε.

You are then inspecting the vanishing or not of the expression $$ (\langle \psi_\epsilon | \sigma_3+3\epsilon \sigma_1 |\psi_\epsilon \rangle - \langle \psi_0 | \sigma_3 |\psi_ 0 \rangle ) - \langle \psi_\epsilon | 3\epsilon \sigma_1 |\psi_\epsilon \rangle ~. $$ Hint: $(2\epsilon \cos 𝜃 \sin 𝜃 ) -6\epsilon \cos 𝜃 \sin 𝜃 = -4\epsilon \cos 𝜃 \sin 𝜃 \neq 0 $.

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