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What is expectation values of this anti-commutator $\langle \{ \Delta \hat x,\Delta \hat p\} \rangle~?$

where the $\Delta \hat p=\hat p-\langle \hat p \rangle$ and $\hat p$ is momentum operator and so $\Delta \hat x=\hat x-\langle \hat x \rangle$ and $\hat x$ is position operator operator.

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This probably isn't needed anymore, but I thought I'd give it a shot. First, $$ \left\{x,p\right\} = \left[x,p\right] + 2 p x = i \hbar + 2 p x. $$ So, $$ \begin{eqnarray} \left<\left\{\Delta x , \Delta p \right\}\right> &=& \left<\left\{x-\left<x\right>,p-\left<p\right>\right\}\right> \\ &=& \left<\left\{x,p\right\} - \left\{\left<x\right>,p\right\} - \left\{x,\left<p\right>\right\} + \left\{\left<x\right>,\left<p\right>\right\}\right> \\ &=& \left<\left\{x,p\right\}\right> - 2 \left<x\right> \left<p\right> - 2 \left<x\right> \left<p\right> + 2\left<x\right>\left<p\right> \\ &=& i \hbar + 2 \left(\left<p x\right> - \left<x\right> \left<p\right>\right) \\ &=& i \hbar + 2 \ cov\left(p,x\right) \end{eqnarray} $$ where the covariance of $p$ and $x$ is used in the last line.

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  • $\begingroup$ @CDCM Yes. Where do you see the commutator not equal to $i \hbar$? $\endgroup$ – Eric Angle Oct 8 '17 at 14:32
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Because of the relations on $x$ and $p$ being for commutators, your anticommutator does not simplify to anything trivial, and is therefore a physically meaningful operator - i.e. the correlation between $x$ and $p$. To get anything more from it you need to specify more information. What are you using this for? where did it come up? in what context? Without this kind of information it is unlikely that you will get a useful answer.

If you're only looking for a place to start looking into it (in which case you should specify it in the question and not phrase it like a specific query) then I recommend you look at squeezed states for the harmonic oscillator. For a symmetric state like the oscillator ground, number, or coherent states, the anticommutator you mention will vanish, but for squeezed states where the squeezing axes do not coincide with the $x$ and $p$ axes of phase space then it will not vanish and give a good measure of how squeezed the state is and in which direction.

diagram of vacuum state and (non-symmetrically) squeezed state in phase space

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