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what is difference between these two expectation values $\langle \hat A \hat B\rangle$ and $\langle \hat B \hat A\rangle$? where the $\hat B$ and $\hat A$ are two operators.

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What are $\hat A$ and $\hat B$ in this case? If they do not commute, then the operator $\hat A \hat B$ and it's opposite aren't Hermitian in general, in which case it may not make sense to talk of their "expectation values" per se. – Niel de Beaudrap Jun 15 '12 at 17:27
up vote 2 down vote accepted

$$\langle \hat{A}\hat{B} \rangle -\langle \hat{B}\hat{A} \rangle = \langle \hat{A}\hat{B}-\hat{B}\hat{A} \rangle$$

So it is simply the expectation of the commutator.

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Note that this difference is the basis of Heisenberg's uncertainty relations. – Arnold Neumaier Jun 15 '12 at 19:02

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