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i've problem with the last terms of uncertainty principle: 1. are these equalities true: $$\langle \{\Delta \hat A,\Delta \hat B\} \rangle=\langle \{\hat A, \hat B \}\rangle$$ $$\langle [\Delta \hat A,\Delta \hat B] \rangle=\langle [\hat A, \hat B] \rangle.$$

  1. what is result of product commutator with anti commutator?
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If you defined $\Delta \hat{A} = \hat{A} - \langle A \rangle$, then yes, the equalities are true... just use the associativity of the sum; the second question it's some kind of something that is not trivial, and I don't if it really has a physical meaning. –  Rodrigo Thomas Jun 16 '12 at 11:44

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If you defined $\Delta \hat A=\hat A -\langle \hat A \rangle$ and then $\Delta \hat B=\hat B -\langle \hat B \rangle$ no , one of the equalities is not true!. $$\langle \{\Delta \hat A,\Delta \hat B\} \rangle\not=\langle \{\hat A, \hat B \}\rangle$$.

but this one is true:

$$\langle [\Delta \hat A,\Delta \hat B] \rangle=\langle [\hat A, \hat B] \rangle.$$

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