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commutators_and_uncertainty_relations [2022/10/06 00:59] – [Properties of Commutators] admincommutators_and_uncertainty_relations [2022/10/06 01:02] (current) – [Uncertainty Relations Between Operators] admin
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 Since $\braket{\psi^{\perp}_B}{\psi^{\perp}_A}$ is the complex conjugate of $\braket{\psi^{\perp}_A}{\psi^{\perp}_B}$, we can rewrite this as Since $\braket{\psi^{\perp}_B}{\psi^{\perp}_A}$ is the complex conjugate of $\braket{\psi^{\perp}_A}{\psi^{\perp}_B}$, we can rewrite this as
 \[ \[
-  \Expect{[\hat{A},\hat{B}]} = 2 \Delta A \Delta B \mathrm{Im} \left ( \braket{\psi^{\perp}_A}{\psi^{\perp}_B} \right ). +  \Expect{[\hat{A},\hat{B}]} = 2i \Delta A \Delta B \mathrm{Im} \left ( \braket{\psi^{\perp}_A}{\psi^{\perp}_B} \right ). 
 \] \]