# When does $\langle \hat A \hat B \rangle =\langle\hat A \rangle \langle \hat B\rangle$?

In ordinary probability theory the relation: $$E[XY]=E[X]E[Y]$$ holds when $X$ and $Y$ are uncorrelated (or independent). The analogous relation in quantum mechanics is: $$\newcommand{\p}{\frac{\partial #1}{\partial #2}} \newcommand{\f}{\frac{ #1}{ #2}} \newcommand{\l}{\left(} \newcommand{\r}{\right)} \newcommand{\mean}{\langle #1 \rangle}\mean{\hat A \hat B}=\mean{\hat A} \mean{\hat B}$$ my question is when in general does this hold?

## 1 Answer

$$\langle \hat{A} \hat{B} \rangle = \langle \Psi | \hat{A} \hat{B} | \Psi \rangle \\ \langle \hat{A}\rangle \langle \hat{B} \rangle = \langle \Psi | \hat{A} | \Psi \rangle \langle \Psi | \hat{B} | \Psi \rangle$$ That means for the equation to hold, you need $$\hat{A} | \Psi \rangle \langle \Psi |\hat{B} = \hat{A} \hat{B}$$

In general the equation holds if $|\Psi \rangle$ is an eigenstate of both of the operators.

• It is sufficient. I edited my post. – Quantumwhisp Dec 9 '16 at 19:33