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


$$\langle \hat{A} \hat{B} \rangle = \langle \Psi | \hat{A} \hat{B} | \Psi \rangle \\ \langle \hat{A}\rangle \langle \hat{B} \rangle = \langle \Psi | \hat{A} | \Psi \rangle \langle \Psi | \hat{B} | \Psi \rangle$$ 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.