Properties of expectation values of quantum operators [closed]

Question

$$\langle \hat A \rangle \langle \hat B \rangle=\langle \hat A\hat B \rangle,$$

$$\langle \hat A \rangle + \langle \hat B \rangle=\langle \hat A + \hat B \rangle,$$

$$\langle \hat A^2 \rangle \langle \hat B^2 \rangle=\langle \hat A^2 \hat B^2 \rangle,$$

$$\langle \hat A^2 \rangle + \langle \hat B^2 \rangle=\langle \hat A^2 + \hat B^2 \rangle,$$ Which one of them is not true!?

Answer 1

The second and the fourth formula are true if $\langle\dots \rangle$ is linear. So most likely, these statements are true in your context.

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For the other two, consider $\hat B:=\hat A$ and notice that $\langle \hat A \rangle^2 =\langle \hat A^2 \rangle$ is not true (there is a thread about this somewhere).

Or consider $\hat B:=\hat A^{-1}$, then $\hat A\hat B$ becomes the unity on the right hand side, but you still have something to calculate on the left hand side. E.g. take a $2$x$2$ diagonal matrix with one big and one small entry in a state $\tfrac{1}{\sqrt{2}}(1,1)$ and see that the left hand side is not $1$ but will depend on the matrix entries.