For an operator $\hat A$, if $\langle \hat A\rangle=0$, does this mean that $\langle \hat A\rangle\langle\hat A\rangle=0\,?$

Question

Consider an arbitrary operator $\hat A$, let's suppose that $\langle \hat A\rangle=0$.

Does this mean that I conclude that $$\langle \hat A^2\rangle=0\,?$$

I used to think that the answer to this was no, we cannot conclude that the expectation value of the operator squared is zero just because the expectation value of the operator is zero.

However, $\hat A^2=\hat A\hat A$, now since $\langle \hat A\rangle=0$ does this mean that $$\langle \hat A\rangle\langle\hat A\rangle=0\,?$$

Answer 1

You are right that $\langle \hat A \rangle = 0$ does not imply $\langle \hat A^2 \rangle = 0$ but it of course implies $\langle \hat A \rangle^2 = \langle \hat A \rangle\langle \hat A \rangle= 0.$