Suppose we have an operator $Q$ with eigenvalue $q$. Expectation value is $\langle Q \rangle$ and dispersion $D(Q) = \sqrt{\langle \left( Q – \langle Q \rangle \right)^2 \rangle} $. I want to find values for the expectation value and dispersion.
Attempt
Expectation value is simply the average value of the observable associated with $Q$ right? Dispersion is the variance of the observable associated with $Q$ right?
$$ \langle Q \rangle = \langle \psi | Q |\psi \rangle = q \langle \psi|\psi \rangle = q $$
$$ D(Q) = \sqrt{\langle \left( Q – \langle Q \rangle \right)^2 \rangle} = \sqrt{ \langle Q^2 -2Q \langle Q\rangle + \langle Q \rangle^2 \rangle } = \sqrt{ \langle Q^2 – 2Qq + q^2 \rangle } $$
Now $\langle Q^2 \rangle = q^2 $ by applying $Q$ twice.
Does this mean that $$D(Q) = 0 $$?
Best Answer
Yes. There's an even easier notation for dispersion, or standard deviation which is
$$D(Q) = \sqrt{ \langle Q^2\rangle - \langle Q\rangle^2 }$$
Both those terms are the same. So $D(Q)$ is zero.