# What does $A - \langle A \rangle$ mean?

I've seen the uncertainty of $$A$$ written as $$(\Delta A)^2 = \langle (A - \langle A \rangle)^2 \rangle.$$ But what does this even mean since $$A$$ is an operator and $$\langle A \rangle$$ is a scalar. How can we compute the difference of the two?

For any vector $$v$$ and scalar $$\alpha$$, we have $$\alpha v = \alpha I v$$, so multiplication by a scalar $$\alpha$$ behaves the same way as the linear operator $$\alpha I$$. Therefore, we interpret $$A - \langle A \rangle$$ as $$A - \langle A \rangle I$$.

Note that

$$\big(Z - \langle Z \rangle \big)^2 = Z^2 - 2Z \langle Z \rangle + \langle Z \rangle^2$$

And therefore $$\langle \big(Z - \langle Z \rangle \big)^2 \rangle = \langle Z^2 - 2Z \langle Z \rangle + \langle Z \rangle^2 \rangle$$

Then by linearity, we have

\begin{align} \langle Z^2 - 2Z \langle Z \rangle + \langle Z \rangle^2 \rangle &= \langle Z^2 \rangle - \langle 2Z \langle Z \rangle \rangle + \langle \langle Z \rangle^2 \rangle \\ &= \langle Z^2 \rangle - (2\langle Z \rangle) \langle Z \rangle + \langle Z \rangle^2 \\ &= \langle Z^2 \rangle - 2 \langle Z \rangle^2 + \langle Z\rangle^2 \\ &= \langle Z^2 \rangle - \langle Z \rangle^2 \end{align}

So therefore,

$$(\Delta Z)^2 = \big(Z - \langle Z \rangle \big)^2 = \langle Z^2 \rangle - \langle Z \rangle^2$$

Which is what you see if you search up Variance