The difference between the two expressions $\langle \hat p^2 \rangle_{\psi}$ and $\langle \hat p \rangle_{\psi}^2$ is defined by the squared uncertainty:
$$\Delta p^2 = {\langle \hat p^2 \rangle_{\psi} – \langle \hat p \rangle_{\psi}^2}$$.
On the other hand, the variance of a random variable is
$$var(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2.$$
How can one represent the uncertainty on the momentum operator through a variance?
Best Answer
Defining the operator $$\Delta A\equiv A-\langle A\rangle$$ the dispersion of an operator A is $$\langle(\Delta A)^{2}\rangle\equiv \langle A^{2}\rangle-\langle A\rangle^{2}$$ The expectation value $\langle(\Delta A)^{2}\rangle$ is called the variance of the operator A. The quantity you call the "squared uncertainty" is actually the variance of the momentum operator.