= Solution
For a normalized <wavefunction> in the required operator domains and a <self-adjoint operator> $A$, its <expectation value> and <standard deviation> are
$$
\langle A\rangle_\psi=\int_{\mathbb R^3}\psi^*(\mathbf x)(A\psi)(\mathbf x)\,d^3x,
$$
$$
\Delta_\psi A=\left[\int_{\mathbb R^3}\psi^*(A-\langle A\rangle_\psi)^2\psi\,d^3x\right]^{1/2}.
$$
Equivalently the squared uncertainty is $\|(A-\langle A\rangle_\psi)\psi\|^2$, which is visibly nonnegative. Since the <expectation value> of a <self-adjoint operator> is real, expansion of the square and $\int|\psi|^2=1$ give
$$
\boxed{\Delta_\psi A=\sqrt{\langle A^2\rangle_\psi-\langle A\rangle_\psi^2}.}
$$
For a state not normalized to one, divide each defining expectation integral by $\int|\psi|^2$ before using this formula.
Back to article page
