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Because $\hat{X}-i\lambda \hat{Y}$ is the adjoint of $\hat{X}+i\lambda \hat{Y}$,Expanding gives the real quadraticIts discriminant must be nonpositive. Since the expectation of a commutator of Hermitian operators is purely imaginary,Taking $\hat{X}=\hat{A}-⟨\hat{A}⟩$ and $\hat{Y}=\hat{B}-⟨\hat{B}⟩$ and then taking square roots gives the stated uncertainty relation. This is the quadratic-norm proof of the Heisenberg uncertainty relation.

Positivity of the squared norm of the supplied state gives, for every real $s$,where the stated zero means were used in the last line. This quadratic in $s$ is nonnegative for every real $s$, so its discriminant is nonpositive:This is the quadratic-norm proof of the Heisenberg uncertainty relation, and proves