Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 1 / 4A / Solution

An operator $Q$ is Hermitian whenon its domain. Hermitian operators represent quantum observables because their expectation values and eigenvalues are real. In a normalized state,

Positivity of the norm of $(Q+i\lambda P)\psi$ for every real $\lambda$ givesThe middle expectation is real because $i[Q,P]$ is Hermitian. The discriminant of this quadratic is nonpositive, soApply this to the centered operators $Q-⟨Q{⟩}_{\psi }$ and $P-⟨P{⟩}_{\psi }$, whose commutator is still $[Q,P]$. This yields the Robertson uncertainty principleNoncommuting observables therefore cannot both have arbitrarily sharply concentrated measurement distributions in the same state.