one defines their commutator as In the case of position and momentum, the commutator is the canonical commutation relation The physical meaning of the non-commutativity can be understood by considering the effect of the commutator on position and momentum eigenstates. Let be a right eigenstate of position with a constant eigenvalue . By definition, this means that Applying the commutator to yields where is the identity operator. Suppose, for the sake of proof by contradiction, that is also a right eigenstate of momentum, with constant eigenvalue . If this were true, then one could write On the other hand, the above canonical commutation relation requires that This implies that no quantum state can simultaneously be both a position and a momentum eigenstate. When a state is measured, it is projected onto an eigenstate in the basis of the relevant observable. For example, if a particle's position is measured, then the state amounts to a position eigenstate. This means that the state is not a momentum eigenstate, however, but rather it can be represented as a sum of multiple momentum basis eigenstates. In other words, the momentum must be less precise. This precision may be quantified by the standard deviations, As in the wave mechanics interpretation above, one sees a tradeoff between the respective precisions of the two, quantified by the uncertainty principle. Heisenberg limit In quantum metrology, and especially interferometry, the Heisenberg limit is the optimal rate at which the accuracy of a measurement can scale with the energy used in the measurement. Typically, this is the measurement of a phase (applied to one arm of a beam-splitter) and the energy is given by the number of photons used in an interferometer. Although some claim to have broken the Heisenberg limit, this reflects disagreement on the definition of the scaling resource. Suitably defined, the Heisenberg limit is a consequence of the basic principles of quantum mechanics and cannot be beaten, although the weak Heisenberg limit can be beaten. Robertson–Schrödinger uncertainty relations The most common general form of the uncertainty principle is the Robertson uncertainty relation. For an arbitrary Hermitian operator we can associate a standard deviation where the brackets indicate an expectation value. For a pair of operators and , we may define their commutator as In this notation, the Robertson uncertainty relation is given by The Robertson uncertainty relation immediately follows from a slightly stronger inequality, the Schrödinger uncertainty