RobertsonSchrödinger

RobertsonSchrödinger, commonly referred to as the Robertson–Schrödinger relation, is a generalized form of the quantum uncertainty principle that extends Heisenberg’s bound to any pair of observables. It is named for Howard P. Robertson, who established a general uncertainty bound in 1936, and Erwin Schrödinger, who contributed a refinement that includes correlations between observables.

For two observables A and B represented by Hermitian operators acting on a quantum state |ψ⟩, define ΔA = A − ⟨A⟩ and ΔB = B − ⟨B⟩, where ⟨·⟩ denotes the expectation value in |ψ⟩. The Robertson–Schrödinger inequality states that

ΔA^2 ΔB^2 ≥ (1/4)|⟨[A,B]⟩|^2 + (1/4)|⟨{ΔA,ΔB}⟩|^2,

where [A,B] is the commutator and {ΔA,ΔB} is the anticommutator of the deviations. The first term on the right-hand side encodes the fundamental limit imposed by the noncommutativity of A and B, while the second term reflects the covariance or correlation between A and B in the state |ψ⟩.

In the special case where the covariance term vanishes, the relation reduces to the Robertson form ΔA ΔB ≥ (1/2)|⟨[A,B]⟩|. The inequality applies broadly in quantum mechanics and underpins analyses in quantum optics, quantum information, and precision metrology, including squeezing and entanglement criteria for continuous-variable systems. It is sometimes referred to as the Schrödinger uncertainty relation.