Seminar of Department of Theoretical Physics - Konrad Szymański (28.4.2026)
Tuesday 28.4.2026 at 14:00, Lecture room F2/125
Konrad Szymański (RCQI FÚ SAV):
Geometry of expectation values of non-commuting observables
Abstract:
Quantum-mechanical expectation values have a simple definition, analogous to the one in standard probability theory: a sum of values weighted by the probabilities of observing each outcome. Naturally, the expectation value is a function of state; and if several observables X1,...,Xk are measured, to each state there corresponds a vector (⟨X1⟩,...,⟨Xk⟩) of expectation values. Considering the set of all such vectors, taken over all quantum states, leads naturally to the study of their geometry; these sets are known as numerical ranges. Their geometric properties reveal peculiar features of quantum mechanics; in particular, the fact that the underlying probabilities cannot arise from a classical random process. In this talk I will give an elementary introduction to the geometry of numerical ranges, present examples for simple quantum-mechanical systems, and show how uncertainty relations can be understood through the lens of this formalism