Cambridge University Press eBooks

Transition-Probability Spaces and Quantum Systems

Enrico G. Beltrametti, Gianni Cassinelli, Peter Carruthers

Published Dec 28, 1984• 0 citations

Abstract

Transition Probabilities between Pure States In physicists' nomenclature the expression “transition probability” generally refers to some dynamical instability, more specifically to a nonzero probability for the system to make a transition from an initial to a final state. Our use of the term is not directly related to dynamical instabilities; rather we follow von Neumann's terminology, and the transition probability between two states is meant to represent, intuitively, a measure of their overlapping. To visualize this notion in an explicit example consider the states of linear polarization of a photon beam, let a be the polarization state filtered by a Nicol prism N 1 , and let β be the state filtered by a Nicol prism N 2 rotated by an angle ϑ with respect to N 1 ; then the transition probability between a and β is the probability for a photon in state α to pass N 2 , which is empirically known to be cos 2 ϑ (Malus law), and equals the probability for a photon in state β to pass N 1 . In the Hilbert-space formulation of quantum mechanics the idea of transition probability we have in mind is simply the modulus squared of the scalar product (see Sections 2.5 and 9.3).

DOI: 10.1017/cbo9781107340725.024

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