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 N1, and let β be the state filtered by a Nicol prism N2 rotated by an angle ϑ with respect to N1; then the transition probability between a and β is the probability for a photon in state α to pass N2, which is empirically known to be cos2 ϑ (Malus law), and equals the probability for a photon in state β to pass N1.

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).