Nonlinear Schrödinger Dynamics and Nonlinear Observables

It is is explained why physical consistency requires substituting linear observables by nonlinear ones for quantum systems with nonlinear time evolution of pure states. The exact meaning and the concrete physical interpretation are described in full detail for a special case of the nonlinear Doebner-Goldin equation. 1 Schrödinger dynamics By Schrödinger dynamics we mean a strongly continuous two-parameter family of mappings β̂t2,t1 of some Hilbert space H of (pure) states onto itself, defined for t1, t2 ∈ IR and satisfying the following conditions: β̂t,t(Ψ) = Ψ ∀ t ∈ IR , Ψ ∈ H , (1) β̂t3,t2 ( β̂t2,t1(Ψ) ) = β̂t3,t1(Ψ) ∀ t1, t2, t3 ∈ IR , Ψ ∈ H \ {0} , (2) β̂t2,t1(cΨ) = cβ̂t2,t1(Ψ) ∀ t1, t2 ∈ IR , c ∈ C , Ψ ∈ H \ {0} . (3) In order to be consistent with the standard (nonrelativistic) interpretation ρΨt def = |Ψt| 2 = probability density for particle position at time t , (4)