Schrödinger Equation - From Linear and Nonlinear Perspectives [Working Title]

Jochen Pade

Angelo Bassi, Detlef Dürr, Günter Hinrichs

The linearity of quantum mechanics leads, under the assumption that the wave function offers a complete description of reality, to grotesque situations famously known as Schrödinger's cat. Ways out are either adding elements of reality or replacing the linear evolution by a nonlinear one. Models of spontaneous wave function collapses took the latter path. The way such models are constructed leaves the question of whether such models are in some sense unique, i.e., whether the nonlinear equations replacing Schrödinger's equation are uniquely determined as collapse equations. Various people worked on identifying the class of nonlinear modifications of the Schrödinger equation, compatible with general physical requirements. Here we identify the most general class of continuous wave function evolutions under the assumption of no-faster-than-light signaling.

Inés de Vega, Daniel Alonso, Pierre Gaspard +1 more

Stochastic Schrodinger equations are used to describe the dynamics of a quantum open system in contact with a large environment, as an alternative to the commonly used master equations. We present a study of the two main types of non-Markovian stochastic Schrodinger equations, linear and nonlinear ones. We compare them both analytically and numerically, the latter for the case of a spin-boson model. We show in this paper that two linear stochastic Schrodinger equations, derived from different perspectives by Diosi, Gisin, and Strunz [Phys. Rev. A 58, 1699 (1998)], and Gaspard and Nagaoka [J. Chem. Phys. 13, 5676 (1999)], respectively, are equivalent in the relevant order of perturbation theory. Nonlinear stochastic Schrodinger equations are in principle more efficient than linear ones, as they determine solutions with a higher weight in the ensemble average which recovers the reduced density matrix of the quantum open system. However, it will be shown in this paper that for the case of a spin-boson system and weak coupling, this improvement does only occur in the case of a bath at high temperature. For low temperatures, the sampling of realizations of the nonlinear equation is practically equivalent to the sampling of the linear ones. We study further this result by analyzing, for both temperature regimes, the driving noise of the linear equations in comparison to that of the nonlinear equations.

Shu Hotta

Iwo Białynicki‐Birula

The problem of the linearity of the Schrödinger equation is described from a historical perspective. It is argued that the Schrödinger picture on which this equation is based cannot be retained in relativistic quantum theory. A closer analysis of realistic experiments might offer a clue how to modify the evolution equation for the state vectors in quantum field theory.

Shu Hotta