Linear and nonlinear analogues of the Schroedinger equation in the contextual approach in quantum mechanics

Andrei Yu. Khrennikov

Waldemar Puszkarz

Andrei Yu. Khrennikov

We constructed the representation of contextual probabilistic dynamics in the complex Hilbert space. Thus dynamics of the wave function can be considered as Hilbert space projections of realistic dynamics in a ``prespace''. The basic condition for representing of the prespace-dynamics is the law of statistical conservation of energy -- conservation of probabilities. Construction of the dynamical representation is an important step in the development of contextual statistical viewpoint to quantum processes. But the contextual statistical model is essentially more general than the quantum one. Therefore in general the Hilbert space projection of the ``prespace'' dynamics can be nonlinear and even irreversible (but it is always unitary). There were found conditions of linearity and reversibility of the Hilbert space dynamical projection. We also found conditions for the conventional Schroedinger dynamics (including time-dependent Hamiltonians). We remark that in general even the Schroedinger dynamics is based just on the statistical conservation of energy; for individual systems the law of conservation of energy can be violated (at least in our theoretical model).

Ervin Kaminski Lenzi, Antonio S. M. de Castro, R. S. Mendes

Iwo Białynicki‐Birula

The problem of linearity of the Schroedinger equation is analyzed in the light of new experiments performed at MIT and the Laue-Langevin Institute. In these experiments slow neutrons were used to place an upper limit on the strength of the logarithmic nonlinear term in the Schroedinger equation introduced in an earlier work by Jerzy Mycielski and me.

акустика Механика