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Schroedinger equation linear

Iwo Białynicki‐Birula

Published Jan 1, 1981• 0 citations
OpenAlex

Abstract

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.

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On the linearity of the Schrödinger equation

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.

Brazilian Journal of Physics•Jun 1, 2005•11 citations
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.
We analyze the free-space propagation of matter waves with a view to placing an upper limit on the strength of possible nonlinear terms in the Schr\"odinger equation. Such additional terms of the form $\ensuremath{\psi}F({|\ensuremath{\psi}|}^{2})$ were introduced by Bialynicki-Birula and Mycielski in order to counteract the spreading of wave packets, thereby allowing solutions which behave macroscopically like classical particles. For the particularly interesting case of a logarithmic nonlinearity of the form $F=\ensuremath{-}\mathit{b}\mathrm{ln}{|\ensuremath{\psi}|}^{2}$, we find that the free-space propagation of slow neutrons places a very stringent upper limit on the magnitude of b\mathit{b}b. Precise measurements of Fresnel diffraction with slow neutrons do not give any evidence for nonlinear effects and allow us to deduce an upper limit for $\mathit{b}<3.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}$ eV about 3 orders of magnitude smaller than the lower bound proposed by the above authors.
A slow-neutron interferometer system has been used to test a nonlinear variant of the Schr\"odinger equation $i\ensuremath{\hbar}\frac{\ensuremath{\partial}\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, t)}{\ensuremath{\partial}t}=[(\ensuremath{-}\frac{{\ensuremath{\hbar}}^{2}}{2m}){\ensuremath{\nabla}}^{2}+U(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, t)]\ensuremath{\psi}\ensuremath{-}b\mathrm{ln}({a}^{3}{|\ensuremath{\psi}|}^{2})\ensuremath{\psi}$. If this equation were correct, then, as Shimony has suggested, repositioning an attenuating plate downstream in a neutron beam would produce a phase modification. No measurable phase shift beyond experimental uncertainty was found and an upper limit of 3.4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ eV for the energy constant bbb was established.