© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical Review A

Neutron optical tests of nonlinear wave mechanics

R. Gähler, A. G. Klein, Anton Zeilinger

Published Apr 1, 1981• 116 citations

Abstract

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 $\mathit{b}$. 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.

DOI: 10.1103/physreva.23.1611

Related papers

Proposed neutron interferometer test of some nonlinear variants of wave mechanics

Abner Shimony

A family of nonlinear variants of the Schr\"odinger equation was defined by Bialynicki-Birula and Mycielski by adding terms of the form $F({|\ensuremath{\psi}|}^{2})$ to the Hamiltonian. It is proposed that this family be tested by observing whether a phase shift occurs when an absorber is moved from one point to another along the path of one of the coherent split beams in a neutron interferometer. If $F$ is $b$ times a logarithmic function, which is the most important case, a null result with apparatus now available would impose an upper bound on $b$ of 1.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}12}$ eV, more than two orders of magnitude smaller than the bound estimated by the above authors on the basis of the Lamb-shift measurement.

Physical Review A•Aug 1, 1979•83 citations

Schroedinger equation linear

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.

•Jan 1, 1981•0 citations

Search for a Nonlinear Variant of the Schrödinger Equation by Neutron Interferometry

C. G. Shull, D. K. Atwood, John Arthur +1 more

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 $b$ was established.

Physical Review Letters•Mar 24, 1980•103 citations

Neutron optics

A. G. Klein, S. A. Werner

A range of phenomena similar or analogous to those of classical optics is exhibited by slow neutrons. The aim of this review is to discuss this similarity and to display the results obtained in a wide range of experimental work on reflection, refraction, diffraction and interference of neutrons. The motivation of such experiments is discussed in the introductory section. The review is divided into three main sections: geometrical optics, wave optics and crystal optics; with subsections that parallel the typical textbook development of classical optics. Special emphasis is placed on the recent developments in neutron interferometry and on the fundamental tests of quantum-mechanical principles made possible by these novel techniques.

Reports on Progress in Physics•Mar 1, 1983•184 citations

Wave-optical experiments with very cold neutrons

R. Gähler, Anton Zeilinger

A series of experiments is reviewed on precision measurements of the diffraction of slow neutrons paralleling those with light: diffraction at an absorbing edge, at an absorbing wire, and at single- and double-slit assemblies. Besides their educational value, the experimental results also provide precise confirmations of the theoretical predictions as calculated via a Fresnel–Kirchhoff approach. The intensity in all these experiments was so low that the formation of the interference patterns individual neutron by individual neutron could easily be observed. The experiments are presented and analyzed in a way suitable for undergraduate teaching.

American Journal of Physics•Apr 1, 1991•47 citations

Proposed neutron interferometer test of some nonlinear variants of wave mechanics

Abner Shimony

Cambridge University Press eBooks•Feb 26, 1993•0 citations

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

\mathit{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 family of nonlinear variants of the Schr\"odinger equation was defined by Bialynicki-Birula and Mycielski by adding terms of the form $F({|\ensuremath{\psi}|}^{2})$ to the Hamiltonian. It is proposed that this family be tested by observing whether a phase shift occurs when an absorber is moved from one point to another along the path of one of the coherent split beams in a neutron interferometer. If

F

is

b

times a logarithmic function, which is the most important case, a null result with apparatus now available would impose an upper bound on

b

of 1.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}12}$ eV, more than two orders of magnitude smaller than the bound estimated by the above authors on the basis of the Lamb-shift measurement.

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

b

was established.

A range of phenomena similar or analogous to those of classical optics is exhibited by slow neutrons. The aim of this review is to discuss this similarity and to display the results obtained in a wide range of experimental work on reflection, refraction, diffraction and interference of neutrons. The motivation of such experiments is discussed in the introductory section. The review is divided into three main sections: geometrical optics, wave optics and crystal optics; with subsections that parallel the typical textbook development of classical optics. Special emphasis is placed on the recent developments in neutron interferometry and on the fundamental tests of quantum-mechanical principles made possible by these novel techniques.

A series of experiments is reviewed on precision measurements of the diffraction of slow neutrons paralleling those with light: diffraction at an absorbing edge, at an absorbing wire, and at single- and double-slit assemblies. Besides their educational value, the experimental results also provide precise confirmations of the theoretical predictions as calculated via a Fresnel–Kirchhoff approach. The intensity in all these experiments was so low that the formation of the interference patterns individual neutron by individual neutron could easily be observed. The experiments are presented and analyzed in a way suitable for undergraduate teaching.