© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical Review Letters

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

C. G. Shull, D. K. Atwood, John Arthur, Michael Horne

Published Mar 24, 1980• 103 citations

Abstract

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.

DOI: 10.1103/physrevlett.44.765

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

Neutron optical tests of nonlinear wave mechanics

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

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.

Physical Review A•Apr 1, 1981•116 citations

Proposed neutron interferometer test of some nonlinear variants of wave mechanics

Abner Shimony

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

Test of the time dependent Schrödinger equation with very slow neutrons

J. Felber, R. Gähler, Robert Golub

Physica B+C•Jul 1, 1988•27 citations

Test of quantum mechanics by neutron interferometers

H. Rauch

Since its advent in 1974, the neutron interferometry technique is a powerful tool for tests of quantum mechanics by matter waves. High order interference pattern could be observed and an upper limit for nonlinear terms in the Schroedinger equation has been given. The 4..pi..-symmetry of a spinor wave function was verified in 1975. Afterwards a precision measurement of this 4..pi..-symmetry effect has been performed which yields, after corrections for the new values of the fundamental constants, a periodicity factor of ..cap alpha.. = 715.87 +- 3.8 deg which does not include the expected 720 deg within its simple error limits. Alternative formulations of quantum mechanics predict a deviation of the 4..pi..-periodicity factor in the region of strong interaction. Therefore, a precision measurement of the influence of strong interactions is proposed where the spinor rotation is observed within the region of additional strong interactions caused by a nonmagnetic material of heavy nuclei inserted alternatively into the beam paths of the interferometer. It is expected to reach an accuracy concerning the additional phase shift in the range of 10/sup -5/ which corresponds to deviation factors within the region of strong interactions of about 20% from the standard formulation. Various factors limiting themore » accuracy are discussed.« less

Hadronic J.; (United States)•Apr 1, 1982•2 citations

Numerical solution of a nonlinear Schrödinger equation for neutron optics experiments

Johann Kamesberger, Anton Zeilinger

Physica B+C•Jul 1, 1988•2 citations

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 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.

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.

Since its advent in 1974, the neutron interferometry technique is a powerful tool for tests of quantum mechanics by matter waves. High order interference pattern could be observed and an upper limit for nonlinear terms in the Schroedinger equation has been given. The 4..pi..-symmetry of a spinor wave function was verified in 1975. Afterwards a precision measurement of this 4..pi..-symmetry effect has been performed which yields, after corrections for the new values of the fundamental constants, a periodicity factor of ..cap alpha.. = 715.87 +- 3.8 deg which does not include the expected 720 deg within its simple error limits. Alternative formulations of quantum mechanics predict a deviation of the 4..pi..-periodicity factor in the region of strong interaction. Therefore, a precision measurement of the influence of strong interactions is proposed where the spinor rotation is observed within the region of additional strong interactions caused by a nonmagnetic material of heavy nuclei inserted alternatively into the beam paths of the interferometer. It is expected to reach an accuracy concerning the additional phase shift in the range of 10/sup -5/ which corresponds to deviation factors within the region of strong interactions of about 20% from the standard formulation. Various factors limiting themore » accuracy are discussed.« less