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 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 LettersMar 24, 1980103 citations

Experiments with Neutron Schroedinger Waves

Muhammad Arif

Sep 1, 19860 citations

Time-dependent Schrodinger equation

Luca Salasnich

Springer eBooksJan 1, 20200 citations

Matter-wave optics in the time domain: Results of a cold-neutron experiment

Th. Hils, J. Felber, R. Gähler +4 more

We report an observation of a change in the energy spectrum of a matter wave due to fast modulation in time. The experiment, carried out with slow neutrons, achieved pulse widths of 33 n, and has been analyzed using time-dependent optics based on the nonrelativistic Schr\"odinger equation. A good fit to the data is obtained. The modulation of a matter wave in time leads to a characteristic energy spread. These phenomena can be seen as a consequence of the time-energy uncertainty relation. In our case, the amplitude modulation is done by a rapid periodic opening and closing of a slit. In the time domain this causes a so-called ``diffraction in time.'' In this work we demonstrate the operation of a time-of-flight spectrometer (FOTOF), and achieve an energy resolution of 20 neV, which to our knowledge represents a new limit for a TOF with slow neutrons.We report an observation of a change in the energy spectrum of a matter wave due to fast modulation in time. The experiment, carried out with slow neutrons, achieved pulse widths of 33 n, and has been analyzed using time-dependent optics based on the nonrelativistic Schr\"odinger equation. A good fit to the data is obtained. The modulation of a matter wave in time leads to a characteristic energy spread. These phenomena can be seen as a consequence of the time-energy uncertainty relation. In our case, the amplitude modulation is done by a rapid periodic opening and closing of a slit. In the time domain this causes a so-called ``diffraction in time.'' In this work we demonstrate the operation of a time-of-flight spectrometer (FOTOF), and achieve an energy resolution of 20 neV, which to our knowledge represents a new limit for a TOF with slow neutrons.

Physical Review ADec 1, 199880 citations

Time-Dependent Schrödinger Equation

MASSIMO RUDAN

Springer eBooksSep 27, 20170 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.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 AApr 1, 1981116 citations

b

was established.

\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.