We investigate the time-dependent Schr\"odinger equation for the case of a plane matter wave incident on an oscillating potential step. Approximate solutions are given for different regimes of the problem. The resulting energy spectra are quantized. This may either be interpreted as a phase modulation of the wave or as a coherent multiphonon exchange. A comparison to the results that apply for a beam of classical particles is made. In particular, the transition from the quantum-mechanical to the classical case is examined. We have performed an experiment with very cold neutrons that clearly revealed the discussed effects. Energy splittings of 2.8 to 9.1 neV of the reflected waves were analyzed with an energy resolution of \ensuremath{\sim}1 neV, thus demonstrating the feasibility of such high-resolution experiments with neutron reflectometry. \textcopyright{} 1996 The American Physical Society.
We investigate the time-dependent Schr"odinger equation for the case of a plane matter wave incident on an oscillating potential step. Approximate solutions are given for different regimes of the problem. The resulting energy spectra are quantized. This may either be interpreted as a phase modulation of the wave or as a coherent multiphonon exchange. A comparison to the results that apply for a beam of classical particles is made. In particular, the transition from the quantum-mechanical to the classical case is examined. We have performed an experiment with very cold neutrons that clearly revealed the discussed effects. Energy splittings of 2.8 to 9.1 neV of the reflected waves were analyzed with an energy resolution of \ensuremath{\sim}1 neV, thus demonstrating the feasibility of such high-resolution experiments with neutron reflectometry. \textcopyright{} 1996 The American Physical Society.
This study defines a neutron as a string with intrinsic vibrational states and numerically clarifies the mechanism by which classical mechanical laws of motion transition into quantum mechanical matter-wave states based on the Ehrenfest Theorem. By deriving the resonance phenomenon—where spatial energy gradients and temporal vibrational terms (cosωt) synchronize within a 3D potential distribution—into a final force equation (Equation 8), this research reinterprets the wave-like characteristics of particles in double-slit experiments as deterministic physical states driven by constructive interference, thereby completing the numerical connectivity between quantum and classical mechanics.
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 investigate the scattering of a quantum-mechanical particle on an oscillating rectangular potential barrier. In the first step, we show the calculation for a classical potential revealing coherent, quantized energy exchange between the particle and potential, respectively, the likewise classical field that generates the latter. In order to explore the quantized energy transfer in more detail, the transition to a quantum field is performed and the Schr\"odinger equation for the composite quantum system consisting of particle and field is solved analytically as well. The result allows a physically very intuitive explanation of the underlying energy exchange processes and explicitly displays the occurring entanglement between particle and field due to their interaction, thus offering the possibility to use this combined system for quantum information experiments. Depending on the initial field state, incoherent or coherent energy exchange is observed. In the limit of high mean photon numbers the classical results can be fully retrieved.
We present a theoretical model of matter-wave diffraction through a material nanostructure. This model is based on the numerical solution of the time-dependent Schrödinger equation, which goes beyond the standard semiclassical approach. In particular, we consider the dispersion force interaction between the atoms and the material, which is responsible for high energy variations. The effect of such forces on the quantum model is investigated, along with a comparison with the semiclassical model. In particular, for atoms at low velocity and close to the material surface, the semiclassical approach fails, while the quantum model accurately describes the expected diffraction pattern. This description is thus relevant for slow and cold atom experiments where increased precision is required, e.g., for metrological applications. Published by the American Physical Society 2024
Following the spirit of de Broglie and Einstein, we think the concepts of matter and radiation can be unified. We know a particle propagates like a wave; its motion is described by certain wave equations. At this point, it is not clear what the wave function represents. Besides the statistical meaning suggested by the Copenhagen interpretation, does the wave function represent any physical motion? For photon, we know it is an electro-magnetic wave. But what about particles with rest mass, such as an electron? To investigate the physical nature of matter wave, we propose that: (1) Like the photon, a particle is an excitation wave of a real physical field. (2) Different types of particles are different excitation modes of the same field. Based on this thinking, we show that the concept of quantum mechanics can be a natural extension of classical mechanics. By critically analyzing the transition from classical physics to quantum physics, we found a new physical meaning for the quantum wave function. This work suggests that various quantum wave equations, including the Klein-Gordon equation, the Dirac equation and the Schrodinger equation, could have a common base relating to the Basic Wave Equation of the matter wave. This work has some interesting implications. It suggests a possible way to explain the origin of visible matters and dark matters in our universe.
Maxwell matter waves emerge from a perspective, complementary to de Broglie's, that matter is fundamentally a wave phenomenon whose particle aspects are revealed by quantum mechanics. Their quantum mechanical description is derived through the introduction of a matter vector potential, having frequency ω0, to Schrodinger's equation for a massive particle. Maxwell matter waves are then seen to be coherent excitations of a single-mode of the matter-wave field. In the classical regime, their mechanics is captured by a matter analog of Maxwell's equations for the electromagnetic field. As such, Maxwell matter waves enable a spectrum of systems that have useful optical analogs, such as resonant matter-wave interferometric sensors and matter-wave parametric oscillators. These waves are associated with a wavelength that is tied to the drive frequency ω0 rather than to the massive particle's energy, as is ordinarily the case with de Broglie matter waves. As a result, simple interferometric measurements lead to different outcomes for the two types of waves. While their apparent departure from de Broglie character is surprising, Maxwell matter waves are wholly consistent with quantum mechanics.