A theorem for the normalization of continuous spectrum stationary states
We present analytic formulae that simplify the evaluation of the normalization of continuous spectrum stationary states in the one-dimensional Schrödinger equation.
quant-ph↗
arXiv subjects
G. Kälbermann
Publications and source records attributed to G. Kälbermann.
Ehrenfest theorem, Galilean invariance and nonlinear Schrödinger equations
Galilean invariant Schrödinger equations possessing nonlinear terms coupling the amplitude and the phase of the wave function can violate the Ehrenfest theorem. An example of this kind is provided. The example leads to the proof of the theorem: A Galilean invariant Schrödinger equation derived from a lagrangian density obeys the Ehrenfest theorem. The theorem holds for any linear or nonlinear lagrangian.
quant-ph↗
Wavepackets falling under a mirror
We depict and analyze a new effect for wavepackets falling freely under a barrier or well. The effect appears for wavepackets whose initial spread is smaller than the combination $\ds \sqrt{\frac{l_g^3}{|z_0|}}$, between the gravitational length scale $\ds l_g = \frac{1}{(2 m^2 g)^{1/3}}$ and the initial location of the packet $z_0$. It consists of a diffractive structure that is generated by the falling and spreading wavepacket and the waves reflected from the obstacle. The effect is enhanced when the Gross-Pitaevskii interaction for positive scattering length is included. The theoretical analysis reproduces the essential features of the effect. Experiments emanating from the findings are proposed.
quant-ph↗
A nonlinear differential approach to the Saffman-Taylor finger
Nonlinear time-dependent differential equations for the Hele-Shaw, Saffman-Taylor problem are derived. The equations are obtained using a separable ansatz expansion for the stream function of the displaced fluid obeying a Darcian flow. Suitable boundary conditions on the stream function, provide a potential term for the nonlinear equation. The limits for the finger widths derived from the potential and boundary conditions are $1>λ>\frac{1}{\sqrt{5}}$, in units of half the width of the Hele-Shaw cell, in accordance with observation. Stationary solutions with no free phenomenological parameters are found numerically. The dependence of asymptotic finger width on the physical parameters of the cell compares satisfactorily with experiment. The correct dispersion relation for the instabilities is obtained from the time dependent equation.
cond-mat↗
Single and double slit scattering of wave packets
The scattering of wave packets from a single slit and a double slit with the Schrödinger equation, is studied numerically and theoretically. The phenomenon of diffraction of wave packets in space and time in the backward region, previously found for barriers and wells, is encountered here also. A new phenomenon of forward diffraction that occurs only for packets thiner than the slit, or slits, is calculated numerically as well as, in a theoretical approximation to the problem. This diffraction occurs at the opposite end of the usual diffraction phenomena with monochromatic waves.
quant-ph↗
Wavepacket diffraction in the Kronig-Penney model
The phenomenon of wavepacket diffraction in space and time is investigated numerically and analytically, for a one-dimensional array of equally spaced finite-depth wells. Theoretical predictions for the lattice at long times and at low scattering energies, coincide exactly with the results for a single well. At intermediate and short times compared to the classical passage time, the pattern shows both a broad diffractive pattern and an interference pattern inside each diffractive peak. The diffractive structure persists for this case to infinite time.
cond-mat↗
Diffraction of wave packets in space and time
The phenomenon of wave packet diffraction in space and time is described. It consists in a diffraction pattern whose spatial location progresses with time. The pattern is produced by wave packet quantum scattering off an attractive or repulsive time independent potential. An analytical formula for the pattern at $t\to\infty$ is derived both in one dimension and in three dimensions. The condition for the pattern to exist is developed. The phenomenon is shown numerically and analytically for the Dirac equation in one dimension also. An experiment for the verification of the phenomenon is described and simulated numerically.
quant-ph↗
Wave packet scattering from an attractive well
Wave packet scattering off an attractive well is investigated in two spatial dimensions numerically. The results confirm what was found previously for the one dimensional case. The wave scattered at large angles is a polychotomous (multiple peak) coherent train. Large angle scattering is extremely important for low impinging velocities and at all impact parameters. The effect disappears for packets more extended than the well. Experiments to detect the polychotomous behavior are suggested.
quant-ph↗
Diffraction of wave packets in space and time in three dimensions
A simple formula for the scattering of wave packets from a square well at long times is derived. The expression shows that the phenomenon of wave packet diffraction in space and time exists in three dimensions also. An experiment for the verification of the phenomenon is described and simulated numerically.
quant-ph↗
The nucleus as a fluid of skyrmions: Energy levels and nucleon properties in the medium
A model of a fluid of skyrmions coupled to a scalar and to the $ø$ meson mean fields is developed. The central and spin-orbit potentials of a skyrmion generated by the fields predict correct energy levels in selected closed shell nuclei. The effect of the meson fields on the properties of skyrmions in nuclei is investigated.
nucl-th↗
Trapping of a model-system for a soliton in a well
The nature of the interaction of a soliton with an attractive well is elucidated using a model of two interacting point particles. The system shows the existence of trapped states at positive kinetic energy, as well as reflection by an attractive impurity, as found when a topological soliton scatters off an attractive well.
math-ph↗
Decay to bound states of a soliton in a well
The decay of a soliton in a trapped state inside a well is shown numerically. Bound states of a kink in an attractive well, both centered and off center are found. Their stability is studied. Unstable soliton solutions inside a repulsive barrier are also found.
math-ph↗
Communication through an extra dimension
If our visible universe is considered a trapped shell in a five-dimensional hyper-universe, all matter in it may be connected by superluminal signals traveling through the fifth dimension. Events in the shell are still causal, however, the propagation of signals proceeds at different velocities depending on the fifth coordinate.
gr-qc↗
Nearness through an extra dimension
It is shown that if our visible universe is a thin trapped shell in a five-dimensional universe, all matter in it may be connected almost instantaneously through the fifth dimension. What appears to be action at a distance is then understood as undetectable ultrafast communication.
gr-qc↗
Coherent polychotomous waves from an attractive well
A novel effect of a wave packet scattering off an attractive one- dimensional well is found numerically and analytically. For a wave packet narrower than the width of the well, the scattering proceeds through a quasi-bound state of almost zero energy. The wave reflected from the well is a polychotomous (multiple peak) monochromatic and coherent train. The transmitted wave is a spreading smooth wave packet. The effect is strong for low average speeds of the packet, and it disappears for wide packets.
quant-ph↗
Trapped States and bound states of a soliton in a well
The nature of the interaction of a soliton with an attractive well is elucidated using a model of two interacting point particles. The model explains the existence of trapped states at positive kinetic energy, as well as reflection by an attractive impurity. The transition from a trapped soliton state to a bound state is studied. Bound states of the soliton in a well are also found.
hep-th↗
The Skyrmion in the nucleus
The energy levels of a skyrmion in nucleus are calculated in a field theory of skyrmions coupled to the dilaton field and the $ω$ meson . The central potential fits well with expectations. The nucleon spin-orbit interaction derived from the omega meson in a rotating frame gives the correct level splittings. The same interaction originating from the Thomas precession effect is negligible. Energy levels are calculated for closed shell nuclei. The meson fields are obtained from a Thomas-Fermi mean field approximation to the nucleus.
nucl-th↗
Soliton tunneling
We present a numerical simulation of the scattering of a topological soliton off finite size attractive impurities, repulsive impurities and a combination of both. The attractive and attractive-repulsive cases show similar features to those found for $δ$ function type of impurities. For the repulsive case, corresponding to a finite width barrier, the soliton behaves completely classically. No tunneling occurs for sub-barrier kinetic energies despite the extended nature of the soliton.
hep-th↗