We consider a very straightforward extension of the usual self-adjoint techniques to solve the quantum mechanical problem of contact interaction potentials (i.e., infinite walls with different boundary conditions and point interactions) moving with constant velocity. From these calculations we obtain general expressions that describe the scattering of wavepackets. Furthermore, we also discuss the cases where the potentials oscillate with constant velocity or have time dependent strengths. Finally, we show, with an explicit example, that the prescription used is not the only possibility to implement self-adjoint extensions for contact interactions. Thus, in principle different models can be realized from such moving systems. Throughout the work we propose some simple physical applications of our results.
We construct an eigenvalue problem by confining many-body system to a bounded domain with the boundary condition that the wave function vanishes. By changing the boundary, however, the eigenvalues of the energy can be varied continuously. The D-matrix is defined for a series of bounded problems with the same value for the ground state energy. The D-matrix is related to the S-matrix, enabling us to calculate the S-matrix at a given energy. The Schrodinger equation for the system is transformed to a diffusion equation by regarding time as imaginary. Initial ensemble, representing an approximate wave function, is evolved, through Monte Carlo simulation of random walks and branching, to the ground state ensemble. The limitations of investigation are: 1. Ingoing and outgoing channels have two fragments. 2. The interaction between the fragments is negligible outside the boundary mentioned above. 3. The particles are bosons or we know the zeros of the wave function. First we consider the scattering of a particle by a potential, which is equivalent to the two-body problem, in one dimension. Here we use the Poschl-Teller potential for which the exact solution is known. We use this case to investigate a new sampling method and study of various parameters. Next we consider three particles in one dimension. Here we take interaction to be a potential well, where at least one of the interactions is attractive so that a two-body bound state is possible.
Using real-time wave packet propagation we consider the scattering properties of a linear collision system consisting of a harmonic oscillator coupled by truncated Morse potentials to two particles. This system is viewed as a model for the collision of a projectile D with a three particle target ABC with interactions such that the possible final states comprise backscattering of the projectile with the target intact, the ‘‘exchange’’ reaction ABC+D→A+BCD and the ‘‘fragmentation’’ reaction ABC+D→A+BC+D. For the types of interaction considered, the exchange reaction was found to proceed primarily via inelastic resonant processes and to be the dominant event for incident energies below the oscillator energy. Fragmentation occurred via nonresonant processes and dominated at higher energy. Other features of the scattering are discussed and compared with simple quantum theory and the description of the system given by classical mechanics.
We study the two-body problem of ultracold fermionic alkaline-earth-metal-(like) atoms in the electronic
1S0 state (
g-state) and
3P0 state (
e-state) which are confined in a quasi-one-dimensional (quasi-1D) tube simultaneously, where in the axial direction the
g-atom experiences a 1D optical lattice and the
e-atom is localized by a harmonic potential. Due to the nuclear-spin exchange interaction between the
g- and
e-atoms, one can use such a quasi-
(1+0)D system to realize the Kondo effect. We suggest two tight-binding models for this system, for the cases that the odd-wave scattering between the
g- and
e-atoms is negligible or not, respectively. Moreover, we give a microscopic derivation for the interatomic interaction parameters of these models, by explicitly calculating the quasi-
(1+0)D low-energy scattering amplitude of the
g- and
e-atoms in this system and matching this exact result with the ones given by tight-binding models. We illustrate our results for the experimental systems of ultracold
173Yb and
171Yb atoms and show the control effect of the confinement potentials on these model parameters. Furthermore, the validity of the simple ``projection approximation'' is examined. In this approximation, one derives the interaction parameters of the tight-binding models by directly projecting the three-dimensional (3D) Huang-Yang pseudopotential on the ground state of the confinement and the lowest band of the optical lattice. This approximation is supposed to be correct when the 3D interatomic scattering length
as is much smaller than the characteristic lengths (CLs) of the confinements. However, we find that for our system this approximation already does not work when
as is only of the order of 10% of the confinement CLs. Using the exact two-atom scattering amplitude, we also calculate the spin-exchanging rate (i.e., the cross section of the spin-exchanging collision between the
g- and
e-atoms) for the recent experiment [L. Riegger et al., Phys. Rev. Lett. 120, 143601 (2018)] of
173Yb atoms in this quasi-
(1+0)D system and investigate finite-quasimomentum effect of the
g-atoms in this experiment. Our results show that this effect is very significant.
A self-contained discussion of nonrelativistic quantum scattering is presented in the case of central potentials in one space dimension, which will facilitate the understanding of the more complex scattering theory in two and three dimensions. The present discussion illustrates in a simple way the concept of partial-wave decomposition, phase shift, optical theorem and There has been great interest in studying tunneling phenomena in a finite superlattice [1], which essentially deals with quantum scattering in one dimension. It has also been emphasized [2] that a study of quantum scattering in one dimension is important for the study of Levinson’s theorem and virial coefficient. Moreover, one-dimensional quantum
The interaction of an atomic system with a single mode of a quantized electromagnetic field is investigated in the dipole approximation. Two approximations that are almost always made in this analysis are the neglect of the diamagnetic term in the Hamiltonian and the dropping of the rotating-wave approximation (RWA) terms that correspond to processes that do not conserve energy. Both approximations involve the neglect of terms that are of the same order of magnitude so that consistency demands that they be considered together. It is demonstrated that, in the dipole approximation, a minor modification of the usual application of quantum electrodynamics to any atomic system can exactly incorporate the effects of the diamagnetic term into a new frequency of the electromagnetic cavity mode; this frequency is slightly different from that of the empty cavity. This result is then applied to the specific case of a charged harmonic oscillator interacting with a single mode. For this simple model an analytic solution can then be found that includes the effects of the diamagnetic term and does not use the RWA. It is found that, for off resonance, the normal modes of the coupled system oscillate at shifted frequencies which are in agreement with the second-order perturbation-theory calculation of the Lamb shift. When the cavity mode is tuned to the atomic oscillator frequency, the perturbation-theory Lamb-shift calculation is no longer valid, and the exact solution derived in this paper predicts a resonant Lamb shift that is significantly larger than the off-resonant single-mode Lamb shift. Furthermore, the frequency spectrum is qualitatively changed from the single frequency of the free atomic system to a doublet spectrum. An analytic solution of this harmonic-oscillator problem can also be found when the RWA terms are dropped from the Hamiltonian. Comparison of the two solutions reveals that the RWA terms contribute a frequency shift of magnitude independent of the field intensity in the simple case of a charged harmonic oscillator.When trajectories of the exact and RWA solutions are compared in phase space, it is found that one effect of the RWA terms is to lift a twofold degeneracy of the phase-space orbits that occurs when the RWA terms are neglected. The harmonic-oscillator problem has several features in common with the interaction of a two-level atom with a single quantized field mode. Both Hamiltonians exhibit inversion symmetry and hence have parity as a good quantum number. Both Hamiltonians contain RWA terms. When these RWA terms are neglected, both Hamiltonians have a new constant of motion that can be identified with the total number of excitations in the combined system.