Scattering of a quantum particle with internal structure is fundamentally different from that of a point particle and shows quantum effects such as the modification of transmission due to tunnelling and trapping of the particle. As in a preceding paper (Shore et al 2014 New J. Phys. 17 013046 ) we consider a model of a symmetric, rigid rotor travelling through an aperture in a thin but impenetrable screen which is perpendicular to both the direction of motion and the rotation axis. We determine the quantum mechanical properties of this two-dimensional geometrical model using a quasi one-dimensional scattering problem with unconventional boundaries. Our calculations rely on finding the Green's function, which has a direct connection to the scattering matrix. Evaluated on a discrete lattice the Hamiltonian is 'dressed' by a self-energy correction that takes into account the open boundary conditions in an exact way. We find that the passage through the aperture can be suppressed or enhanced as a result of the rotational motion. These effects manifest themselves through resonances in the transmission probability as a function of incident energy and symmetry of the incident wavefunction. We determine the density-of-states to reveal the mode structure of resonant states and to exhibit the lifetimes of temporary trapping within the aperture.
Scattering of a quantum particle with internal structure is fundamentally different from that of a point particle and shows quantum effects such as the modification of transmission due to tunnelling and trapping of the particle. As in a preceding paper (Shore et al 2014 New J. Phys. 17 013046 ) we consider a model of a symmetric, rigid rotor travelling through an aperture in a thin but impenetrable screen which is perpendicular to both the direction of motion and the rotation axis. We determine the quantum mechanical properties of this two-dimensional geometrical model using a quasi one-dimensional scattering problem with unconventional boundaries. Our calculations rely on finding the Green's function, which has a direct connection to the scattering matrix. Evaluated on a discrete lattice the Hamiltonian is 'dressed' by a self-energy correction that takes into account the open boundary conditions in an exact way. We find that the passage through the aperture can be suppressed or enhanced as a result of the rotational motion. These effects manifest themselves through resonances in the transmission probability as a function of incident energy and symmetry of the incident wavefunction. We determine the density-of-states to reveal the mode structure of resonant states and to exhibit the lifetimes of temporary trapping within the aperture.
Classically, rigid objects with elongated shapes can fit through apertures only when properly aligned. Quantum-mechanical particles which have internal structure (e.g. a diatomic molecule) also are affected during attempts to pass through small apertures, but there are interesting differences with classical structured particles. We illustrate here some of these differences for ultra-slow particles. Notably, we predict resonances that correspond to prolonged delays of the rotor within the aperture—a trapping phenomenon not found classically.
We study the quantum slit diffraction problem in three dimensions. In the treatment of diffraction of particles by a slit, it is usually assumed that the motion perpendicular to the slit is classical. Here we take into account the effect of the quantum nature of the motion perpendicular to the slit using the Green function approach [18]. We treat the diffraction of a Gaussian wave packet for general boundary conditions on the shutter. The difference between the standard and our three-dimensional slit diffraction models is analogous to the diffraction in time phenomenon introduced in [16]. We derive corrections to the standard formula for the diffraction pattern, and we point out situations in which this might be observable. In particular, we discuss the diffraction in space and time in the presence of gravity.
We study quasi-one-dimensional scattering of one and two particles with short-range interactions on a discrete lattice model in two dimensions. One of the directions is tightly confined by an arbitrary trapping potential. We obtain the collisional properties of these systems both at finite and zero Bloch quasimomenta, considering as well finite sizes and transversal traps that support a continuum of states. This is made straightforward by using the exact ansatz for the quasi-one-dimensional states from the beginning. In the more interesting case of genuine two-particle scattering, we find that more than one confinement-induced resonances appear due to the nonseparability of the center-of-mass and relative coordinates on the lattice. This is done by solving its corresponding Lippmann-Schwinger-like equation. We characterize the effective one-dimensional interaction and compare it with a model that includes only the effect of the dominant, broadest resonance, which amounts to a single-pole approximation for the interaction coupling constant.
The double-slit experiment is typically presented as evidence for the non-classical nature of microscopic particles, especially when performed in the single-particle regime. However, the interference pattern that emerges on the detection screen is generated by wave propagation through two structured apertures, each acting as a narrow resonant cavity supporting longitudinal modes of a scalar field. In this work we derive the outgoing fields from the slits directly from boundary conditions and show that their superposition yields the standard far-field intensity profile of diffraction theory. The remaining aspect of the experiment — namely discrete, localized detection events — is accounted for by a minimal discrete-event model in which the continuous classical field drives quantized energy transfers at the detector. This approach is fully classical, requires no collapse postulate or nonlocality, and remains compatible with all known data. It clarifies which features of the experiment arise from field propagation and which arise from the discrete nature of detection.
The Schrödinger equation is reformulated as a universal continuity equation, which connects between changes in the particles probability density distribution to probability current densities (fluxes). The formulation of particle conservation in terms of stationary fluxes enables one to associate stationary wave functions also to open quantum systems characterized by stationary particle currents. These functions are (improper) solutions of the stationary Schrödinger equation, obtained under scattering boundary conditions. These boundary conditions can be fulfilled for any positive asymptotic kinetic energy, hence, the energy spectrum of the scattering states is continuous. We demonstrate flux calculations in scattering through a one-dimensional potential energy well/barrier, focusing on transmission and reflection probabilities. Nonclassical phenomena such as transmission at energies below a potential energy barrier (quantum tunneling), or reflections at energies above a potential energy well are analyzed. The phenomenon of full transmission through a double barrier structure (resonant tunneling) is introduced in the context of nanoscale transport.
In this paper, the resonance transmission of discrete time quantum walk is studied when it walks on one-dimensional lattice in which two-phase defects or a piece of phase defects exists. The quasi energy of discrete time quantum walk has a unique dispersion relation with the momentum, from which we first discuss the wave velocity direction versus the values of momentum, and distinguish the incident wave and the reflected wave. The gap between two energy bands depends on the parameters of coincident operator, so the phase defects, which break down the translation invariance of quantum walk on uniform lattices, can be regarded as an analogue of quantum potential. Then we use the condition of energy conversion at the boundary points to obtain the transmission rate and discuss its variation with the incident momentum for different strengths and widths of defects in detail. The multiple resonant peaks are observed due to the enhanced interference effect. Different resonant behaviors are shown when the strength of defect is less or greater than /2, correspondingly the resonances occur in a wide region of incident momentum or the sharp resonant peaks appear at discrete values of momentum. Under the condition of strong defect strength, i.e., approaching to , the qualitative relation between the number of resonant peaks and the widths of defect region is given. The number of resonant peaks is 2(N-1) when the two phase defects are located at N sites symmetric about the origin, while the number is 2N when a piece of phase defects is located at -N to N sites. In the case of a piece of phase defects, we also present the phase diagram in parameter space of (k, ) to show the discrete time of quantum walk propagating or tunneling through the defect region. In terms of this phase diagram, the variations of transmission rate with the incident momentum are reasonably explained. One special phenomenon is that the quantum walk is almost totally reflected in the tunneling case except for =/2 and k being slightly off -/2. Moreover, this behavior seems little affecting the defect strength, just similar to a classical particle. As a result of this research, we hope to deepen the insight of the quantum walk and provide methods to control the spreading of quantum walk through artificial defects.