By using an exact solution to the time-dependent Schrödinger equation with a point source initial condition, we investigate both the time and spatial dependence of quantum waves in a step potential barrier. We find that for a source with energy below the barrier height, and for distances larger than the penetration length, the probability density exhibits a {\it forerunner} associated with a non-tunneling process, which propagates in space at exactly the semiclassical group velocity. We show that the time of arrival of the maximum of the {\it forerunner} at a given fixed position inside the potential is exactly the traversal time, $τ$. We also show that the spatial evolution of this transient pulse exhibits an invariant behavior under a rescaling process. This analytic property is used to characterize the evolution of the {\it forerunner}, and to analyze the role played by the time of arrival, $3^{-1/2}τ$, found recently by Muga and Büttiker [Phys. Rev. A {\bf 62}, 023808 (2000)].
We derive an exact analytic solution to a Klein-Gordon equation for a step potential barrier with cutoff plane wave initial conditions, in order to explore wave evolution in a classical forbidden region. We find that the relativistic solution rapidly evanesces within a depth $2x_p$ inside the potential, where $x_p$ is the penetration length of the stationary solution. Beyond the characteristic distance $2x_p$, a Sommerfeld-type precursor travels along the potential at the speed of light, $c$. However, no spatial propagation of a main wavefront along the structure is observed. We also find a no…
We analytically compute the time evolution of an initial infinite plane wave in the presence of a 1-dimensional square quantum barrier. This calculation generalizes the analysis of the shutter problem and sets the basis for the calculation of the transmission of general wave packets, aiming to work out the explicit contribution of the resonant (Gamow) states. The method relies mainly on the analytical properties of the Green function. The role of separate boundary conditions on the Green function and on the evolution equation is highlighted. As in previous works on related problems, only the d…
Path integral method in quantum mechanics provides a new thinking for barrier option pricing. For proportional double-barrier step (PDBS) options, the option price changing process is analogous to a particle moving in a finite symmetric square potential well. We have derived the pricing kernel of PDBS options with time dependent interest rate and volatility. Numerical results of option price as a function of underlying asset price are shown as well. Path integral method can be easily generalized to the pricing of PDBS options with curved boundaries.
In recent years, graphene and other two-dimensional Dirac materials like silicene, germanene, etc. have been studied from different points of view: from mathematical physics, condensed matter physics to high energy physics. In this study, we utilize both supersymmetric quantum mechanics (SUSY-QM) and transfer matrix method (TTM) to examine electronic transport in two-dimensional Dirac materials under the influences of multi-step deformation as well as multi-step Fermi velocity barrier. The effects of multi-step effective mass and multi-step applied fields are also taken into account in our inv…
We study the evolution of a wave packet impinging onto a one dimensional potential barrier. The transmission and reflection times discussed in the literature for stationary states do not correspond to the times required for the emergence of a transmitted or a reflected packet. We propose new definitions for the interaction (dwell) time and the transmission and reflection times which are suitable for packets and fit better the actual time evolution of the packet.
We reanalyze the problem of a 1D Dirac single particle colliding with the electrostatic potential step of height $V_{0}$ with a positive incoming energy that tends to the limit point of the so-called Klein energy zone, i.e., $E\rightarrow V_{0}-\mathrm{m}c^{2}$, for a given $V_{0}$. In such a case, the particle is actually colliding with an impenetrable barrier. In fact, $V_{0}\rightarrow E+\mathrm{m}c^{2}$, for a given relativistic energy $E\,(<V_{0})$, is the maximum value that the height of the step can reach and that ensures the perfect impenetrability of the barrier. Nevertheless, we note…
A method of a non-stationary description of tunneling of a particle through the one-dimensional and spherically symmetric rectangular barriers on the basis of analisis of multiple internal reflections of wave packets in relation on the barrier boundaries, named as the Method of multiple internal reflections, is presented at the first time. For the one-dimensional problem the applicability of this method is proved, its specific features are analyzed. For the spherically symmetric problem the amplitudes of the transmitted and reflected wave packets, times of tunneling and reflection in relation …
The entanglement between $SU(2) \otimes SU(2)$ internal degrees of freedom of parity and helicity for reflected and transmitted waves of Dirac-like particles scattered by a potential step along an arbitrary direction on the $x-y$ plane is quantified. Diffusion ($E \geq V$) and Klein zone ($V \geq E$) energy regimes are considered. It has been shown that, for $SU(2) \otimes SU(2)$ polarized structures of helicity eigenstates impinging the barrier, the local interaction with a {\em step} potential destroys the {\em parity-spin} separability. The framework presented here can be straightforwardly …
We analyze the quantum-mechanical behavior of a system described by a one-dimensional asymmetric potential constituted by a step plus (i) a linear barrier or (ii) an exponential barrier. We solve the energy eigenvalue equation by means of the integral representation method, classifying the independent solutions as equivalence classes of homotopic paths in the complex plane. We discuss the structure of the bound states as function of the height U_0 of the step and we study the propagation of a sharp-peaked wave packet reflected by the barrier. For both the linear and the exponential barrier we …
The Moller operators and the asociated Lippman-Schwinger equations obtained from different partitionings of the Hamiltonian for a step-like potential barrier are worked out, compared and related.
Quantum walks are well-known for their ballistic dispersion, traveling $Θ(t)$ away in $t$ steps, which is quadratically faster than a classical random walk's diffusive spreading. In physical implementations of the walk, however, the particle may need to tunnel through a potential barrier to hop, and a naive calculation suggests this could eliminate the ballistic transport. We show by explicit calculation, however, that such a loss does not occur. Rather, the $Θ(t)$ dispersion is retained, with only the coefficient changing, which additionally gives a way to detect and quantify the hopping erro…
The transmission of fermions of mass m and energy E through an electrostatic potential barrier of rectangular shape (i.e. supporting an infinite electric field), of height U> E + m - due to the many-body nature of the Dirac equation evidentiated by the Klein paradox - has been widely studied. We exploit here the analytical solution, given by Sauter for the linearly rising potential step, to show that the tunnelling rate through a more realistic trapezoidal barrier is exponentially depressed, as soon as the length of the regions supporting a finite electric field exceeds the Compton wavelenght …
The time evolution of plane waves in the presence of a 1-dimensional square quantum barrier is considered. Comparison is made between the cases of an infinite and a cut-off (shutter) initial plane wave. The difference is relevant when the results are applied to the analysis of the tunnelling regime. This work is focused on the analytical calculation of the time-evolved solution and highlights the contribution of the resonant (Gamow) states. PACS numbers: 11.10.Ef, 11.10.Lm, 04.60
We consider reflection and transmission of 2D quantum wavepackets with phase vortices (also known in optics as spatiotemporal vortex pulses) at potential step-like, delta-function, and rectangular barriers. The presence of a vortex significantly modifies the Wigner time delays and Goos-Hänchen shifts, previously studied for Gaussian-like wavepackets. In particular, the scattered wavepackets undergo non-zero time delays and lateral shifts even for purely real scattering coefficients, when the standard Wigner and Artmann formulae vanish. We derive analytical expressions for the vortex-induced ti…
We present a new potential barrier that presents the phenomenon of superradiance when the reflection coefficient $R$ is greater than one. We calculated the transmission and reflection coefficients for three different regions. The results are compared with those obtained for the hyperbolic tangent potential barrier and the step potential barrier. We also present the solution of the Klein-Gordon equation with the Lambert-W potential barrier in terms of the Heun Confluent functions.
We analyze the behavior of a quantum system described by a one-dimensional asymmetric potential consisting of a step plus a harmonic barrier. We solve the eigenvalue equation by the integral representation method, which allows us to classify the independent solutions as equivalence classes of homotopic paths in the complex plane. We then consider the propagation of a wave packet reflected by the harmonic barrier and obtain an expression for the interaction time as a function of the peak energy. For high energies we recover the classical half-period limit.
Exact reflection and transmission coefficients for supersymmetric shape-invariant potentials barriers are calculated by an analytical continuation of the asymptotic wave functions obtained via the introduction of new generalized ladder operators. The general form of the wave function is obtained by the use of the F-matrix formalism of Froman and Froman which is related to the evolution of asymptotic wave function coefficients.
Quantum many-body systems out of equilibrium pose some of the most intriguing questions in physics. Unfortunately, numerically keeping track of time evolution of states under Hamiltonian dynamics constitutes a severe challenge for all known methods. Prominently, tensor network methods are marred by an entanglement blowup, which allows to simulate systems following global quenches only to constant time. In this work, we present a scheme that allows to significantly extend the simulation time for interacting fermionic or equivalent spin systems. In the past when keeping track of evolution in one…
A method is derived to solve the massless Dirac-Weyl equation describing electron transport in a mono-layer of graphene with a scalar potential barrier U(x,t), homogeneous in the y-direction, of arbitrary x- and time dependence. Resonant enhancement of both electron backscattering and currents, across and along the barrier, is predicted when the modulation frequencies satisfy certain resonance conditions. These conditions resemble those for Shapiro-steps of driven Josephson junctions. Surprisingly, we find a non-zero y-component of the current for carriers of zero momentum along the y-axis.
From the exact solution of the Dirac-Weyl equation we find unusual currents j_y running in y-direction parallel to a time-dependent scalar potential barrier W(x,t) placed upon a monolayer of graphene, even for vanishing momentum component p_y. In their sine-like dependence on the phase difference of wave functions, describing left and right moving Dirac fermions, these currents resemble Josephson currents in superconductors, including the occurance of Shapiro steps at certain frequencies of potential oscillations. The Josephson-like currents are calculated for several specific time-dependent b…
We study the real time dynamics of a quantum system with potential barrier coupled to a heat-bath environment. Employing the path integral approach an evolution equation for the time dependent density matrix is derived. The time evolution is evaluated explicitly near the barrier top in the temperature region where quantum effects become important. It is shown that there exists a quasi-stationary state with a constant flux across the potential barrier. This state generalizes the Kramers flux solution of the classical Fokker-Planck equation to the quantum regime. In the temperature range explore…
We examine an analytical expression for the survival probability for the time evolution of quantum decay to discuss a regime where quantum decay is nonexponential at all times. We find that the interference between the exponential and nonexponential terms of the survival amplitude modifies the usual exponential decay regime in systems where the ratio of the resonance energy to the decay width, is less than 0.3. We suggest that such regime could be observed in semiconductor double-barrier resonant quantum structures with appropriate parameters.
We analyze the dynamical evolution of the resonant tunneling of an ensemble of electrons through a double barrier in the presence of the self-consistent potential created by the charge accumulation in the well. The intrinsic nonlinearity of the transmission process is shown to lead to oscillations of the stored charge and of the transmitted and reflected fluxes. The dependence on the electrostatic feedback induced by the self-consistent potential and on the energy width of the incident distribution is discussed.
Process of formation of the universe with its further expansion in the first evolution stage is investigated in the framework of Friedmann-Robertson-Walker metrics on the basis of quantum model, where a new type of matter is introduced, which energy density is dependent on velocity of the expansion. It is shown that such an improvement of the model forms potential barrier for the flat universe at $k=0$ (in contrast with generalized Chaplygin gas model). Peculiarities of wave function are analyzed in details, which is calculated by fully quantum (non-semiclassic) approach, for the different bar…
