Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

physica status solidi (b)

On Relativistic Transmissions through a Symmetrical Pair of Delta-Barriers or Delta-Wells

I. Yanetka

Published Aug 1, 2002• 7 citations
Publisher

Abstract

The solution of the one-dimensional Dirac equation is examined for the symmetrical double delta-function potential. A formula for the relativistic transmission coefficient is derived from the solution. The maximum condition as well as the minimum condition for the transmission coefficient is determined. The transmission coefficient is shown to be equal to unity in its maximum. In its minimum, it is smaller than the transmission coefficient for the single delta-function potential of the twofold strength. Further, some differences between the relativistic and non-relativistic transmission coefficient are presented. It is found that the wave numbers, at which the maxima and minima occur according to the non-relativistic treatment, are shifted by the relativistic treatment. Unlike the non-relativistic transmission coefficient, the relativistic transmission coefficient does not approach unity in the range of high energies. It oscillates between unity and a constant. The value of this constant depends on the strength of the delta-interaction. Thus, the relativistic corrections to the transmission coefficient for the symmetrical double delta-function potential are found to represent significant shifts in its characteristics.

DOI
10.1002/1521-3951(200208)232:2<196::aid-pssb196>3.0.co;2-4

Related papers

On the Transmission Coefficient for the Double δ'-Function Potential

I. Yanetka

The unbound-state solution of the Schrödinger equation is examined for the potential that is defined as the sum of two δ'-functions of non-equal strengths.The analytical expression for the transmission coemcient is derived from the solution.The transmission coemcient has an absolute maximum at the zero wave number.Further, the transmission coemcient is shown to exhibit relative maxima and minima.Moreover, it is proved that the transmission coemcient in its relative maxima is larger and in its relative minima is smaller than the transmission coemcient for the corresponding single δ' -functiοn potential.

Acta Physica Polonica A•Jun 1, 2000•2 citations

Transmission Coefficients of One-dimensional Symmetric Double Rectangular Potential andδPotential

Luo Zhi-Quan

The scattering of a free particle by one - dimensional symmetric double rectangular potential(ODSDRP) andδpotential barriers is analyzed.Bound - states energy levels of the free particle in the ODSDRP andδpotential wells are discussed.Transmission coefficients and equation of bound -states energy levels are strictly derived.

Journal of China West Normal University•Jan 1, 2011•0 citations

Revisiting double Dirac delta potential

Zafar Ahmed, Sachin Kumar, Mayank Sharma +1 more

Abstract We study a general double Dirac delta potential to show that this is the simplest yet still versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission ( T = 1). Perfect transmission energies turn out to be the critical property of symmetric and anti-symmetric cases wherein these discrete energies are found to correspond to the eigenvalues of a Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission (or zero reflectivity, R ( E ) ) at energy E = 0 is non-intuitive. However, this has been found earlier and called the ‘threshold anomaly’. Here we show that it is a critical phenomenon and we can have 0 ≤ R ( 0 ) < 1 when the parameters of the double delta potential satisfy an interesting condition. We also invoke a zero-energy and zero curvature eigenstate ( ψ ( x ) = Ax + B ) of the delta well between two symmetric rigid walls for R ( 0 ) = 0 . We resolve that the resonant energies and the perfect transmission energies are different and they arise differently.

arXiv:1603.07726•European Journal of Physics•Jun 10, 2016•20 citations

Application of a quantum wave impedance approach for a set of delta-potentials in 1D

Orest Hryhorchak

An application of the quantum wave impedance method for the analytical study of quantum-mechanical systems containing a set of delta-wells and delta-barriers is considered in this study.The algorithm for determining both transmission and reflection coefficients for a set of delta-barriers was proposed.Similarly, the scheme for analytically deriving the expressions for energy levels and wave functions of bound states for a set of delta-wells was developed.In both cases, the solution for the quantum wave impedance function in the region with a constant potential [1] and the matching conditions at the origins of deltafunctions were used.As an illustration of the application of the proposed approach, the quantum-mechanical systems with symmetric double-barrier potential and double-well potential were investigated.In the case of a single delta-well, the eigenenergy value as well as the expression for the eigenfunction were obtained directly using the non-linear differential equation for the quantum wave impedance function.The results demonstrate the effectiveness of the quantum wave impedance method in handling quantum-mechanical systems with deltalike potentials, offering a robust framework for further studies in this area.

Visnyk of the Lviv University Series Physics•Jan 1, 2024•0 citations

On Relativistic Singular Harmonic-Oscillator Potentials

Francisco Domínguez-Adame, Enrique Maciá

The Dirac equation for a harmonic-oscillator potential plus a Lorentz scalar delta-shaped term has been solved. This singular potential changes the usual oscillator energy levels. Unlike the nonrelativistic treatment, the particle cannot fall into the centre for any negative value of the delta-function strength. The apparent occurrence of double degenerate energy levels for some limiting cases is found.

Europhysics Letters (EPL)•Apr 15, 1989•11 citations

Relativistic phase time in delta-potential scattering of Dirac fermions in 1D and 2D systems

Jorge Villavicencio, Alberto Hernández-Maldonado, I. K. A. ROMERO MALDONADO

Abstract We present a unified analytical study of the transmission phase time for Dirac particles scattered by delta-function potentials in one and two dimensions. For the one-dimensional case, we obtain a closed-form expression for the transmission relativistic phase time and show that it coincides with both the reflection phase time and the self-interference delay, in agreement with general predictions for symmetric, zero-width scatterers. The resulting delay alternates periodically between positive and negative values as the coupling strength increases, and converges to the known non-relativistic expression in the appropriate limit. This delay-advance pattern can be interpreted through the spectral structure of the Dirac Hamiltonian, where electron and positron bound-state sectors correlate with time advances and delays, respectively, reflecting the underlying nature of the delta interaction. In the two-dimensional case, we derive an exact expression for the transmission phase time in graphene and examine its angular and coupling dependence. The delay exhibits a periodic, butterfly-like interference pattern governed by resonant transmission and Klein tunnelling, vanishing at normal incidence and at integer values of the reduced coupling strength. These features reveal how angular interference and relativistic dynamics shape temporal signatures of quantum transport in Dirac materials. The closed-form analytical expressions for the relativistic phase time in one dimension and in two dimensions (graphene), which constitute the main results of this work. Our results provide analytic insight into phase-time phenomena in relativistic systems and provide compact benchmarks for time-resolved quantum transport in narrow-gap and Dirac-like media.

Physica Scripta•Nov 26, 2025•0 citations
The solution of the one-dimensional Dirac equation is examined for the symmetrical double delta-function potential. A formula for the relativistic transmission coefficient is derived from the solution. The maximum condition as well as the minimum condition for the transmission coefficient is determined. The transmission coefficient is shown to be equal to unity in its maximum. In its minimum, it is smaller than the transmission coefficient for the single delta-function potential of the twofold strength. Further, some differences between the relativistic and non-relativistic transmission coefficient are presented. It is found that the wave numbers, at which the maxima and minima occur according to the non-relativistic treatment, are shifted by the relativistic treatment. Unlike the non-relativistic transmission coefficient, the relativistic transmission coefficient does not approach unity in the range of high energies. It oscillates between unity and a constant. The value of this constant depends on the strength of the delta-interaction. Thus, the relativistic corrections to the transmission coefficient for the symmetrical double delta-function potential are found to represent significant shifts in its characteristics.
The unbound-state solution of the Schrödinger equation is examined for the potential that is defined as the sum of two δ'-functions of non-equal strengths.The analytical expression for the transmission coemcient is derived from the solution.The transmission coemcient has an absolute maximum at the zero wave number.Further, the transmission coemcient is shown to exhibit relative maxima and minima.Moreover, it is proved that the transmission coemcient in its relative maxima is larger and in its relative minima is smaller than the transmission coemcient for the corresponding single δ' -functiοn potential.
Abstract We study a general double Dirac delta potential to show that this is the simplest yet still versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission ( T = 1). Perfect transmission energies turn out to be the critical property of symmetric and anti-symmetric cases wherein these discrete energies are found to correspond to the eigenvalues of a Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission (or zero reflectivity, R ( E ) ) at energy E = 0 is non-intuitive. However, this has been found earlier and called the ‘threshold anomaly’. Here we show that it is a critical phenomenon and we can have 0 ≤ R ( 0 ) < 1 when the parameters of the double delta potential satisfy an interesting condition. We also invoke a zero-energy and zero curvature eigenstate ( ψ ( x ) = Ax + B ) of the delta well between two symmetric rigid walls for R ( 0 ) = 0 . We resolve that the resonant energies and the perfect transmission energies are different and they arise differently.
An application of the quantum wave impedance method for the analytical study of quantum-mechanical systems containing a set of delta-wells and delta-barriers is considered in this study.The algorithm for determining both transmission and reflection coefficients for a set of delta-barriers was proposed.Similarly, the scheme for analytically deriving the expressions for energy levels and wave functions of bound states for a set of delta-wells was developed.In both cases, the solution for the quantum wave impedance function in the region with a constant potential [1] and the matching conditions at the origins of deltafunctions were used.As an illustration of the application of the proposed approach, the quantum-mechanical systems with symmetric double-barrier potential and double-well potential were investigated.In the case of a single delta-well, the eigenenergy value as well as the expression for the eigenfunction were obtained directly using the non-linear differential equation for the quantum wave impedance function.The results demonstrate the effectiveness of the quantum wave impedance method in handling quantum-mechanical systems with deltalike potentials, offering a robust framework for further studies in this area.
Abstract We present a unified analytical study of the transmission phase time for Dirac particles scattered by delta-function potentials in one and two dimensions. For the one-dimensional case, we obtain a closed-form expression for the transmission relativistic phase time and show that it coincides with both the reflection phase time and the self-interference delay, in agreement with general predictions for symmetric, zero-width scatterers. The resulting delay alternates periodically between positive and negative values as the coupling strength increases, and converges to the known non-relativistic expression in the appropriate limit. This delay-advance pattern can be interpreted through the spectral structure of the Dirac Hamiltonian, where electron and positron bound-state sectors correlate with time advances and delays, respectively, reflecting the underlying nature of the delta interaction. In the two-dimensional case, we derive an exact expression for the transmission phase time in graphene and examine its angular and coupling dependence. The delay exhibits a periodic, butterfly-like interference pattern governed by resonant transmission and Klein tunnelling, vanishing at normal incidence and at integer values of the reduced coupling strength. These features reveal how angular interference and relativistic dynamics shape temporal signatures of quantum transport in Dirac materials. The closed-form analytical expressions for the relativistic phase time in one dimension and in two dimensions (graphene), which constitute the main results of this work. Our results provide analytic insight into phase-time phenomena in relativistic systems and provide compact benchmarks for time-resolved quantum transport in narrow-gap and Dirac-like media.