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

arXiv:0706.1630

An analytic study of the ionization from an ultrathin quantum well in a weak electrostatic field

Ilki Kim

Published Jun 12, 2007• 0 citations
arXivHTMLPDF

Abstract

We consider the time evolution of a particle bound by an attractive one-dimensional delta-function potential (at x = 0) when a uniform electrostatic field (F) is applied. We explore explicit expressions for the time-dependent wavefunction ψ_F(x,t) and the ionization probability {\mathcal{P}}(t), respectively, in the weak-field limit. In doing so, ψ_F(0,t) is a key element to their evaluation. We obtain a closed expression for ψ_F(0,t) which is an excellent approximation of the exact result being a numerical solution of the Lippmann-Schwinger integral equation. The resulting probability density |ψ_F(0,t)|^2, as a simple alternative to {\mathcal{P}}(t), is also in good agreement to its counterpart from the exact one. In doing this, we also find a new and useful integral identity of the Airy function.

DOI
10.48550/arxiv.0706.1630

Related papers

Electrostatic-field-induced dynamics in an ultrathin quantum well

Ilki Kim

We consider the time evolution of a particle subjected to both a uniform electrostatic field F and a one-dimensional delta-function potential well. We derive the propagator K_F(x,t|x',0) of this system, directly leading to the wavefunction psi_F(x,t), in which its essential ingredient K_F(0,t|0,0), accounting for the ionization-recombination in the bound-continuum transition, is exactly expressed in terms of the multiple hypergeometric functions F(z_1,z_2,...,z_n). And then we obtain the ingredient K_F(0,t|0,0) in an appropriate approximation scheme, expressed in terms of the generalized hypergeometric functions p_F_q(z) being much more transparent to physically interpret and much more accessible in their numerical evaluation than the functions F(z_1,z_2,...,z_n).

arXiv:1205.2312•May 10, 2012•0 citations

Exact Ionization Amplitudes for a Delta-Function Well in an Arbitrary-Strength DC Electric Field

Ilki Kim, Gerald J. Iafrate

We investigate finite-time field-induced ionization from a one-dimensional attractive delta-function well in a uniform dc electric field of arbitrary strength. Our aim is to determine the physical bound-state survival amplitude \(a_b(t)\) at any observation time without constructing the complete time-dependent propagator or wavefunction. In the gauge-equivalent Kramers--Henneberger representation, the field appears as motion of the contact point, and the resulting dynamics reduces to a closed Volterra equation. Exact endpoint-phase factorization organizes the full chronological rescattering history into a relative-time convolution hierarchy and an exact resolvent. By introducing the accumulated bound-state amplitude and reorganizing its two-time domain in relative and complementary center times, the final contact contribution becomes an explicit boundary integral. All spatial integrations are carried out analytically. The field-driven contact-free term is obtained in closed form using the Faddeeva function, while the direct and repeated-rescattering terms are expressed as explicitly evaluable time integrals. The result applies at arbitrary dc-field strength and finite time, with no weak-field expansion, rescattering truncation, or asymptotic-time approximation. It is verified through the exact field-free limit and an independent numerical solution of the original physical Volterra equation. The formulation gives an exact nonperturbative solution of a fundamental ionization model and reveals otherwise hidden analytical structure in driven quantum dynamics.

quant-pharXiv:2609.16507•Sep 16, 2026

Decay in a uniform field: an exactly solvable model

R. M. Cavalcanti, Paola Giacconi A, Roberto Soldati

We investigate the time evolution of the decay (or ionization) probability of a D -dimensional model atom ( D = 1, 2, 3) in the presence of a uniform (i.e., static and homogeneous) background field. The model atom consists in a non-relativistic point particle in the presence of a point-like attractive well. It is shown that the model exhibits infinitely many resonances leading to possible deviations from the naive exponential decay law of the non-decay (or survival) probability of the initial atomic quantum state. Almost stable states exist due to the presence of the attractive interaction, no matter how weak it is. Analytic estimates as well as numerical evaluation of the decay rates are explicitly given and discussed.

arXiv:quant-ph/0307232•Journal of Physics A Mathematical and General•Nov 18, 2003•25 citations

An exact treatment of the Dirac delta function potential in the Schrödinger equation

Daniel A. Atkinson, Horace W. Crater

This paper presents several cases in which the effects of the addition of a delta function potential on bound states can be computed exactly. In the case of the one dimensional Schrödinger equation, a Green’s function technique is used to compute an exact implicit expression that gives the effect of the one−dimensional delta function potential on the bound states eigenvalues. It is also shown that the weak coupling limit of the implicit expression agrees with the perturbative treatment of the delta function. It is pointed out that this technique cannot be applied to obtain an exact treatment of the delta function potential δ3(r) in the three−dimensional Schrödinger equation. Instead, an alternate definition of the delta function in three dimensions as a limit of a sequence of square well potentials is used. Unlike the one dimensional case, the exact effect on the eigenvalue does not agree in the limit of weak coupling with the perturbative results. This indicates that the three−dimensional delta function potential should not be used to approximate strongly coupled short range potentials for which perturbation theory is inadequate.

American Journal of Physics•Apr 1, 1975•62 citations

Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a One-Dimensional Potential Exhibiting Non-Exponential Decay at All Times

A. Petridis, L. P. Staunton, Jon Vermedahl +1 more

The departure at large times from exponential decay in the case of resonance wavefunctions is mathematically demonstrated. Then, exact, analytical solutions to the time-dependent Schrödinger equation in one dimension are developed for a time-independent potential consisting of an infinite wall and a repulsive delta function. The exact solutions are obtained by means of a superposition of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions for which the resulting integrals can be evaluated exactly. Square-integrability and the boundary conditions are satisfied. The simplest of the obtained solutions is presented and the probability for the particle to be found inside the potential well as a function of time is calculated. The system exhibits non-exponential decay for all times; the probability decreases at large times as . Other exact solutions found exhibit power law behavior at large times. The results are generalized to all normalizable solutions to this problem. Additionally, numerical solutions are obtained using the staggered leap-frog algorithm for select potentials exhibiting the prevalence of non-exponential decay at short times.

Journal of Modern Physics•Jan 1, 2010•6 citations

Electron delocalization and transfer induced by a time-dependent potential: Exact treatment of a simple model—formalism

J.-M. Lopez-Castillo, A.-M. Daré, Jean‐Paul Jay‐Gerin +1 more

A quantum one-particle model for electron delocalization and transfer induced by a time-dependent potential is studied with a formalism based on the Lippman-Schwinger approach and fractional as well as operational calculus methods. The potential examined consists of a line of an arbitrary number of short-range (\ensuremath{\delta} function) potentials whose amplitudes have an arbitrary functional dependence on time. The temporal evolution of the ionization probability, the energy spectrum, and the time of transfer are obtained analytically and calculated numerically through a stable and accuracy-controllable technique without any assumption regarding the strength of the perturbing potential.

Physical Review A•Jan 1, 1991•1 citation
We consider the time evolution of a particle bound by an attractive one-dimensional delta-function potential (at x = 0) when a uniform electrostatic field (F) is applied. We explore explicit expressions for the time-dependent wavefunction ψ_F(x,t) and the ionization probability {\mathcal{P}}(t), respectively, in the weak-field limit. In doing so, ψ_F(0,t) is a key element to their evaluation. We obtain a closed expression for ψ_F(0,t) which is an excellent approximation of the exact result being a numerical solution of the Lippmann-Schwinger integral equation. The resulting probability density |ψ_F(0,t)|^2, as a simple alternative to {\mathcal{P}}(t), is also in good agreement to its counterpart from the exact one. In doing this, we also find a new and useful integral identity of the Airy function.
We consider the time evolution of a particle subjected to both a uniform electrostatic field F and a one-dimensional delta-function potential well. We derive the propagator K_F(x,t|x',0) of this system, directly leading to the wavefunction psi_F(x,t), in which its essential ingredient K_F(0,t|0,0), accounting for the ionization-recombination in the bound-continuum transition, is exactly expressed in terms of the multiple hypergeometric functions F(z_1,z_2,...,z_n). And then we obtain the ingredient K_F(0,t|0,0) in an appropriate approximation scheme, expressed in terms of the generalized hypergeometric functions p_F_q(z) being much more transparent to physically interpret and much more accessible in their numerical evaluation than the functions F(z_1,z_2,...,z_n).
We investigate finite-time field-induced ionization from a one-dimensional attractive delta-function well in a uniform dc electric field of arbitrary strength. Our aim is to determine the physical bound-state survival amplitude ab(t)a_b(t)ab​(t) at any observation time without constructing the complete time-dependent propagator or wavefunction. In the gauge-equivalent Kramers--Henneberger representation, the field appears as motion of the contact point, and the resulting dynamics reduces to a closed Volterra equation. Exact endpoint-phase factorization organizes the full chronological rescattering history into a relative-time convolution hierarchy and an exact resolvent. By introducing the accumulated bound-state amplitude and reorganizing its two-time domain in relative and complementary center times, the final contact contribution becomes an explicit boundary integral. All spatial integrations are carried out analytically. The field-driven contact-free term is obtained in closed form using the Faddeeva function, while the direct and repeated-rescattering terms are expressed as explicitly evaluable time integrals. The result applies at arbitrary dc-field strength and finite time, with no weak-field expansion, rescattering truncation, or asymptotic-time approximation. It is verified through the exact field-free limit and an independent numerical solution of the original physical Volterra equation. The formulation gives an exact nonperturbative solution of a fundamental ionization model and reveals otherwise hidden analytical structure in driven quantum dynamics.
We investigate the time evolution of the decay (or ionization) probability of a D -dimensional model atom ( D = 1, 2, 3) in the presence of a uniform (i.e., static and homogeneous) background field. The model atom consists in a non-relativistic point particle in the presence of a point-like attractive well. It is shown that the model exhibits infinitely many resonances leading to possible deviations from the naive exponential decay law of the non-decay (or survival) probability of the initial atomic quantum state. Almost stable states exist due to the presence of the attractive interaction, no matter how weak it is. Analytic estimates as well as numerical evaluation of the decay rates are explicitly given and discussed.
This paper presents several cases in which the effects of the addition of a delta function potential on bound states can be computed exactly. In the case of the one dimensional Schrödinger equation, a Green’s function technique is used to compute an exact implicit expression that gives the effect of the one−dimensional delta function potential on the bound states eigenvalues. It is also shown that the weak coupling limit of the implicit expression agrees with the perturbative treatment of the delta function. It is pointed out that this technique cannot be applied to obtain an exact treatment of the delta function potential δ3(r) in the three−dimensional Schrödinger equation. Instead, an alternate definition of the delta function in three dimensions as a limit of a sequence of square well potentials is used. Unlike the one dimensional case, the exact effect on the eigenvalue does not agree in the limit of weak coupling with the perturbative results. This indicates that the three−dimensional delta function potential should not be used to approximate strongly coupled short range potentials for which perturbation theory is inadequate.
The departure at large times from exponential decay in the case of resonance wavefunctions is mathematically demonstrated. Then, exact, analytical solutions to the time-dependent Schrödinger equation in one dimension are developed for a time-independent potential consisting of an infinite wall and a repulsive delta function. The exact solutions are obtained by means of a superposition of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions for which the resulting integrals can be evaluated exactly. Square-integrability and the boundary conditions are satisfied. The simplest of the obtained solutions is presented and the probability for the particle to be found inside the potential well as a function of time is calculated. The system exhibits non-exponential decay for all times; the probability decreases at large times as . Other exact solutions found exhibit power law behavior at large times. The results are generalized to all normalizable solutions to this problem. Additionally, numerical solutions are obtained using the staggered leap-frog algorithm for select potentials exhibiting the prevalence of non-exponential decay at short times.
A quantum one-particle model for electron delocalization and transfer induced by a time-dependent potential is studied with a formalism based on the Lippman-Schwinger approach and fractional as well as operational calculus methods. The potential examined consists of a line of an arbitrary number of short-range (\ensuremath{\delta} function) potentials whose amplitudes have an arbitrary functional dependence on time. The temporal evolution of the ionization probability, the energy spectrum, and the time of transfer are obtained analytically and calculated numerically through a stable and accuracy-controllable technique without any assumption regarding the strength of the perturbing potential.