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:2206.04593 · Acta Polytechnica

Exact solutions for time-dependent complex symmetric potential well

Boubakeur Khantoul, Abdelhafid Bounames

Published May 2, 2023• 6 citations
arXivHTMLPDF

Abstract

Using the pseudo-invariant operator method, we investigate the model of a particle with a time-dependent mass in a complex time-dependent symmetric potential well V (x, t) = if (t) |x|. The problem is exactly solvable and the analytic expressions of the Schrödinger wavefunctions are given in terms of the Airy function. Indeed, with an appropriate choice of the time-dependent metric operators and the unitary transformations, for each region, the two corresponding pseudo-Hermitian invariants transform into a well-known time-independent Hermitian invariant which is the Hamiltonian of a particle confined in a symmetric linear potential well. The eigenfunctions of the last invariant are the Airy functions. Then, the phases obtained are real for both regions and the general solution to the problem is deduced.

DOI
10.14311/ap.2023.63.0132

Related papers

Exact time dependence of solutions to the time-dependent Schrödinger equation

Max A. Lohe

Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.

Journal of Physics A Mathematical and Theoretical•Dec 9, 2008•65 citations

On the pseudo-Hermitian invariant method for the time-dependent Non-Hermitian Hamiltonians

Boubakeur Khantoul, Abdelhafid Bounames, Mustapha Maamache

We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators $H(t)$ that generate a real phase in their time-evolution. This involves the use of invariant operators $I_{PH}(t)$ that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, $H(t)$ is not generally quasi-Hermitian and does not define an observable of the system but $I_{PH}(t)$ obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.

•Oct 28, 2016•3 citations

A Real Expectation Value of the Time-dependent Non-Hermitian Hamiltonians*

F. Kecita, Abdelhafid Bounames, Mustapha Maamache

Abstract With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent   -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the  ( t ) PT −inner product, where  ( t ) is a obtained from the charge conjugation operator  through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the  ( t ) PT normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.

arXiv:2112.13535•Physica Scripta•Nov 26, 2021•3 citations

Non-Hermitian Hamiltonian beyond PT symmetry for time-dependent S U ( 1 , 1 ) and S U ( 2 ) systems — Exact solution and geometric phase in pseudo-invariant theory

Nadjat Amaouche, Maroua Sekhri, Rahma Zerimeche +2 more

In this paper we investigate time-dependent non-Hermitian Hamiltonians, which consist of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1) and SU(2) systems in a unified manner. Then, we derive the Lewis–Riesenfeld (LR) phase, which can be separated into the dynamic and the geometrical phases. The analytical results are well consistent with those of the corresponding Hermitian Hamiltonians reported in the literature.

Physics Open•Nov 21, 2022•4 citations

Quantum Jaynes–Cummings model for a two-level system with effects of parametric time-dependences

Mounira Berrehail, Nadjet Benchiheub, Salah Menouar +1 more

An approach to exact quantum solutions of the time-dependent two energy level Jaynes–Cummings model with an imaginary photon process is represented in this work. The Lewis–Riesenfeld invariant treatment and the unitary transformation method are used for this purpose. The original Schrödinger equation is reduced to an equivalent solvable one through unitary transformations by using suitable unitary operators. The reduced equation corresponds to a simpler Hamiltonian which is written as a linear combination of the generators of the reduced-dimensional SU(2) algebra. A Hermitian invariant operator is constructed based on the same algebraic formulation and its instantaneous eigenfunctions are obtained. By utilizing such eigenfunctions, the complete quantum wave functions of the system are evaluated. Such wave functions are necessary when we analyze the quantum characteristics of the system.

Lithuanian Journal of Physics•Apr 24, 2022•2 citations

The pseudo Hermitian invariant operator and time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry

Walid Koussa, Naima Mana, Oum Kaltoum Djeghiour +1 more

We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator. Finally some interesting physical applications are suggested and discussed.

Journal of Mathematical Physics•Jul 1, 2018•18 citations
Using the pseudo-invariant operator method, we investigate the model of a particle with a time-dependent mass in a complex time-dependent symmetric potential well V (x, t) = if (t) |x|. The problem is exactly solvable and the analytic expressions of the Schrödinger wavefunctions are given in terms of the Airy function. Indeed, with an appropriate choice of the time-dependent metric operators and the unitary transformations, for each region, the two corresponding pseudo-Hermitian invariants transform into a well-known time-independent Hermitian invariant which is the Hamiltonian of a particle confined in a symmetric linear potential well. The eigenfunctions of the last invariant are the Airy functions. Then, the phases obtained are real for both regions and the general solution to the problem is deduced.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.
We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators H(t)H(t)H(t) that generate a real phase in their time-evolution. This involves the use of invariant operators IPH(t)I_{PH}(t)IPH​(t) that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, H(t)H(t)H(t) is not generally quasi-Hermitian and does not define an observable of the system but IPH(t)I_{PH}(t)IPH​(t) obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.
Abstract With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent   -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the  ( t ) PT −inner product, where  ( t ) is a obtained from the charge conjugation operator  through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the  ( t ) PT normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.
In this paper we investigate time-dependent non-Hermitian Hamiltonians, which consist of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1) and SU(2) systems in a unified manner. Then, we derive the Lewis–Riesenfeld (LR) phase, which can be separated into the dynamic and the geometrical phases. The analytical results are well consistent with those of the corresponding Hermitian Hamiltonians reported in the literature.
An approach to exact quantum solutions of the time-dependent two energy level Jaynes–Cummings model with an imaginary photon process is represented in this work. The Lewis–Riesenfeld invariant treatment and the unitary transformation method are used for this purpose. The original Schrödinger equation is reduced to an equivalent solvable one through unitary transformations by using suitable unitary operators. The reduced equation corresponds to a simpler Hamiltonian which is written as a linear combination of the generators of the reduced-dimensional SU(2) algebra. A Hermitian invariant operator is constructed based on the same algebraic formulation and its instantaneous eigenfunctions are obtained. By utilizing such eigenfunctions, the complete quantum wave functions of the system are evaluated. Such wave functions are necessary when we analyze the quantum characteristics of the system.
We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator. Finally some interesting physical applications are suggested and discussed.