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

Journal of Physics A Mathematical and General

Unitary transformation approach to the exact solutions of time-dependent quantum systems withSU(1, 1) dynamical group

ShengLi Zhang, Fuli Li

Published Sep 21, 1996• 6 citations
Publisher

Abstract

A unitary transformation is introduced to determine the exact solution of the time-dependent quantum system. As examples, two quantum systems with Hamiltonian including the quadratic term of generators of SU(1,1) are solved analytically.
DOI
10.1088/0305-4470/29/18/038

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

Unitary transformation approach to the exact solution for a class of time-dependent nonlinear Hamiltonian systems

Mustapha Maamache

By performing unitary transformations on the time-dependent Schrödinger equation, the exact solution for a class of nonlinear Hamiltonian systems is obtained and it is shown that these time-dependent problems are related to the associated time- independent problems. In addition, the evolution operator is derived. Nonadiabatic Berry’s phase is calculated on the basis of the exact solution. The theory is applied to some illustrative examples.

Journal of Mathematical Physics•Jan 1, 1998•28 citations

Unitary transformation approach to the cyclic evolution of SU(1, 1) and SU(2) time-dependent systems and geometrical phases

Mustapha Maamache

Using the time-dependent unitary transformation instead of the invariant operator, the solutions of SU(1, 1) and SU(2) time-dependent quantum systems are obtained. It is shown that the evolution operator is decomposed into the product form of two unitary operators in such a way that one of them has the same periodicity as the Hamiltonian and the other correspond to the Floquet operator which gives the cyclic states and their associated phases over one period of the evolution. The non-adiabatic Berry's (or Aharonov-Anandan) phases are determined totally by such a unitary transformation.

Journal of Physics A Mathematical and General•Aug 14, 1998•19 citations

Quantum Dynamics

Uri Peskin

Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.

Cambridge University Press eBooks•May 11, 2023•0 citations

Quantum canonical transformations and exact solution of the Schrödinger equation

Ali Mostafazadeh

Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly time-dependent Hamiltonians of the form H(t)=R(t)⋅J, where R is an arbitrary real vector-valued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case R=(1,0,ν(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)=R(t)⋅J there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang–Mills gauge theory) is also pointed out.

arXiv:quant-ph/9611052•Journal of Mathematical Physics•Jul 1, 1997•17 citations

NON-UNITARY TRANSFORMATION OF QUANTUM TIME-DEPENDENT NON-HERMITIAN SYSTEMS

Mustapha Maamache

We provide a new perspective on non-Hermitian evolution in quantum mechanics by emphasizing the same method as in the Hermitian quantum evolution. We first give a precise description of the non unitary transformation and the associated evolution, and collecting the basic results around it and postulating the norm preserving. This cautionary postulate imposing that the time evolution of a non Hermitian quantum system preserves the inner products between the associated states must not be read naively. We also give an example showing that the solutions of time-dependent non Hermitian Hamiltonian systems given by a linear combination of SU(1,1) and SU(2) are obtained thanks to time-dependent non-unitary transformation.

Acta Polytechnica•Dec 30, 2017•14 citations
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.
Using the time-dependent unitary transformation instead of the invariant operator, the solutions of SU(1, 1) and SU(2) time-dependent quantum systems are obtained. It is shown that the evolution operator is decomposed into the product form of two unitary operators in such a way that one of them has the same periodicity as the Hamiltonian and the other correspond to the Floquet operator which gives the cyclic states and their associated phases over one period of the evolution. The non-adiabatic Berry's (or Aharonov-Anandan) phases are determined totally by such a unitary transformation.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly time-dependent Hamiltonians of the form H(t)=R(t)⋅J, where R is an arbitrary real vector-valued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case R=(1,0,ν(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)=R(t)⋅J there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang–Mills gauge theory) is also pointed out.
We provide a new perspective on non-Hermitian evolution in quantum mechanics by emphasizing the same method as in the Hermitian quantum evolution. We first give a precise description of the non unitary transformation and the associated evolution, and collecting the basic results around it and postulating the norm preserving. This cautionary postulate imposing that the time evolution of a non Hermitian quantum system preserves the inner products between the associated states must not be read naively. We also give an example showing that the solutions of time-dependent non Hermitian Hamiltonian systems given by a linear combination of SU(1,1) and SU(2) are obtained thanks to time-dependent non-unitary transformation.