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

Physical Review A

Unitary transformations for the time-dependent quantum oscillator

Alla N. Seleznyova

Published Feb 1, 1995• 29 citations
Publisher

Abstract

An approach to study the time-dependent quantum oscillator with the Schr\"odinger operator S(\ensuremath{\Omega}(t))\ensuremath{\equiv}1/2[${\mathbf{p}}^{2}$+${\mathrm{\ensuremath{\Omega}}}^{2}$(t)${\mathbf{x}}^{2}$]-i(\ensuremath{\partial}/\ensuremath{\partial}t) is presented. A family of unitary operators {${\mathbf{W}}_{1\mathit{z}}$(t)} is found such that S(\ensuremath{\Omega}(t))=${\mathbf{W}}_{1\mathit{z}}$(t)S(\ensuremath{\omega})${\mathbf{W}}_{1\mathit{z}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$(t), where z=z(t) is a certain function and \ensuremath{\omega} is a positive number. Wave vectors, the boson creation and destruction operators, the evolution operator, and invariants for the time-dependent oscillator are obtained by means of {${\mathbf{W}}_{1\mathit{z}}$(t)}. The approach is applied to calculate transition probabilities, Berry phases, and to study coherent states of the time-dependent oscillator.

DOI
10.1103/physreva.51.950

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

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

Unitary transformation approach to the exact solution for the singular oscillator

Mustapha Maamache

By performing unitary transformations, the exact solution of the time-dependent singular oscillator is obtained. The invariant operator and the auxiliary equation are rigorously established. The non-adiabatic Berry's phase is calculated.

Journal of Physics A Mathematical and General•Jun 7, 1996•39 citations

Unitary approach to the quantum forced harmonic oscillator

D. Velasco-Martínez, Victor Guadalupe Ibarra‐Sierra, Juan Carlos Sandoval-Santana +2 more

In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.

arXiv:1409.0236•Aug 31, 2014•1 citation

Elementary derivation of the quantum propagator for the harmonic oscillator

Jiushu Shao

Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.

American Journal of Physics•Sep 21, 2016•9 citations

Time-dependent quantum systems and the invariant Hermitian operator

Y. Lai, J.-Q. Liang, Harald J W Müller-Kirsten +1 more

We study the time evolution of quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators. The invariant Hermitian operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. With the help of the invariant Hermitian operator we obtain not only the exact solutions of the Schr\"odinger equation but also the time-evolution operator. The adiabatic and nonadiabatic Berry phases are calculated with the exact solutions. \textcopyright{} 1996 The American Physical Society.

Physical Review A•May 1, 1996•93 citations
An approach to study the time-dependent quantum oscillator with the Schr"odinger operator S(\ensuremath{\Omega}(t))\ensuremath{\equiv}1/2[p2{\mathbf{p}}^{2}p2+\ensuremathΩ2{\mathrm{\ensuremath{\Omega}}}^{2}\ensuremathΩ2(t)x2{\mathbf{x}}^{2}x2]-i(\ensuremath{\partial}/\ensuremath{\partial}t) is presented. A family of unitary operators {W1z{\mathbf{W}}_{1\mathit{z}}W1z​(t)} is found such that S(\ensuremath{\Omega}(t))=W1z{\mathbf{W}}_{1\mathit{z}}W1z​(t)S(\ensuremath{\omega})W1z\ifmmode∘\else\textdegree\fi{\mathbf{W}}_{1\mathit{z}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}W1z\ifmmode∘\else\textdegree\fi​(t), where z=z(t) is a certain function and \ensuremath{\omega} is a positive number. Wave vectors, the boson creation and destruction operators, the evolution operator, and invariants for the time-dependent oscillator are obtained by means of {W1z{\mathbf{W}}_{1\mathit{z}}W1z​(t)}. The approach is applied to calculate transition probabilities, Berry phases, and to study coherent states of the time-dependent oscillator.
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.
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.
In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.
Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.
We study the time evolution of quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators. The invariant Hermitian operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. With the help of the invariant Hermitian operator we obtain not only the exact solutions of the Schr\"odinger equation but also the time-evolution operator. The adiabatic and nonadiabatic Berry phases are calculated with the exact solutions. \textcopyright{} 1996 The American Physical Society.