Journal ArticleDOI
Exact time dependence of solutions to the time-dependent Schrödinger equation
Reads0
Chats0
TLDR
In this article, the Schrodinger equation with an exact time dependence is derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations.Abstract:
Solutions of the Schrodinger 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 Schrodinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.read more
Citations
More filters
Journal ArticleDOI
Shortcuts to adiabaticity: Concepts, methods, and applications
David Guéry-Odelin,Andreas Ruschhaupt,Anthony Kiely,E. Torrontegui,S. Martínez-Garaot,J. G. Muga +5 more
TL;DR: Shortcuts to adiabaticity (STA) as mentioned in this paper is a systematic approach to accomplish the same final state transfer in a faster manner, which is used for atomic and molecular physics.
Book ChapterDOI
Shortcuts to Adiabaticity
E. Torrontegui,S. Ibáñez,S. Martínez-Garaot,Michele Modugno,A. del Campo,David Guéry-Odelin,Andreas Ruschhaupt,Xi Chen,J. G. Muga +8 more
TL;DR: Shortcuts to adiabaticity as discussed by the authors are alternative fast processes which reproduce the same final populations, or even the same last state, as the adiabiabatic process in a finite, shorter time.
Journal ArticleDOI
Classical and Quantum Shortcuts to Adiabaticity for Scale-Invariant Driving
TL;DR: It is possible to design and implement dissipationless ''shortcuts'' for quantum systems as discussed by the authors, which is a natural nonequilibrium process that takes a system from one equilibrium state to another in a short time always involves dissipation.
Journal ArticleDOI
Quantum supremacy of many-particle thermal machines
TL;DR: In this paper, a nonadiabatic quantum heat engine operating an Otto cycle with a many-particle working medium, consisting of an interacting Bose gas confined in a time-dependent harmonic trap, is presented.
Journal ArticleDOI
Quantum Supremacy of Many-Particle Thermal Machines
TL;DR: In this article, a nonadiabatic quantum heat engine operating an Otto cycle with a many-particle working medium, consisting of an interacting Bose gas confined in a time-dependent harmonic trap, is presented.
References
More filters
Book
Generalized Coherent States and Their Applications
TL;DR: In this paper, the authors define the notion of generalized coherent states and define a generalization of the Coherent State Representation T?(g) of the Heisenberg-Weyl Group.
Journal ArticleDOI
Quantised Singularities in the Electromagnetic Field
TL;DR: The steady progress of physics requires for its theoretical formulation a mathematics that gets continually more advanced as discussed by the authors, and it seems likely that this process of increasing abstraction will continue in the future and that advance in physics is to be associated with a continual modification and generalisation of the axioms at the base of the mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation.
Journal ArticleDOI
Making sense of non-Hermitian Hamiltonians
TL;DR: In this article, an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space?time reflection ( ) symmetry.
Journal ArticleDOI
An Exact Quantum Theory of the Time‐Dependent Harmonic Oscillator and of a Charged Particle in a Time‐Dependent Electromagnetic Field
H. R. Lewis,W. B. Riesenfeld +1 more
TL;DR: The theory of explicitly time-dependent invariants for quantum systems whose Hamiltonians are explicitly time dependent was developed in this article, where the authors derived a simple relation between eigenstates of such an invariant and solutions of the Schrodinger equation.
Related Papers (5)
An Exact Quantum Theory of the Time‐Dependent Harmonic Oscillator and of a Charged Particle in a Time‐Dependent Electromagnetic Field
H. R. Lewis,W. B. Riesenfeld +1 more