Journal ArticleDOI
A direct approach to finding exact invariants for one‐dimensional time‐dependent classical Hamiltonians
H. Ralph Lewis,P. G. L. Leach +1 more
TLDR
For a classical Hamiltonian H=(1/2)p2+V(q,t) with an arbitrary time-dependent potential V(qs,t), exact invariants that can be expressed as series in positive powers of ǫ p, I(qp,p,t)=∑∞n=0pnfn(qs),t, are examined in this article.Abstract:
For a classical Hamiltonian H=(1/2) p2+V(q,t) with an arbitrary time‐dependent potential V(q,t), exact invariants that can be expressed as series in positive powers of p, I(q,p,t)=∑∞n=0pnfn(q,t), are examined. The method is based on direct use of the equation dI/dt=∂I/∂t +[I,H] =0. A recursion relation for the coefficients fn(q,t) is obtained. All potentials that admit an invariant quadratic in p are found and, for those potentials, all invariants quadratic in p are determined. The feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed. The systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation.read more
Citations
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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.
Book ChapterDOI
Time-dependent density functional theory
R. M. Dreizler,H. Köhl +1 more
TL;DR: Time-dependent density functional theory (TDDFT) has developed rapidly since its beginnings in 1984 [472, 473] as discussed by the authors, and it has been applied extensively in the literature.
Journal ArticleDOI
The Ermakov equation: A commentary
P. G. L. Leach,S.K. Andriopoulos +1 more
TL;DR: A short history of the Ermakov equation with an emphasis on its discovery by the west and the subsequent boost to research into invariants for nonlinear systems is given in this paper.
Journal ArticleDOI
Feynman path integrals: Some exact results and applications
D.C. Khandekar,Suresh V. Lawande +1 more
TL;DR: The status of exactly solvable problems within the path integral formulation of non-relativistic quantum mechanics is reviewed in this paper, where some applications of these exact results are presented.
References
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Journal ArticleDOI
Classical and Quantum Systems with Time-Dependent Harmonic-Oscillator-Type Hamiltonians
Journal ArticleDOI
Class of Exact Invariants for Classical and Quantum Time‐Dependent Harmonic Oscillators
TL;DR: In this article, a class of exact invariants for oscillator systems whose Hamiltonians are H=(1/2e)[p 2 + Ω 2 (t)q 2 ] is given in closed form in terms of a function ρ(t) which satisfies e 2 d 2 ρ/dt 2 + ǫ 2 (T)ρ−ρ −3 = 0.
Journal ArticleDOI
More exact invariants for the time-dependent harmonic oscillator
John R. Ray,James L. Reid +1 more
TL;DR: In this paper, an infinite number of new invariants and related auxiliary equations for the time-dependent harmonic oscillator were derived, and the number of auxiliary equations was shown to be infinite.
Journal ArticleDOI
A Note on the Time-Dependent Harmonic Oscillator
C. J. Eliezer,A. Gray +1 more
TL;DR: In this article, a physical meaning to the origin of the invariant is presented, and the relationship between the solutions of the two equations is pursued, and some particular cases when $\Omega $ has certain values are discussed.
Journal ArticleDOI
Symmetry groups and conserved quantities for the harmonic oscillator
TL;DR: The complete eight-parameter symmetry group of the one-dimensional harmonic oscillator is investigated in this paper using the fact that the system is describable by a variational principle.
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