Newtonian trajectories and quantum waves in expanding force fields

Berry, Michael; Klein, G.

doi:10.1088/0305-4470/17/9/016

Cited by 58 publications

(77 citation statements)

References 6 publications

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“…As a further application we turn to the expanding potential problem ( [260]- [262]), where we shall work from the amplitude-modulus to the phase. The time dependent potential is of the form:…”

Section: Wave Packetsmentioning

confidence: 99%

Complex States of Simple Molecular Systems

Englman¹,

Yahalom²

2002

Advances in Chemical Physics

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A review is given of phase properties in molecular wave functions, composed of a number of (and, at least, two) electronic states that become degenerate at some nearby values of the nuclear configuration. Apart from discussing phases and interference in classical (non-quantal) systems, including light-waves, the review looks at the constructability of complex wave functions from observable quantities ("the phase problem"), at the controversy regarding quantum mechanical phase-operators, at the modes of observability of phase and at the role of phases in some non-demolition measurements. Advances in experimental and (especially) theoretical aspects of Aharonov-Bohm and topological (Berry) phases are described, including those involving two-electron and relativistic systems. Several works in the phase control and revivals of molecular wave-packets are cited as developments and applications of complex-function theory. Further topics that this review touches on are: coherent states, semiclassical approximations and the Maslov index. The interrelation between time and the complex state is noted in the contexts of time delays in scattering, of time-reversal invariance and of the existence of a molecular time-arrow.When the stationary Born-Oppenheimer description for nearly degenerate state is regarded as "embedded" in a broader dynamic formulation, namely through solution of the time dependent Schrödinger equation, the wave function becomes necessarily complex. The analytic behavior of this function in the complex time-plane can be exploited to gain information about the connection between phases and moduli of the componentamplitudes. One form of this connection are a pair of reciprocal (Kramers-Kronig type) relations in the complex time-domain. These show that certain phase changes in a component amplitude (such as, e.g., lead to a non-zero Berry-phase) require changes in the amplitude-moduli, too, and imply degeneracies of the electronic state.The subject of conical degeneracy, or intersection, of adjacent potential surface is extended to nonlinear nuclear-electronic coupling, which can then result in multiple conical degeneracies on a nuclear coordinate plane. We treat cases of double and fourfold intersections, the latter under trigonal symmetry, and employ an analytic-graphical phase tracing method to obtain the resulting Berry phases as the system circles around some or all of the degeneracy points. These phases can take up values that are all integral multiples of π, with integers that vary with the physical situation (or with the model postulated for it).A further type of invariance, with respect to gauge transformation, is also tied to the complex form of the wave function. For a many-component state this leads to a consideration of tensorial (Yang-Mills type) fields F for molecular systems. It is shown that it is the truncation of the Born-Oppenheimer electron-nuclear superposition that generates the molecular Yang-Mills fields.The equations of motion for the nuclear degrees of freedom are derived from a Lagr...

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Section: Wave Packetsmentioning

confidence: 99%

Complex States of Simple Molecular Systems

Englman¹,

Yahalom²

2002

Advances in Chemical Physics

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“…In our previous work [13] we have proposed a simple metastable system with a moving potential which has height and width scaled in a specific way introduced by Berry and Klein [14]. In that model we found that a small but finite nondecay probability could persist at large time limit for an expanding potential.…”

Section: Introductionmentioning

confidence: 95%

Quantum metastability in time-periodic potentials

Lee¹,

Ho²

2005

Annals of Physics

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In this paper we investigate quantum metastability of a particle trapped in between an infinite wall and a square barrier, with either a time-periodically oscillating barrier (Model A) or bottom of the well (Model B). Based on the Floquet theory, we derive in each case an equation which determines the stability of the metastable system. We study the influence on the stability of two Floquet states when their Floquet energies (real part) encounter a direct or an avoided crossing at resonance. The effect of the amplitude of oscillation on the nature of crossing of Floquet energies is also discussed. It is found that by adiabatically changing the frequency and amplitude of the oscillation field, one can manipulate the stability of states in the well. By means of a discrete transform, the two models are shown to have exactly the same Floquet energy spectrum at the same oscillating amplitude and frequency. The equivalence of the models is also demonstrated by means of the principle of gauge invariance.

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“…We assume that the potential V (x, t) is of the scaling form proposed by Berry and Klein [11], namely,…”

Section: Schrödinger Equation With a Scaling Potentialmentioning

confidence: 99%

“…This class of potentials was introduced by Berry and Klein [11]. Potentials in this class have their heights and widths scaled in a specific way so that one can transform the potential into a stationary one.…”

Section: Introductionmentioning

confidence: 99%

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Quantum metastability in a class of moving potentials

Lee

2002

Phys. Rev. A

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In this paper we consider quantum metastability in a class of moving potentials introduced by Berry and Klein. Potential in this class has its height and width scaled in a specific way so that it can be transformed into a stationary one. In deriving the non-decay probability of the system, we argue that the appropriate technique to use is the less known method of scattering states. This method is illustrated through two examples, namely, a moving delta-potential and a moving barrier potential. For expanding potentials, one finds that a small but finite non-decay probability persists at large times. Generalization to scaling potentials of arbitrary shape is briefly indicated.

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Newtonian trajectories and quantum waves in expanding force fields

Cited by 58 publications

References 6 publications

Complex States of Simple Molecular Systems

Complex States of Simple Molecular Systems

Quantum metastability in time-periodic potentials

Quantum metastability in a class of moving potentials

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