Constrained quantum systems as an adiabatic problem

Wachsmuth, Jakob; Teufel, Stefan

doi:10.1103/physreva.82.022112

Cited by 19 publications

(24 citation statements)

References 31 publications

(143 reference statements)

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“…As mentioned in the notes to Sect. 1.1, the thin tube limit can use potentials instead of Dirichlet conditions; one can regard the problem alternatively as an adiabatic approximation [WT10]. Section 1.7 The repulsive effect of torsion was first pointed out in [CBr96].…”

Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

146

133

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More information about this series at http://www.springer.com/series/720The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs in Physics (TMP) publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals that the series is a suitable publication platform for both the mathematical and the theoretical physicist. The wider scope of the series is reflected by the composition of the editorial board, comprising both physicists and mathematicians. The books, written in a didactic style and containing a certain amount of elementary background material, bridge the gap between advanced textbooks and research monographs. They can thus serve as basis for advanced studies, not only for lectures and seminars at graduate level, but also for scientists entering a field of research.

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Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

146

133

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“…The analysis in this section is along the line of what is usually known as constrained quantum system in the literature. A partial list of references is [21][22][23][24][25][26][27][28]. Here one considers a nonrelativistic classical system in an ambient space with a potential that tries to confine the motion into a submanifold.…”

Section: Isrn High Energy Physicsmentioning

confidence: 99%

“…In the discussion of Section 4.2.1 we assumed that starting from the direct product coordinate system = ( 1 , 2 ) (see (28), (29)) on M × M one can arrive at another, namely, = ( , ), such that the transformed vielbein components are given, up to a constant conformal transformation, by (32) up to quadratic order in . Here we will explicitly construct in a region whose overlap with the diagonal submanifold is sufficiently small.…”

Section: Existence Of ( )-Systemmentioning

confidence: 99%

On a Semiclassical Limit of Loop Space Quantum Mechanics

Mukhopadhyay

2013

ISRN High Energy Physics

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Following earlier work, we view two-dimensional nonlinear sigma model as single particle quantum mechanics in the free loop space of the target space. In a natural semiclassical limit of this model, the wavefunction localizes on the submanifold of vanishing loops. One would expect that the semiclassical expansion should be related to the tubular expansion of the theory around the submanifold and effective dynamics on the submanifold is obtainable using Born-Oppenheimer approximation. We develop a framework to carry out such an analysis at the leading order. In particular, we show that the linearized tachyon effective equation is correctly reproduced up to divergent terms all proportional to the Ricci scalar. The steps are as follows: first we define a finite dimensional analogue of the loop space quantum mechanics (LSQM) where we discuss its tubular expansion and how that is related to a semiclassical expansion of the Hamiltonian. Then we study an explicit construction of the relevant tubular neighborhood in loop space using exponential maps. Such a tubular geometry is obtained from a Riemannian structure on the tangent bundle of target space which views the zero-section as a submanifold admitting a tubular neighborhood. Using this result and exploiting an analogy with the toy model, we arrive at the final result for LSQM.

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“…Then there are multiple values of u 1 corresponding to the same point along the curve, and it must be ensured that the adiabatic-to-diabatic basis transformation is still unique at each point (cf. [62]). …”

Section: Transverse Basis Transformations and Gauge-theoretical Smentioning

confidence: 99%

“…Going beyond this lowest order, the almost invariant subspace is modified by a prescribed "tilt" admixing other modes [61]. Assuming a particular scaling behavior of the various length scales, the first few orders of this adiabatic perturbation theory expansion for the quantum waveguide problem have been worked out in [62,63] (see also the recent work Ref. [64]).…”

Section: Introductionmentioning

confidence: 99%

Nonadiabatic couplings and gauge-theoretical structure of curved quantum waveguides

Stockhofe

Schmelcher

2014

Phys. Rev. A

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We investigate the quantum mechanics of a single particle constrained to move along an arbitrary smooth reference curve by a confinement that is allowed to vary along the waveguide. The Schrödinger equation is evaluated in the adapted coordinate frame and a transverse-mode decomposition is performed, taking into account both curvature and torsion effects and the possibility of a cross-section potential that changes along the curve in an arbitrary way. We discuss the adiabatic structure of the problem, and examine nonadiabatic couplings that arise due to the curved geometry, the varying transverse profile, and their interplay. The exact multimode matrix Hamiltonian is taken as the natural starting point for few-mode approximations. Such approximate equations are provided, and it is worked out how these recover known results for twisting waveguides and can be applied to other types of waveguide designs. The quantum waveguide Hamiltonian is recast into a form that clearly illustrates how it generalizes the Born-Oppenheimer Hamiltonian encountered in molecular physics. In analogy to the latter, we explore the local gauge structure inherent to the quantum waveguide problem and suggest the usefulness of diabatic states, giving an explicit construction of the adiabatic-to-diabatic basis transformation.

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Constrained quantum systems as an adiabatic problem

Cited by 19 publications

References 31 publications

Quantum Waveguides

Quantum Waveguides

On a Semiclassical Limit of Loop Space Quantum Mechanics

Nonadiabatic couplings and gauge-theoretical structure of curved quantum waveguides

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