Passage-time distributions from a spin-boson detector model

The passage-time distribution for a spread-out quantum particle to traverse a specific region is calculated using a detailed quantum model for the detector involved. That model, developed and investigated in earlier works, is based on the detected particle's enhancement of the coupling between a collection of spins (in a metastable state) and their environment. We treat the continuum limit of the model, under the assumption of the Markov property, and calculate the particle state immediately after the first detection. An explicit example with 15 boson modes shows excellent agreement between the discrete model and the continuum limit. Analytical expressions for the passage-time distribution as well as numerical examples are presented. The precision of the measurement scheme is estimated and its optimization discussed. For slow particles, the precision goes like $E^{-3/4}$, which improves previous $E^{-1}$ estimates, obtained with a quantum clock model.Comment: 11 pages, 6 figures; minor changes, references corrected; accepted for publication in Phys. Rev.

Closed Path Integrals and Renormalisation in Quantum Mechanics

We suggest a closed form expression for the path integral of quantum transition amplitudes. We introduce a quantum action with renormalized parameters. We present numerical results for the $V \sim x^{4}$ potential. The renormalized action is relevant for quantum chaos and quantum instantons.Comment: Revised text, 1 figure added; Text (LaTeX file), 1 Figure (ps file

Analysis of a three-component model phase diagram by Catastrophe Theory

We analyze the thermodynamical potential of a lattice gas model with three components and five parameters using the methods of Catastrophe Theory. We find the highest singularity, which has codimension five, and establish its transversality. Hence the corresponding seven-degree Landau potential, the canonical form Wigwam or $A_6$, constitutes the adequate starting point to study the overall phase diagram of this model.Comment: 16 pages, Latex file, submitted to Phys. Rev.

On the exactness of the Semi-Classical Approximation for Non-Relativistic One Dimensional Propagators

For one dimensional non-relativistic quantum mechanical problems, we investigate the conditions for all the position dependence of the propagator to be in its phase, that is, the semi-classical approximation to be exact. For velocity independent potentials we find that: (i) the potential must be quadratic in space, but can have arbitrary time dependence. (ii) the phase may be made proportional to the classical action, and the magnitude (``fluctuation factor'') can also be found from the classical solution. (iii) for the driven harmonic oscillator the fluctuation factor is independent of the driving term.Comment: 7 pages, latex, no figures, published in journal of physics

Transition-Event Durations in One Dimensional Activated Processes

Despite their importance in activated processes, transition-event durations -- which are much shorter than first passage times -- have not received a complete theoretical treatment. We therefore study the distribution of durations of transition events over a barrier in a one-dimensional system undergoing over-damped Langevin dynamics.Comment: 39 pages, 11 figure

Topological quenching of the tunnel splitting for a particle in a double-well potential on a planar loop

The motion of a particle along a one-dimensional closed curve in a plane is considered. The only restriction on the shape of the loop is that it must be invariant under a twofold rotation about an axis perpendicular to the plane of motion. Along the curve a symmetric double-well potential is present leading to a twofold degeneracy of the classical ground state. In quantum mechanics, this degeneracy is lifted: the energies of the ground state and the first excited state are separated from each other by a slight difference ¿E, the tunnel splitting. Although a magnetic field perpendicular to the plane of the loop does not influence the classical motion of the charged particle, the quantum-mechanical separation of levels turns out to be a function of its strength B. The dependence of ¿E on the field B is oscillatory: for specific discrete values Bn the splitting drops to zero, indicating a twofold degeneracy of the ground state. This result is obtained within the path-integral formulation of quantum mechanics; in particular, the semiclassical instanton method is used. The origin of the quenched splitting is intuitively obvious: it is due to the fact that the configuration space of the system is not simply connected, thus allowing for destructive interference of quantum-mechanical amplitudes. From an abstract point of view this phenomenon can be traced back to the existence of a topological term in the Lagrangian and a nonsimply connected configuration space. In principle, it should be possible to observe the splitting in appropriately fabricated mesoscopic rings consisting of normally conducting metal

Renormalisation in Quantum Mechanics

We study a recently proposed quantum action depending on temperature. We construct a renormalisation group equation describing the flow of action parameters with temperature. At zero temperature the quantum action is obtained analytically and is found free of higher time derivatives. It makes the quantum action an ideal tool to investigate quantum chaos and quantum instantons.Comment: replaced version with new figs. Text (LaTeX), 3 Figs. (ps

Path Integral Approach for Spaces of Non-constant Curvature in Three Dimensions

In this contribution I show that it is possible to construct three-dimensional spaces of non-constant curvature, i.e. three-dimensional Darboux-spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that in the two three-dimensional Darboux spaces, which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In \threedDI we find seven coordinate systems which separate the Schr\"odinger equation. For the second space, \threedDII, all coordinate systems of flat three-dimensional Euclidean space which separate the Schr\"odinger equation also separate the Schr\"odinger equation in \threedDII. I solve the path integral on \threedDI in the $(u,v,w)$-system, and on \threedDII in the $(u,v,w)$-system and in spherical coordinates

Comments on the Sign and Other Aspects of Semiclassical Casimir Energies

The Casimir energy of a massless scalar field is semiclassically given by contributions due to classical periodic rays. The required subtractions in the spectral density are determined explicitly. The so defined semiclassical Casimir energy coincides with that obtained using zeta function regularization in the cases studied. Poles in the analytic continuation of zeta function regularization are related to non-universal subtractions in the spectral density. The sign of the Casimir energy of a scalar field on a smooth manifold is estimated by the sign of the contribution due to the shortest periodic rays only. Demanding continuity of the Casimir energy under small deformations of the manifold, the method is extended to integrable systems. The Casimir energy of a massless scalar field on a manifold with boundaries includes contributions due to periodic rays that lie entirely within the boundaries. These contributions in general depend on the boundary conditions. Although the Casimir energy due to a massless scalar field may be sensitive to the physical dimensions of manifolds with boundary, its sign can in favorable cases be inferred without explicit calculation of the Casimir energy.Comment: 39 pages, no figures, references added, some correction

Exactly solvable path integral for open cavities in terms of quasinormal modes

We evaluate the finite-temperature Euclidean phase-space path integral for the generating functional of a scalar field inside a leaky cavity. Provided the source is confined to the cavity, one can first of all integrate out the fields on the outside to obtain an effective action for the cavity alone. Subsequently, one uses an expansion of the cavity field in terms of its quasinormal modes (QNMs)-the exact, exponentially damped eigenstates of the classical evolution operator, which previously have been shown to be complete for a large class of models. Dissipation causes the effective cavity action to be nondiagonal in the QNM basis. The inversion of this action matrix inherent in the Gaussian path integral to obtain the generating functional is therefore nontrivial, but can be accomplished by invoking a novel QNM sum rule. The results are consistent with those obtained previously using canonical quantization.Comment: REVTeX, 26 pages, submitted to Phys. Rev.