A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities
نویسندگان
چکیده
It is shown that tunnelling splittings in ergodic double wells and resonant widths in ergodic metastable wells can be approximated as easily-calculated matrix elements involving the wavefunction in the neighbourhood of a certain real orbit. This orbit is a continuation of the complex orbit which crosses the barrier with minimum imaginary action. The matrix element is computed by integrating across the orbit in a surface of section representation, and uses only the wavefunction in the allowed region and the stability properties of the orbit. When the real orbit is periodic, the matrix element is a natural measure of the degree of scarring of the wavefunction. This scarring measure is canonically invariant and independent of the choice of surface of section, within semiclassical error. The result can alternatively be interpretated as the autocorrelation function of the state with respect to a transfer operator which quantises a certain complex surface of section mapping. The formula provides an efficient numerical method to compute tunnelling rates while avoiding the need for the exceedingly precise diagonalisation endemic to numerical tunnelling calculations.
منابع مشابه
The Effect of Scars on the Statistics of Transition Probabilities of Classically Chaotic Quantum Systems∗
We study the statistical properties of generalized intensities (squared matrix elements of Hermitian operators) for the hydrogen atom in strong magnetic fields in a range of parameters where the classical analogue of the system exhibits completely chaotic dynamics. In this way we extend previous work by Prosen and Robnik on the statistics of generalized intensities in billiard systems in the tr...
متن کاملRegular Tunnelling Sequences in Mixed Systems
We show that the pattern of tunnelling rates can display a vivid and regular pattern when the classical dynamics is of mixed chaotic/regular type. We consider the situation in which the dominant tunnelling route connects to a stable periodic orbit and this orbit is surrounded by a regular island which supports a number of quantum states. We derive an explicit semiclassical expression for the po...
متن کاملThe Effects of Boat Propeller Scarring on Nekton Growth in Subtropical Seagrass Meadows
—An increasing boating population has led to extensive propeller scarring in many shallow seagrass meadows, and research has focused on relating scarring to nekton abundance; however, little information exists on the impacts on habitat functionality. In this study we moved beyond simple measures of faunal density as an indicator of habitat quality by comparing the growth rates of common estuari...
متن کاملScarring effects on tunneling in chaotic double-well potentials.
The connection between scarring and tunneling in chaotic double-well potentials is studied in detail through the distribution of level splittings. The mean level splitting is found to have oscillations as a function of energy, as expected if scarring plays a role in determining the size of the splittings, and the spacing between peaks is observed to be periodic of period 2 pi Planck's over 2 pi...
متن کاملA numerical approach for variable-order fractional unified chaotic systems with time-delay
This paper proposes a new computational scheme for approximating variable-order fractional integral operators by means of finite element scheme. This strategy is extended to approximate the solution of a class of variable-order fractional nonlinear systems with time-delay. Numerical simulations are analyzed in the perspective of the mean absolute error and experimental convergence order. To ill...
متن کامل