Bounding the Bogoliubov coefficients

While over the last century or more considerable effort has been put into the problem of finding approximate solutions for wave equations in general, and quantum mechanical problems in particular, it appears that as yet relatively little work seems to have been put into the complementary problem of establishing rigourous bounds on the exact solutions. We have in mind either bounds on parametric amplification and the related quantum phenomenon of particle production (as encoded in the Bogoliubov coefficients), or bounds on transmission and reflection coefficients. Modifying and streamlining an approach developed by one of the present authors [M. Visser, Phys. Rev. A 59 (1999) 427-438, arXiv:quant-ph/9901030], we investigate this question by developing a formal but exact solution for the appropriate second-order linear ODE in terms of a time-ordered exponential of 2×2 matrices, then relating the Bogoliubov coefficients to certain invariants of this matrix. By bounding the matrix in an appropriate manner, we can thereby bound the Bogoliubov coefficients.     

Analytic bounds on transmission probabilities

We develop some new analytic bounds on transmission probabilities (and the related reflection probabilities and Bogoliubov coefficients) for generic one-dimensional scattering problems. To do so we rewrite the Schrödinger equation for some complicated potential whose properties we are trying to investigate in terms of some simpler potential whose properties are assumed known, plus a (possibly large) “shift” in the potential. Doing so permits us to extract considerable useful information without having to exactly solve the full scattering problem.     

Investigation on Identifying Implicit Learning Event from EEG Signal Using Multiscale Entropy and Artificial Bee Colony

The way people learn will play an essential role in the sustainable development of the educational system for the future. Utilizing technology in the age of information and incorporating it into how people learn can produce better learners. Implicit learning is a type of learning of the underlying rules without consciously seeking or understanding the rules; it is commonly seen in small children while learning how to speak their native language without learning grammar. This research aims to introduce a processing system that can systematically identify the relationship between implicit learning events and their Encephalogram (EEG) signal characteristics. This study converted the EEG signal from participants while performing cognitive task experiments into Multiscale Entropy (MSE) data. Using MSE data from different frequency bands and channels as features, the system explored a wide range of classifiers and observed their performance to see how they classified the features related to participants’ performance. The Artificial Bee Colony (ABC) method was used for feature selection to improve the process to make the system more efficient. The results showed that the system could correctly identify the differences between participants’ performance using MSE data and the ABC method with 95% confidence.