Wei Khim Ng

This paper studies how the GeoGebra 3D Calculator with augmented reality (AR) serves as a tool for visualizing 3D graphs from functions of two variables in a mid-sized multivariable calculus class. We assessed this AR tool's usability as a content delivery system. In our study, 39 students in the AR group received instructions on using the GeoGebra 3D Calculator with AR tool, while another 28 students in the control group worked exclusively with PowerPoint slides. Statistical testing reveals that the GeoGebra 3D Calculator with AR proves more effective than PowerPoint slides in terms of usability for teaching and learning multivariable calculus. However, the students' attitudes and engagement do not show significant differences between the AR and control groups. This preliminary study highlights the crucial role of innovative educational technologies like the GeoGebra 3D Calculator with AR in enhancing the visualization of 3D graphs and other multivariable calculus topics. It suggests an advancement in the evolution of digital tools in education. The increased usability observed in the AR group indicates a promising direction for future educational strategies, especially in fields that require strong spatial visualization skills.

We examine the nonperturbative effect of maximum momentum on the relativistic wave equations. In momentum representation, we obtain the exact eigen-energies and wave functions of one-dimensional Klein Cordon and Di rac equation with linear confining potentials, and the Dirac oscillator. Bound state solutions are only possible when the strength of scalar potential is stronger than vector potential. The energy spectrum of the systems studied is bounded from above, whereby classical characteristics are observed in the uncertainties of position and momentum operators. Also, there is a truncation in the maximum number of bound states that is allowed. Some of these quantum-gravitational features may have future applications.

Following the Gazeau-Klauder approach, we construct generalized coherent states (GCS) as the quantum simulator to examine the deformed quantum mechanics, which exhibits an intrinsic maximum momentum. We study deformed harmonic oscillators and compute their probability distribution and entropy of states exactly. Also, a particle in an infinite potential box is studied perturbatively. In particular, unlike usual quantum mechanics, the present deformed case increases the entropy of the Planck scale quantum optical system. Furthermore, for simplicity, we obtain the modified uncertainty principle (MUP) with the perturbative treatment up to leading order. MUP turns out to increase generally. However, for certain values of gamma (a parameter of GCS), it is possible that the MUP will vanish and hence will exhibit the classical characteristic. This is interpreted as the manifestation of the intrinsic high-momentum cutoff at lower momentum in a perturbative treatment. Although the GCS saturates the minimal uncertainty in a simultaneous measurement of physical position and momentum operators, thus constituting the squeezed states, complete coherency is impossible in quantum gravitational physics. The Mandel Q number is calculated, and it is shown that the statistics can be Poissonian and super-/sub-Poissonian depending on gamma. The equation of motion is studied, and both Ehrenfest's theorem and the correspondence principle are recovered. Fractional revival times are obtained through the autocorrelation, and they indicate that the superposition of a classical-like subwave packet is natural in GCS. We also contrast our results with the string-motivated (Snyder) type of deformed quantum mechanics, which incorporates a minimum position uncertainty rather than a maximum momentum. With the advances of quantum optics technology, it might be possible to realize some of these distinguishing quantum-gravitational features within the domain of future experiments.

Information measures for relativistic quantum spinors are constructed to satisfy various postulated properties such as normalization invariance and positivity. Those measures are then used to motivate generalized Lagrangians meant to probe shorter distance physics within the maximum uncertainty framework. The modified evolution equations that follow are necessarily nonlinear and simultaneously violate Lorentz invariance, supporting previous heuristic arguments linking quantum nonlinearity with Lorentz violation, The nonlinear equations also break discrete symmetries. We discuss the implications of our results for physics in the neutrino sector and cosmology.

We investigate potential quantum nonlinear corrections to Dirac's equation through its sub-leading effect on neutrino oscillation probabilities. Working in the plane-wave approximation and in the mu - tau sector, we explore various classes of nonlinearities, with or without an accompanying Lorentz violation. The parameters in our models are first delimited by current experimental data before they are used to estimate corrections to oscillation probabilities. We find that only a small subset of the considered nonlinearities has the potential to be relevant at higher energies and thus possibly detectable in future experiments. A falsifiable prediction of our models is an energy-dependent effective mass-squared, generically involving fractional powers of the energy.

We first review a method to generate nonlinear Dirac equations. The method demands the nonlinear extensions preserve several physical properties like locality, Hermiticity, Poincaré invariance and separability. The last constraint results in nonlinear extensions of non-polynomial type. A class of nonlinear extensions that simultaneously violate Lorentz invariance is also constructed. We then review, using the classes of nonlinear extensions with or without violation of Lorentz symmetry, the sub-leading modifications to the neutrino oscillation probabilities in the νµ-ντ sector. The parameters in our models are bounded using the current experimental data. These are then used to estimate corrections to the oscillation probabilities and the corresponding energies at which the corrections will be sizeable. Thus one may test quantum nonlinearities in future higher energy experiments.

We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincare invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.

We obtain novel nonlinear Schrödinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential singularities brought forward by the nonlinear terms and suggests how to regularise previous equations studied in the literature. The enhancement of contributions coming from the regularised singularities suggests that the obtained equations might be useful for future precision tests of quantum nonlinearity.