Right eigenvalue equation in quaternionic quantum mechanics

We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations. Our discussion is also extended to complex linear operators, whose spectrum is characterized by 2n complex eigenvalues. We show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products. Finally, we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution.Comment: 24 pages, AMS-Te

Solving simple quaternionic differential equations

The renewed interest in investigating quaternionic quantum mechanics, in particular tunneling effects, and the recent results on quaternionic differential operators motivate the study of resolution methods for quaternionic differential equations. In this paper, by using the real matrix representation of left/right acting quaternionic operators, we prove existence and uniqueness for quaternionic initial value problems, discuss the reduction of order for quaternionic homogeneous differential equations and extend to the non-commutative case the method of variation of parameters. We also show that the standard Wronskian cannot uniquely be extended to the quaternionic case. Nevertheless, the absolute value of the complex Wronskian admits a non-commutative extension for quaternionic functions of one real variable. Linear dependence and independence of solutions of homogeneous (right) H-linear differential equations is then related to this new functional. Our discussion is, for simplicity, presented for quaternionic second order differential equations. This involves no loss of generality. Definitions and results can be readily extended to the n-order case.Comment: 9 pages, AMS-Te

The Development of COLESTVIAModel as An Effort of Internanalization of Character Values in Social Studies

ABSTRACT Purpose: This research is based on a condition of Social Studies instruction in Junior High schools in Surakarta that is still far from the expectation. It focuses on the cognitive aspect and lack of affective aspects. Therefore, it is important to develop an instruction throughCOLESTVIA Model by combining STAD type, Tournament and VIA as an effort in internalization of character values in Social Studies instruction. The development of character values is concerned with cognitive development and the result of social interaction. It is believed that by integrating the character values, it can strengthen the character and personality for the Junior High school students in Surakarta. Method:The Developmentof COLESTVIA Model is done through development research by three main steps, namely Preliminary Study, Development, and Model Pilot. The data was collected through Observational Technique, Interview, Questionaire and Doccumentation. The Data Analysis for the Preliminary Study uses qualitative, while the Model Pilot was by means of experiment using quantitative Approach as to answer the research questions, namely : (1) How is the performance of the Social Studies instruction of the grade VIII of Junior High School students in Surakarta, (2) How is the development of COLESTVIA Model that can integrate the character values, and (3) How is the effectiveness of The DevelopmentCOLESTVIA Model as an attempt of the internalization of the character values in Social Studies instruction in grade VIII Junior High school.T- test was used to know the difference between the model of COLESTVIA instruction and conventional model. Findings: The result of the research shows after performing the try out in SMP 19 through Classroom Action Research (CAR); the larger scaled-trial in SMP 2 and private Junior High School A, (experiment), SMP 3 and private Junior High School B (control), and the effectiveness of trial in SMP 19, SMP 24 and private Junior High School C (experiment) and SMP 10, SMP 25 and private Junior High School D (contol), proved that thatCOLESTVIA instruction is able to improve the competency and the SMP students’ character values in various groups of schools. The conclusion of the research is that the model of COLESTVIA instruction fulfill the principles and proves significantly the increase of the SMP students’ character values in Surakarta compared with the various conventional models of instruction

Quaternions and Special Relativity

We reformulate Special Relativity by a quaternionic algebra on reals. Using {\em real linear quaternions}, we show that previous difficulties, concerning the appropriate transformations on the $3+1$ space-time, may be overcome. This implies that a complexified quaternionic version of Special Relativity is a choice and not a necessity.Comment: 17 pages, latex, no figure

Quaternionic potentials in non-relativistic quantum mechanics

We discuss the Schrodinger equation in presence of quaternionic potentials. The study is performed analytically as long as it proves possible, when not, we resort to numerical calculations. The results obtained could be useful to investigate an underlying quaternionic quantum dynamics in particle physics. Experimental tests and proposals to observe quaternionic quantum effects by neutron interferometry are briefly reviewed.Comment: 21 pages, 16 figures (ps), AMS-Te

The octonionic eigenvalue problem

By using a real matrix translation, we propose a coupled eigenvalue problem for octonionic operators. In view of possible applications in quantum mechanics, we also discuss the hermiticity of such operators. Previous difficulties in formulating a consistent octonionic Hilbert space are solved by using the new coupled eigenvalue problem and introducing an appropriate scalar product for the probability amplitudes.Comment: 21 page

Quaternionic Electroweak Theory and CKM Matrix

We find in our quaternionic version of the electroweak theory an apparently hopeless problem: In going from complex to quaternions, the calculation of the real-valued parameters of the CKM matrix drastically changes. We aim to explain this quaternionic puzzle.Comment: 8, Revtex, Int. J. Theor. Phys. (to be published

Quaternionic Electroweak Theory

We explicitly develop a quaternionic version of the electroweak theory, based on the local gauge group $U(1, q)_{L}\mid U(1, c)_{Y}$. The need of a complex projection for our Lagrangian and the physical significance of the anomalous scalar solutions are also discussed.Comment: 12 pages, Revtex, submitted to J. Phys.

Quaternionic eigenvalue problem

We discuss the (right) eigenvalue equation for $\mathbb{H}$, $\mathbb{C}$ and $\mathbb{R}$ linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows to translate the quaternionic problem into an {\em equivalent} real or complex counterpart. Interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics.Comment: 13 pages, AMS-Te

Amplification of coupling for Yukawa potentials

It is well known that Yukawa potentials permit bound states in the Schrodinger equation only if the ratio of the exchanged mass to bound mass is below a critical multiple of the coupling constant. However, arguments suggested by the Darwin term imply a more complex situation. By numerically studying the Dirac equation with a Yukawa potential we investigate this amplification effect.Comment: 7 pages, 2 figure