947,925 research outputs found
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
Dymanics of Generalized Coherent States
We show that generalized coherent states follow Schr\"{o}dinger dynamics in
time-dependent potentials. The normalized wave-packets follow a classical
evolution without spreading; in turn, the Schr\"{o}dinger potential depends on
the state through the classical trajectory. This feedback mechanism with
continuous dynamical re-adjustement allows the packets to remain coherent
indefinetely.Comment: 8 pages, plain latex, no figure
Magnetisation reversal mechanism in Co-Cr media for perpendicular magnetic recording
In this study Co---Cr thin films with perpendicular anisotropy are investigated. Three films with values for Hc of 11, 90 and 170 kA/m have been selected for this paper. Besides the coercivily several other parameters such as the Hc/Hk, Cr-segregation, domain structure, column sizes, etc. were studied by VSM, SEM, NMR, MFM, AFM and selective etching. The anomalous Hall effect (AHE) has been used to record the hysteresis curves of submicron Hall crosses. This very sensitive technique in combination with e-beam lithography and ion-beam etching resulted in the recording of AHE hysteresis loops with dimensions of the Hall crosses as small as 0.3 × 0.3 ¿m2. The AHE loops of three samples, with less than 60 columns, show different micromagnetic properties. Only the sample with Hc1 = 90 kA/m shows clear steps in the curves above the noise level. The largest steps correspond with the reversal of one column and the total number of steps was five times the number of columns for this sample. The different reversal mechanisms observed by the AHE are related to the differences in structure, coercivity and domain size
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
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