Abstract. We transform the time-dependent Schrödinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time-evolution of exact wave functions of generalized harmonic oscillators is determined in terms of solutions of certain Ermakov and Riccati-type systems. In addition, we show that the classical Arnold transformation is naturally connected with Ehrenfest's theorem for generalized harmonic oscillators.
Dedicated to Margarita A. Man'ko and Vladimir I. Man'ko on the occasion of their 75 + 75 = 150 birthday for their great contributions to science and pedagogy.Abstract. We critically analyze the concept of photon helicity and its connection with the Pauli-Lubański vector from the viewpoint of the complex electromagnetic field, E + iH, sometimes attributed to Riemann but studied by Weber, Silberstein and Minkowski. To this end, a complex covariant form of Maxwell's equations is used.
We analyze basic relativistic wave equations for the classical fields, such as Dirac's equation, Weyl's two-component equation for massless neutrinos, and the Proca, Maxwell, and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubański vector and the Casimir operators of the Poincaré group. In general, in this group-theoretical approach, the above wave equations arise in certain overdetermined forms, which can be reduced to the conventional ones by a Gaussian elimination. A connection between the spin of a particle/field and consistency of the corresponding overdetermined system is emphasized in the massless case.
Physical laws should have mathematical beauty. " -P. A. M. DiracAbstract. We consider a complex covariant form of the macroscopic Maxwell equations, in a moving medium or at rest, following the original ideas of Minkowski. A compact, Lorentz invariant, derivation of the energy-momentum tensor and the corresponding differential balance equations are given. Conservation laws and quantization of the electromagnetic field will be discussed in this covariant approach elsewhere.
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