By Alexandru Popa
Quantum and Classical Connections in Modeling Atomic, Molecular and Electrodynamic structures is meant for scientists and graduate scholars attracted to the rules of quantum mechanics and utilized scientists attracted to actual atomic and molecular types. it is a connection with these operating within the new box of relativistic optics, in issues with regards to relativistic interactions among very extreme laser beams and debris, and relies on 30 years of analysis. the newness of this paintings contains actual connections among the houses of quantum equations and corresponding classical equations used to calculate the lively values and the symmetry homes of atomic, molecular and electrodynamical platforms, in addition to delivering purposes utilizing equipment for calculating the symmetry homes and the full of life values of platforms and the calculation of homes of excessive harmonics in interactions among very severe electromagnetic fields and electrons.
- Features targeted motives of the theories of atomic and molecular structures, in addition to wave houses of desk bound atomic and molecular systems
- Provides periodic ideas of classical equations, semi-classical tools, and theories of platforms composed of very extreme electromagnetic fields and debris
- Offers versions and techniques in keeping with 30 years of research
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Extra info for Theory of Quantum and Classical Connections in Modeling Atomic, Molecular and Electrodynamical Systems
The four-dimensional wave vectors are denoted, respectively, by ðωL =c; kLx kLy ; kLz Þ and ðω0L =c; k0Lx0 ; k0Ly0 ; k 0Lz0 Þ in the systems S and S0 . 142) where β 0 and γ 0 are given by Eq. 61). 143) Since the relations between the spaceÀtime four-dimensional vectors in the systems S and S0 are ct0 5 γ0 ðct 1 jβ 0 jzÞ, x0 5 x, y0 5 y, and z0 5 γ 0 ðz 1 jβ 0 jctÞ, it is easy to see that these relations and Eqs. 142) verify Eq. 143). 149) from Jackson (1999), written in the International System, to calculate the Lorentz transformations of the field.
168) With the aid of Eqs. 170) 0 where vy0 5 0. With the aid of Eq. 172) with β 0y0 5 0 and β 0x0 2 1 β 0z0 2 5 1 2 γ02 . 174) We substitute Eq. 173) in Eqs. 172), and, after substitution, multiply Eqs. 172), respectively, by cos ξ and 2sin ξ. The sum of these equations leads to Eq. 175). Similarly, by multiplication of the same equations by sin ξ and cos ξ, we obtain Eq. 176). 176) We multiply Eqs. 176), respectively, by β u and β v . 177) Connection Between KleinÀGordon and Relativistic HamiltonÀJacobi Equations 53 The differentiation of the phase η0 , given by Eq.
Due to its potential importance, we analyze now this interaction. We consider that an electromagnetic field described by Eqs. 5), which propagates in the oz direction, collides with a relativistic electron beam which propagates in the negative sense of the oz axis. 57) Since the phase of the electromagnetic field is a relativistic invariant, it is convenient to calculate the motion of the electron in the inertial system, denoted by S 0 ðt0 ; x0 ; y0 ; z0 Þ, in which the initial velocity of 35 Connection Between KleinÀGordon and Relativistic HamiltonÀJacobi Equations the electron is zero.