Approximation semi-classical

Equation (A3.13.54) legitimates the use of this semi-classical approximation of the molecule-field interaction in the low-pressure regime. Since /7j(t) is explicitly time dependent, the time evolution operator is more... 

V. P. Maslov and M. V. Fedoriuk, Semi-classical Approximations in Quantum Mechanics, Reidel, Boston, 1981. 

Maslov, V. R, Fedoriuk, M. V. Semi-Classical Approximation in Quantum Mechanics. D. Reidel Publishing Company, Dordrecht, Boston, London (1981)... 

That is, the semi-classical approximation to the photon absorption rate is equivalent to a Landau-Zener treatment of the probability of hopping from Vj -i-hco to Vf induced by the electronic coupling perturbation p, f (s,0,Q). 

By introducing the simplest semi-classical approximation to the propagators, in which the nuclear motion kinetic energy is assumed to commute with the anion and neutral potential energy functions and with the non BO coupling operators, one obtains... 

The quantum-mechanical ionization cross section is derived using one of several approximations—for example, the Born, Ochkur, two-state, or semi-classical approximations—and numerical computations (Mott and Massey, 1965). In some cases, a binary encounter approximation proves useful, which means that scattering between the incident particle and individual electrons is considered classically, followed by averaging over the quantum-mechanical velocity distribution of the electrons in the atom (Gryzinski, 1965a-c). However, Born s approximation is the most widely used one. This is discussed in the following paragraphs. 

However, a variety of spectroscopic features can only be explained by the so-called semi-classical approximation, in which the solid material is described by its quantum response, while the propagating radiation is still considered classically. In this case, the classical model of the oscillator must be modified to take into account the fact that solids can only absorb or emit energy quanta according to discrete energy levels. This model will also be used in Chapter 4, as well as in Chapters 5, 6, and 7. 

Semi-classical approximations. In classical formulae, quantum effects may be accounted for to low order. For example, the the Wigner-Kirkwood expansion of the pair distribution function may be used [136, 302],... 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

The function tr describes the relative translational motion of the photofragments. This motion can be described in the semi-classical approximation (except in the turning point region) by an oscillating wavefunction for which the number of oscillations increases with an increase of the relative momentum. An increase of the number of oscillations results in a decrease of the FC factor. Hence, the semiclassical behavior of the translational wavefunction makes a transition to a state with large momentum less favorable. Because of conservation of energy, the resulting state is characterized by a small vibrational quantum number. 

As known from nuclear physics, a direct extension of quantal mean-field is delicate [36]. Fortunately enough, in the high excitation regime, where semi-classical approximations are likely to become acceptable, Boltzmann-like kinetic equations, offer an efficient alternative. They thus have been extensively used in nuclear physics for describing heavy-ion collisions (VUU,... 

The simplest variational ansatz we can make is the narrowly semi-classical approximation of (6.6), with effective p(x) ... 

The cause of the failure of the classical equation is illustrated in Fig. 3 for a single oscillator. The classical W = C /hv is the full line through the origin, but the exact count is a step function, shown by the dotted line. The broken line, which bisects the steps at their mid-points, is an improved smoothed function with which to represent the number of states of the oscillator, as originally pointed out by Marcus and Rice [4] who named it the semi-classical approximation. 

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