Classical mechanics particle encounters

The effect of the nuclear mass was already mentioned in the introduction. In the Furry picture which is employed in the calculations of QED effects on bound electron states a static external field is assumed which corresponds to an infinitely heavy nucleus. In a non-relativistic approximation its finite mass is encountered by the reduced mass correction similar to the two-body problem in classical mechanics. In a relativistic treatment, however, this approach is oversimplyfied. Recently Artemyev et al. [42, 43] almost solved the whole problem by considering the nucleus as a simple Dirac particle with spin 1 /2, mass M and charge Ze. The interaction of the two Dirac particles electron and nucleus leads to a quasipotential equation in the center-of-mass system,... 

Then, T = 0.5 + 0.28 = 0.78 and R = 0.5- 0.28 = 0.22, and in contrast to the prior specific case, there is a noticeable probability that the incoming particle will be reflected back. This is at odds with the experience in our macroscopic, classical mechanical world where an object encountering a potential will not be stopped or reflected if it has energy in excess of the potential barrier. In the quantum world, there is a real probability of being reflected even if a particle s energy is more than the potential barrier. 

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. 

Molecular dynamics is the study of basic principles of chemical change. Its underlying theory is either classical or quantum mechanics. Much of the original insight into molecular encounters stemmed from a classical picture where the atoms were imagined to be positioned in three-dimensional coordinate space. This picture is in conflict with the quantum mechanical viewpoint that particles do not possess a definite position. A true quantum image of the particle is now understood as a blurred object able to interfere with itself. 

As was mentioned above, the value of AE is determined by the barrier shape. The characteristic values of AE determining the behavior of the system near the top and at the bottom of the barrier are generally different so that in the general case we get two parameters which can be described in terms of the energy levels of a particle in the potential well formed by inverting the barrier (as shown in Figure 3.6). These parameters are AE, the distance from the bottom of the well to the lowest level in it, and AEj, the distance from the upper level in the well to the level of a free particle outside the well. The criterion for the classical behavior is AEq < kT, and for the quantum-mechanical behavior, AEj > 8.6kT. Naturally, the relation between AE and AEj depends on the barrier shape. Finally, we shall formulate the criterion for the classical and quantum-mechanical behavior for a frequently encountered symmetrical barrier formed by the intersection of two parabolic terms with the same natural frequency o) the system behaves quantum. mechanically for ho) > 2.8 kT, and classically for "ho) < kT(l ia)/E ) (here E is the barrier height). 

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