Modified wave number approximation

In his notation, /3 is the nonsphericity parameter, similar to Parker s e [equation (44)], and 0 is the angle between the H2 molecular axis and the line joining the centers of mass of the colliding particles. In the modified wave-number approximation, the quantity k — J(J + 1 )/ / in the Hamiltonian is replaced by k 2 = k 2 — J(J + 1)1 Rl, where Rc is assumed to be a constant approximately equal to the distance of closest approach. Defining Fm(Jc ) = QozOc IttRI, we write the distorted-wave expression (see Jonkman et al. [73]) as... 

The JSA is essentially a molecular version of the modified wave number approximation of atomic physics. For this perspective, see ... 

The argument of the function G is a length because the wave number has the dimension cm If we divide the function G(x) by the exponential term at the right-hand side of Eq. (20.17), then the contribution of the bandwidth will be filtered off. The approximate bandwidth should be known initially. The back transformation of a so-modified function should be a 5-function, or a series of 5-functions, in the... 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

Although wave equations are readily composed for more-electron atoms, they are impossible to solve in closed form. Approximate solutions for many-electron atoms are all based on the assumption that the same set of hydrogen-atom quantum numbers regulates their electronic configurations, subject to the effects of interelectronic repulsions. The wave functions are likewise assumed to be hydrogen-like, but modified by the increased nuclear charge. The method of solution is known as the self-consistent-field procedure. 

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