Read e-book online Asymptotic Methods in Quantum Mechanics: Application to PDF

By Professor S. H. Patil, Professor K. T. Tang (auth.)

ISBN-10: 3642573177

ISBN-13: 9783642573170

ISBN-10: 3642631371

ISBN-13: 9783642631375

Asymptotic tools in Quantum Mechanics is an in depth dialogue of the final homes of the wave features of many particle platforms. specific emphasis is put on their asymptotic behaviour, because the outer sector of the wave functionality is so much delicate to exterior interplay. The research of those neighborhood homes is helping in developing easy and compact wave capabilities for classy platforms. It additionally is helping in constructing a large figuring out of other elements of quantum mechanics. As functions, wave services with right asymptotic varieties are used to systematically generate a wide information base for susceptibilities, polarizabilities, interactomic potentials and nuclear densities of many atomic, molecular and nuclear systems.

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G. laser fields, and in the analysis of dispersion coefficients. 2 Dispersion Coefficients The dominant interaction between two spherically symmetric atoms at large separation R is given in terms of dispersion coefficients. Let two atoms be on the z axis, atom A at the origin and atom B at a distance R from the origin. Let electron 1 be at a distance r from the nucleus of A and electron 2 be at a distance r' from the nucleus of B. "p R 1 IR - rl _ 1 IR + r'l + 1 IR - r + r'l . 38) When there are more than one electrons, this term has to be summed over all the pairs, one from each atom.

I' + m)! 42) m which follows from taking coef. of x 21 in (x + y)2H21' = L[coef. of xHm in (x + y/H'] m [coef. of x l - m in (x + y)lH'] . 43) This leads to (2£ + 2i')! (£ + i')! 44) so that finally we have 8E(2) = + 2£')! (2£' + 1)! 45) When more than one electron is involved, the multipolar potential terms rlyt0 are to be summed over the electrons of the atom. One usually writes the expression for this energy as a two-body, long-range interaction potential in the form 48 4. 46) where n-2 C2n (AB) = (2n - 2)!

24). J k=O (47r)2(n + I)! ] . 2ni +n ;+4-k(n- + 1- £ - k)! 2 Wave Functions Satisfying Cusp, Coalescence & Asymptotic Conditions 25 The calculated values of the coefficients of Ci (L) for L = 1,2, and the polarizabilities are given in table 1. The polarizabilities are again in good agreement with the accurate values. 383. 78). 19) In particular, for £1 + £2 + £3 = 4, we have a hyperpolarizability B defined as B = -2 L t(£l, £2, £3), £1 + £2 + £3 = 4 = -2t(l, 2,1) - 4t(l, 1, 2) . 12). 2 Wave Functions Satisfying Cusp, Coalescence and Asymptotic Conditions The wave function we have described in the previous section has the advantage of great simplicity.

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Asymptotic Methods in Quantum Mechanics: Application to Atoms, Molecules and Nuclei by Professor S. H. Patil, Professor K. T. Tang (auth.)


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