Atomic structure theory: lectures on atomic physics by Walter R. Johnson

By Walter R. Johnson

Atomic constitution idea is a textbook for college students with a history in quantum mechanics. The textual content is designed to provide hands-on event with atomic constitution calculations. fabric coated comprises angular momentum equipment, the principal box Schrödinger and Dirac equations, Hartree-Fock and Dirac-Hartree-Fock equations, multiplet constitution, hyperfine constitution, the isotope shift, dipole and multipole transitions, uncomplicated many-body perturbation thought, configuration interplay, and correlation corrections to matrix parts. Numerical tools for fixing the Schrödinger and Dirac eigenvalue difficulties and the (Dirac)-Hartree-Fock equations are given besides. B-spline foundation units are used to hold out sums bobbing up in higher-order many-body calculations. Illustrative difficulties are supplied, including options. FORTRAN courses enforcing the numerical tools within the textual content are included.

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18) that are regular at the origin are the confluent hypergeometric functions [31, chap. VI]: a(a + 1) x2 a(a + 1)(a + 2) x3 a + + ··· F (a, b, x) = 1 + x + b b(b + 1) 2! b(b + 1)(b + 2) 3! a(a + 1) · · · (a + k − 1) xk + + ··· . b(b + 1) · · · (b + k − 1) k! 20) for large |x|. 20) vanishes. Since Γ (b) = 0, we must require Γ (a) = ∞ to obtain normalizable solutions. The function Γ (a) = ∞ when a vanishes or when a is a negative integer. Thus, normalizable wave functions are only possible when a = −nr with nr = 0, 1, 2, · · · .

20 1 Angular Momentum required calculations efficiently. There are several ways of introducing graphical rules for angular momentum summations [28, 29, 51]. Here, we follow those introduced by Lindgren and Morrison [30]. The basic graphical element is a line segment labeled at each end by a pair of angular momentum indices jm. The segment with j1 m1 at one end and j2 m2 at the other end is the graphical representation of δj1 j2 δm1 m2 ; thus, j 1 m1 j 2 m2 = δj1 j2 δm1 m2 . 99) A directed line segment, which is depicted by attaching an arrow to a line segment, is a second important graphical element.

The √ number. The relation λ = −2E leads immediately to the energy eigenvalue equation λ2 Z2 E = En = − n = − 2 . 21) 2 2n There are n distinct radial wave functions corresponding to En . These are the functions Pn (r) with = 0, 1, · · · , n − 1. 22) where Nn is a normalization constant. This constant is determined by requiring ∞ Nn2 dr (2Zr/n)2 +2 −2Zr/n e F 2 (−n + + 1, 2 + 2, 2Zr/n) = 1 . 23) 0 This integral can be evaluated analytically to give Nn = Z(n + )! (n − − 1)! 1 n(2 + 1)! 24) The radial functions Pn (r) for the lowest few states are found to be: P10 (r) = 2Z 3/2 re−Zr , 1 1 P20 (r) = √ Z 3/2 re−Zr/2 1 − Zr , 2 2 1 P21 (r) = √ Z 5/2 r2 e−Zr/2 , 2 6 2 2 2 P30 (r) = √ Z 3/2 re−Zr/3 1 − Zr + Z 2 r2 3 27 3 3 1 8 P31 (r) = √ Z 5/2 r2 e−Zr/3 1 − Zr , 6 27 6 4 P32 (r) = √ Z 7/2 r3 e−Zr/3 .

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