By Walter R. Johnson
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Extra resources for Atomic structure theory: lectures on atomic physics
S + 1)! 34), a b = s γ λ−µ+γ s µ+s−γ s+1 . a! (b − a)! b! 35) designates the binomial coeﬃcient. 3 Numerical Solution to the Radial Equation Since analytical solutions to the radial Schr¨ odinger equation are known for only a few central potentials, such as the Coulomb potential or the harmonic oscillator potential, it is necessary to resort to numerical methods to obtain solutions in practical cases. We use ﬁnite diﬀerence techniques to ﬁnd numerical solutions to the radial equation on a ﬁnite grid covering the region r = 0 to a practical inﬁnity, a∞ , a point determined by requiring that P (r) be negligible for r > a∞ .
130) δJl δM m . 131) Therefore, one may write rˆ YJM (θ, φ) = J YJJ−1M (θ, φ) − 2J + 1 J +1 YJJ+1M (θ, φ) . 132) This vector is in the direction rˆ and is, therefore, referred to as a longitudinal vector spherical harmonic. Following the notation of , we introduce (−1) (−1) YJM (θ, φ) = rˆ YJM (θ, φ). The vector YJJM (θ, φ) is orthogonal to YJM (θ, φ), and is,therefore, transverse. The combination J +1 YJJ−1M (θ, φ) + 2J + 1 J YJJ+1M (θ, φ) 2J + 1 (−1) is also orthogonal to YJM (θ, φ) and gives a second transverse spherical vector.
111) satisﬁes the boundary condition φ(0) = 1. From ρ(R) = 0, it follows that φ(X) = 0 at X = R/ξ. By choosing the initial slope appropriately, we can ﬁnd solutions to the Thomas-Fermi equation that satisfy the two boundary conditions for a wide range of values X. 98) is satisﬁed. 112) r 2 = 4πr2 ρ(r) . dr Z From this equation, it follows that N (r), the number of electrons inside a sphere of radius r, is given by r d2 φ(r) dr dr2 0 r dφ = r −φ dr 0 dφ =r − φ(r) + 1 . 115) Evaluating this expression at r = R, we obtain the normalization condition 54 2 Central-Field Schr¨ odinger Equation X dφ dx =− X Z −N .