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The Hydrogen Atom
Published in Vinod Kumar Khanna, Introductory Nanoelectronics, 2020
The associated Legendre polynomials satisfy the orthogonality condition for fixed m∫0πPkm(cosθ){Plm(cosθ)}sinθdθ=22l+1(l+|m|)!(l−|m|)!δk,l
The Theory of Atom of Hydrogen
Published in Mikhail G. Brik, Chong-Geng Ma, Theoretical Spectroscopy of Transition Metal and Rare Earth Ions, 2019
Mikhail G. Brik, Chong-Geng Ma
Taking into account the recurrence relation for the associated Legendre polynomials tPlm=l+m2l+1Pl−1m(t)+l−m+12l+1Pl+1m(t) and their orthogonal properties ∫−11PkmPlmdt=2(l+m)!(2l+1)(l−m)!δkl, we immediately see that the integral in Eq. (3.44) is not zero if l′ = l ±1. This is the selection rule for the orbital quantum number l: An electric dipole transition between two states can be realized if and only if the orbital quantum numbers l in these states differ by unity: Δl = ±1.
Physical analysis of multichannel sound field recording and reconstruction
Published in Bosun Xie, Spatial Sound, 2023
The elevation dependence of spherical harmonic functions is determined by the associated Legendre polynomials PlmcosαS. Higher-order associated Legendre polynomials represent the rapid variation with elevation. To obtain smooth transition characteristics, an elevation-bandlimited truncation up to the (L − 1) order can be applied to Q-order horizontal signals. That is, the function in the following equation is used to replace YllσΩS on the right side of Equation (9.3.31): YllσΩS→PL−1mαScoslβSσ=1sinlβSσ=2.
On free vibration of piezoelectric nanospheres with surface effect
Published in Mechanics of Advanced Materials and Structures, 2018
Bin Wu, Weiqiu Chen, Chuanzeng Zhang
Substituting Eq. (17) into Eq. (2) and then into Eqs. (1) and (3), we obtain the governing equations of the spherical core in terms of ψ, G, w, and Φ, which can be solved by assuming where is the imaginary unit; Smn(θ, ϕ) = Pnm(cos θ)exp (imϕ) are spherical harmonics and Pmn (cos θ) are the associated Legendre polynomials; n and m are integers; ω is the circular frequency; and ξ = r/r0 is the dimensionless radial coordinate. The four dimensionless unknown functions Un(ξ), Vn(ξ), Wn(ξ), and Xn(ξ) in Eq. (18) satisfy where is the dimensionless frequency; a prime denotes the differentiation with respect to ξ; pi, qi, and fi are the dimensionless material constants defined in Ding and Chen [47], which are omitted here for brevity.