Coulomb wave function
coulomb wave function symbol, coulomb wave function collapseIn mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after CharlesAugustin de Coulomb They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument
Contents
 1 Coulomb wave equation
 11 Partial wave expansion
 2 Properties of the Coulomb function
 3 Further reading
 4 References
Coulomb wave equation
The Coulomb wave equation for a single charged particle is the Schrödinger equation with Coulomb potential
− ∇ 2 2 + Z r ψ k → r → = k 2 2 ψ k → r → , }}+}\right\psi _}}=}}\psi _}}\,,}where Z = Z 1 Z 2 Z_} is the product of the charges of the particle and of the field source in units of the elementary charge, Z = − 1 for hydrogen atom and k 2 } is proportional to the asymptotic energy of the particle The solution – Coulomb wave function – can be found by solving this equation in parabolic coordinates
ξ = r + r → ⋅ k ^ , ζ = r − r → ⋅ k ^ k ^ = k → / k }\cdot },\quad \zeta =r}\cdot }\qquad }=}/k\,}Depending on the boundary conditions chosen the solution has different forms Two of the solutions are
ψ k → ± r → = 1 2 π 3 / 2 Γ 1 ± i η e − π η / 2 e i k → ⋅ r → M ∓ i η , 1 , ± i k r − i k → ⋅ r → , }^}=}}\Gamma 1\pm i\eta e^e^}\cdot }}M\mp i\eta ,1,\pm ikri}\cdot }\,,}where M a , b , z ≡ 1 F 1 a ; b ; z _\!F_a;b;z} is the confluent hypergeometric function, η = Z / k and Γ z is the gamma function The two boundary conditions used here are
ψ k → ± r → → 1 2 π 3 / 2 e i k → ⋅ r → r → ⋅ k → → ∓ ∞ , }^}\rightarrow }}e^}\cdot }}\qquad }\cdot }\rightarrow \mp \infty \,,}which correspond to k → }} oriented planewave asymptotic state before or after its approach of the field source at the origin, respectively The functions ψ k → ± }^} are related to each other by the formula
ψ k → + = ψ − k → − ∗ }^=\psi _}}^\,}Partial wave expansion
The wave function ψ k → r → }}} can be expanded into partial waves ie with respect to the angular basis to obtain angleindependent radial functions w ℓ η , ρ \eta ,\rho } Here ρ = k r
ψ k → r → = 1 2 π 3 / 2 1 r ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ 4 π − i ℓ w ℓ η , ρ Y ℓ m r ^ Y ℓ m ∗ k ^ }}=}}}\sum _^\sum _^4\pi i^w_\eta ,\rho Y_^}Y_^}\,}A single term of the expansion can be isolated by the scalar product with a specific angular state
ψ k ℓ m r → = ∫ ψ k → r → Y ℓ m k → d k ^ = R k ℓ r Y ℓ m r ^ , R k ℓ r = 2 π − i ℓ 1 r w ℓ η , ρ }=\int \psi _}}Y_^}d}=R_rY_^},\qquad R_r=}}i^}w_\eta ,\rho }\sqrt} part seems off, wrong norm factor, orthonormality down below not true like this Please check it The equation for single partial wave w ℓ η , ρ \eta ,\rho } can be obtained by rewriting the laplacian in the Coulomb wave equation in spherical coordinates and projecting the equation on a specific spherical harmonic Y ℓ m r ^ ^}}
d 2 w ℓ d ρ 2 + 1 − 2 η ρ − ℓ ℓ + 1 ρ 2 w ℓ = 0 w_}}}+\left1}}}\rightw_=0\,}The solutions are also called Coulomb partial wave functions Putting x = 2 i ρ changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments Two special solutions called the regular and irregular Coulomb wave functions are denoted by F ℓ η , ρ \eta ,\rho } and G ℓ η , ρ \eta ,\rho } , and defined in terms of the confluent hypergeometric function by
F ℓ η , ρ = 2 ℓ e − π η / 2  Γ ℓ + 1 + i η  2 ℓ + 1 ! ρ ℓ + 1 e ∓ i ρ M ℓ + 1 ∓ i η , 2 ℓ + 2 , ± 2 i ρ \eta ,\rho =e^\Gamma \ell +1+i\eta }}\rho ^e^M\ell +1\mp i\eta ,2\ell +2,\pm 2i\rho \,}The two possible sets of signs are related to each other by the Kummer transform
Properties of the Coulomb function
The radial parts for a given angular momentum are orthonormal,
∫ 0 ∞ R k ℓ ∗ r R k ′ ℓ r r 2 d r = δ k − k ′ ^R_^rR_rr^dr=\delta kk'}and for Z = − 1 they are also orthogonal to all hydrogen bound states
∫ 0 ∞ R k ℓ ∗ r R n ℓ r r 2 d r = 0 ^R_^rR_rr^dr=0}due to being eigenstates of the same hermitian operator the hamiltonian with different eigenvalues
Further reading
 Bateman, Harry 1953, Higher transcendental functions PDF, 1, McGrawHill
 Jaeger, J C; Hulme, H R 1935, "The Internal Conversion of γ Rays with the Production of Electrons and Positrons", Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences, The Royal Society, 148 865: 708–728, Bibcode:1935RSPSA148708J, doi:101098/rspa19350043, ISSN 00804630, JSTOR 96298
 Slater, Lucy Joan 1960, Confluent hypergeometric functions, Cambridge University Press, MR 0107026
References
 ^ Hill, Robert N 2006, Drake, Gordon, ed, Handbook of atomic, molecular and optical physics, Springer New York, pp 153–155
 ^ Landau, L D; Lifshitz, E M 1977, Course of theoretical physics III: Quantum mechanics, Nonrelativistic theory 3rd ed, Pergamon Press, p 569
 ^ Thompson, I J 2010, "Coulomb Functions", in Olver, Frank W J; Lozier, Daniel M; Boisvert, Ronald F; Clark, Charles W, NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 9780521192255, MR 2723248
 ^ Abramowitz, Milton; Stegun, Irene Ann, eds 1983 "Chapter 14" Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables Applied Mathematics Series 55 Ninth reprint with additional corrections of tenth original printing with corrections December 1972; first ed Washington DC; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications p 538 ISBN 9780486612720 LCCN 6460036 MR 0167642 LCCN 6512253
 ^ Formánek, Jiří 2004, Introduction to quantum theory I in Czech 2nd ed, Prague: Academia, pp 128–130
 ^ Landau, L D; Lifshitz, E M 1977, Course of theoretical physics III: Quantum mechanics, Nonrelativistic theory 3rd ed, Pergamon Press, pp 668–669
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Coulomb wave function
Coulomb wave function
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