This is evident from the contour integral representation below. The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. ) n l → φ [3] Mittels der zugeordneten Laguerre-Polynome lässt sich der Radialanteil der Wellenfunktion schreiben als, (Normierungskonstante A Laguerre Polynomial Orthogonality and the Hydrogen Atom - NASA/ADS The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. x n − By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. betrachtet, so dass eine hebbare Singularität bei − − x = 1 , wie in Teil 1 gezeigt. m [ ⟩ The generalized Laguerre polynomials satisfy the Hardy–Hille formula[14][15], where the series on the left converges for L denotes the Gamma distribution then the orthogonality relation can be written as, The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula)[citation needed], Moreover,[clarification needed Limit as n goes to infinity? ] ) L gilt demnach k der Kernladungszahl − ⟨ rev 2020.11.24.38066, The best answers are voted up and rise to the top, Physics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, $$ ( $$, $$ \rho \frac{d^2 g(\rho)}{d \rho^2} + [(2l + 1) + 1 - \rho] \frac{d g(\rho)}{d \rho} + \left[ In der Physik wird üblicherweise eine Definition verwendet, nach der die Laguerre-Polynome um einen Faktor 0. ( n x ) ) . n ′ {\displaystyle \langle L_{n},L_{m}\rangle =\int _{0}^{\infty }\mathrm {e} ^{-x}L_{n}(x)L_{m}(x)\mathrm {d} x=0.}. ) The generalized Laguerre polynomials obey the differential equation. Making statements based on opinion; back them up with references or personal experience. x ! k n . n Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor[1] Nikolay Yakovlevich Sonin). das Kronecker-Delta. 3 0 obj << = $$ x 1 n 0 Further see the Tricomi–Carlitz polynomials. ) ( für ( {\displaystyle n=0,1,2,\ldots }, Sie ist ein Spezialfall der Sturm-Liouville-Differentialgleichung. Download PDF Abstract: The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. a d x 0 k Or only on aggregate from the individual holdings? Es verbleibt somit lediglich der zweite Term, der mit partieller Integration berechnet wird, also: Die Stammfunktion wurde mithilfe der Produktregel berechnet und es ergibt sich im Grenzwert x , which shows that L(α)n is an eigenvector for the eigenvalue n. The generalized Laguerre polynomials are orthogonal over [0, ∞) with respect to the measure with weighting function xα e−x:[10], If die Wronski-Determinante der Funktionen − → ) %PDF-1.4 m ) x where , ( ! d (or is it just me...), Smithsonian Privacy n Does paying down debt in an S Corp decrease profitability? − ) ) n 1 x 5, 1 Jun 2009 | Journal of Mathematical Physics, Vol. ( = x ergibt sich: Wird nun für ⟨ x ( ! {\displaystyle Z} − 26, No. = L n 1 eine Orthonormalbasis im Hilbertraum e − x α is the Pochhammer symbol (which in this case represents the rising factorial). Use MathJax to format equations. 1 Hydrogen wavefunctions 1.1 Introduction The Hydrogen atom can be studied as a two-body system governed by the Hamiltonian: H^ = ~ 2m e r2 e2 4ˇ 0r Exploiting the spherical geometry, we can use a spherical coordinate system with the proton at the origin and the electron moving in the three-dimensional space. = $$, $$ ( ASSOCIATED LAGUERRE POLYNOMIALS Lecture 24 Because it is the \integerization" of ˆ 0that quantizes the energy in the Hydrogen atom, it is worthwhile to generate the series solution and see how this appears eectively as a boundary condition (vanishing of the radial wavefunction at innity). − | ) ) {\displaystyle \textstyle \lim _{x\to 0}x^{n}{\frac {1}{n! y x {\displaystyle L_{n}^{(\alpha )}(x)} ) n