Gaussian Curvature in a p-orbital, Hydrogen-like Atoms

Transcription

1 Advanced Studies in Theoretica Physics Vo. 9, 015, no. 6, HIKARI Ltd, Gaussian Curvature in a p-orbita, Hydrogen-ike Atoms Sandro-Jose Berrio-Guzman Ana-Magnoia Marin-Ramirez Ruben-Dario Ortiz-Ortiz Copyright c 015 Sandro-Jose Berrio-Guzman, Ana-Magnoia Marin-Ramirez and Ruben- Dario Ortiz-Ortiz. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. Abstract When soving the Schrödinger equation for the hydrogen atom which is h m e ψ + V ψ = Eψ setting ψ(r, θ, φ) = R(r) Y (θ, φ), the anguar soution Y (θ, φ) is caed Spherica harmonics by means of Euer s identity the soid harmonics are obtained, which are the geometric references of orbitas, the gaussian curvature wi be cacuated for p-orbitas. Mathematics Subject Cassification: 4A0, 5A05 Keywords: Gaussian curvature, Hydrogen-ike atom, Orbita, Spherica harmonics, Soid harmonics

2 8 S. J. Berrio, A. M. Marin and R. D. Ortiz 1 Introduction Some properties on surfaces such as eiptica points and the definitions of Gaussian curvature and mean curvature can be found in [1]. In [] there are some comments about orbitas. In [4] was found the anguar soution of the Schrödinger equation using Legendre poynomias. It was studied antibound states for the Schrödinger equation in []. In this paper we show that the surface of the p-orbita for the hydrogen atom is eiptica. Preiminaries The hydrogen-ike atom is a idea system where the atom is supposed to have ony an eectron, the anguar soution of the Schrödinger equation to this system in a spherica coordinate system, namey is given by (see [4]) x = r sin θ cos φ 0 r (1) y = r sin θ sin φ 0 θ π () z = r cos θ 0 φ π () Y,m (θ, φ) = Θ,m (θ)φ m (φ) = ( + m )! P m (cos θ) exp(imφ) Where p m are the associated Legendre poynomias, when m = 0 the equation is rea, if m 0 the foowing inear combinations are considered 1 (Y,m + Y, m ) and 1 i (Y,m Y, m ) or in the same way π ( + m )! P m (cos θ) cos(mφ) and π ( + m )! P m (cos θ) sin(mφ)

3 Gaussian curvature in a p-orbita Hydrogen-ike atoms 8 Geometric argument By means of the gauss map a surface can be described (see [1]), in the case of a p-orbita, take = 1 and m = 0, so the anguar function is p z = cos θ (4) (5) Setting N =, repacing (4) in (1), () and (), an orthogona parametrization for p z, is obtained, which is ( sin(θ) Z(θ, φ) = N cos φ, sin(θ) ) 1 + cos θ sin φ, (6) 4 Main Resut Theorem 4.1. The Gaussian curvature for p z is K z = 16π and the mean curvature is H z = Proof. For Z(θ, φ) Z θ (θ, φ) = N (cos(θ) cos φ, cos(θ) sin φ, sin(θ)) ( Z φ (θ, φ) = N sin(θ) sin φ, sin(θ) ) cos φ, 0 Z θθ (θ, φ) = N ( sin(θ) cos φ, sin(θ) sin φ, cos(θ)) ( Z φφ (θ, φ) = N sin(θ) cos φ, sin(θ) ) sin φ, 0 Z θφ (θ, φ) = N ( cos(θ) sin φ, cos(θ) cos φ, 0) So, the coefficients of the first fundamenta form are given by E = Z θ, Z θ = N F = Z θ, Z φ = 0 G = Z φ, Z φ = N sin (θ) 4

4 84 S. J. Berrio, A. M. Marin and R. D. Ortiz a norma mapping of the surface is given by, η(θ, φ) = Z θ Z φ Z θ Z φ = (sin(θ) cos φ, sin(θ) sin φ, cos(θ)) so, the coefficients of the second fundamenta form are e = η, Z θθ = N f = η, Z θφ = 0 g = η, Z φφ = N sin (θ) consequenty the curvature for p z is and the mean curvature is K z = eg f EG F = 4 N = 16π H z = 1 eg ff + ge = 1 ( N) N sin (θ) N sin (θ) 4 EG F N 4 sin (θ) 4 = N = For this orbita the curvature is positive for any vaue of θ and φ hence every point of the surface is an eiptica one. Orbitas can not be observed, given that are just a mathematica mode (see []), however the notion of atomic orbita is very important in the study of chemistry and aso the anguar functions as it was showed, and here the geometry has been reated to it by using the gauss map, since an intrinsic property of a surface (in this case an spherica harmonic) is the Gaussian curvature which describes the whoe surface (an atomic p-orbita). Exampe 4.. Given that every point of the surface is an eiptica one, then for each p = Z(θ 0, φ 0 ) there is a neighborhood V of p = Z(θ 0, φ 0 ) such that a the points in V beong to the same side of the tangent pane T p (Z(θ, φ)) Acknowedgements. The authors express their deep gratitude to Universidad de Cartagena for partia financia support. References [1] M. P. do Carmo, Differentia geometry of curves and surfaces, Prentice- Ha, Inc, New Jersey, [] A. M. Marin, R. D. Ortiz and E. E. Caro, Antibound State for the Discrete Schrödinger Equation, Advanced Studies in Theoretica Physics, 9 (), (015),

5 Gaussian curvature in a p-orbita Hydrogen-ike atoms 85 [] E. R. Scerri, Have orbitas reay been observed?, Journa of Chemica Education, 77 (0), (000). [4] E. Steiner, The chemistry maths book, Oxford university press, New York, 008. Received: February 8, 015; Pubished: March 9, 015