Theory of the Hydrogen Atom - Introduction to Modern Physics - Lecture Notes, Study notes of Physics

Download Theory of the Hydrogen Atom - Introduction to Modern Physics - Lecture Notes and more Study notes Physics in PDF only on Docsity! 1 7.2. Schrödinger equation in spherical coordinates / Theory of the hydrogen atom done first by Schrödinger himself, he discovered how naturally quantization occurs in wave mechanics: “It has its basis in the requirement that a certain spatial function be finite and single values.” electron is considered to be a point charge that goes around the nucleus, which is also considered to be a point charge and assumed to be stationary, the great distance between electron and proton with respect to their actual size and the 1836 times larger mass of the proton makes these assumptions entirely feasible potential energy is electrostatic potential energy U = r e 0 2 4  U is a function of r rather than x, y, z, but r = 222 zyx  as our electrostatic potential function depends on r, so we can not substitute it directly into 3D Schrödinger equation that is valid for a wave function that depends on x,y,z two alternatives: express U as a function of x,y,z or express Schrödinger equation in spherical polar coordinates, latter is by far easier, especially as we want to separate the variables later and get a physical interpretation in the form of quantum numbers on as we did for 3D infinite square well so it is naturally to use spherical coordinates as it is a spherical problem docsity.com 2 “kinds” of latitude, angle θ is zenith angle = arc cos 222 zyx z  angle  is azimuth angle = arc tan x y “kinds” of meridians longitude Schrödinger equation in spherical polar coordinates 0)( 2 sin 1 )(sin sin 1 )( 1 22 2 222 2 2                     UE m rrr r rr  now the potential energy function U = r e 0 2 4  can be substituted and we multiply whole equation by r2 sin2θ 0) 4 ( sin2 )(sinsin)(sin 0 2 2 22 2 2 22                       E r erm r r r  docsity.com 5 0] sin )1([(sin sin 1 2 2      lmll d d d d 0] )1( ) 4 ( 2 [)( 1 2 0 2 2 2 2    R r ll E r em dr dR r dr d r  each of these equations is an ordinary differential equation for a dingle function of a single variable !!! Only the equation for R depends on the potential energy U(r) !!! – that is as far as Schrödinger equation can be simplified for the hydrogen atom now we put physical meaning to ml and l which turn out to be quantum numbers note that we have only ml in the first equation l and ml in the second equation l and E in the third equation, E will again be total energy that is quantized docsity.com 6 first equation for azimuthal part  has simple solution  limAe )( as this is a component-wave function of the full wave function  , we know that it must have a ingle value at any given point in space as it is only the function of an angle , we will have identical values of  for any  and +2π so it must be true that )2()(   ll imim AeAe which can only happen when ml = 0 or a positive or negative integer, ± 1, ± 2, ± 3, …. we call ml magnetic quantum number second equation for zenithal part Θ has a solution provided constant l is an integer, equal or greater than the absolute value of ml  we express this requirement as a condition on ml ml = 0, ± 1, ± 2, ± 3, docsity.com 7 we call l orbital quantum number solution to 3rd equation for radial R part requires two conditions 1. that E has one of the values En that are identical to energy levels that could be calculated from the Bohr formula!!! 2 1 2222 4 ) 1 ( 32 n E n me E o n   n = 1, 2,3, so the correct part of the Bohr model (which described may spectral lines correctly) is contained in Schrödinger model we call n principal quantum number the principal quantum number 2. n must be equal or greater than l + 1 we express this requirement as a condition on l l = 0, 1, 2, 3, (n - 1) docsity.com