The Hydrogen Atom 1-Quantum Physics and Mechanics-Lecture Slides, Slides of Quantum Mechanics

Download The Hydrogen Atom 1-Quantum Physics and Mechanics-Lecture Slides and more Slides Quantum Mechanics in PDF only on Docsity! Homework this week: • HW #17 Tuesday Nov 8 by 10pm • HW #18 Thursday Nov 10 by 10pm Announcements Quantum mechanics docsity.com Quantum mechanics The hydrogen atom What is the density of probability of the electron? docsity.com kr  Asymptotic behaviors The hydrogen atom The radial equation 2 0 2 2 ( 1) 1 d u l l u d             2mE k     2 2 d u u d  0  2 2 2 ( 1)d u l l u d    u Ae  1lu B  Phys 451 2 0 2 02 me k    docsity.com The hydrogen atom 1 ( )lu e v   Peeling off the asymptotic behaviors   2 02 2( 1 ) 2( 1) 0 d v dv l l v d d             Power expansion 0 ( ) jj j v c     Recursion formula: 0 1 2( 1) 2 ( 1)( 2 2) 1 j j j j l c c c j j l j           Phys 451 docsity.com The hydrogen atom 1 ( )lu e v   The series must terminate 0 ( ) jj j v c     02( 1) 0 ( 1)( 2 2) j l j j l        max 02( 1)j l    Principal quantum number 0 max 1 2 n j l      Phys 451 docsity.com Quantum mechanics The hydrogen atom 2 2 0 1 4 me k n        1 ~ tan k dis ce Bohr radius 2 100 2 4 0.529 10a m me     na k 1  docsity.com Quantum mechanics The hydrogen atom 2 1 n E En  na kn 1  Energies levels Stationary states   ),()(,,  mlnlnlm YrRr  n: principal quantum number l: azimuthal quantum number 1 nl m: magnetic quantum number lm  Degeneracy of nth energy level: 1 0 2 1 n l l    docsity.com Quantum mechanics Quiz 24b A. 5 B. 9 C. 11 D. 25 E. 50 What is the degeneracy of the 5th energy band of the hydrogen atom? 1 2 0 2 1 n l l n     docsity.com Quantum mechanics The hydrogen atom 0 ( ) jj j v c        jj c ljj nlj c 221 )1(2 1    Equivalent to associated Laguerre polynomials )2()( 12 1    l lnLv   )(1)( xL dx d xL q p pp pq         qx q x q xe dx d exL       )(Pb 4.12 docsity.com Quantum mechanics The hydrogen atom How to find the stationary states?   ),()(,,  mlnlnlm YrRr  na kn 1 Step1: determine the principal quantum number n Step 2: set the azimutal quantum number l (0, 1, …n-1) Step 3: Calculate the coefficients cj in terms of c0 (from the recursion formula, at a given l and n) Step 4: Build the radial function Rnl(r) and normalize it (value of c0) Step 5: Multiply by the spherical harmonics (tables) and obtain 2l +1 functions nlm for given (n,l) ),( mlY (Step 6): Eventually, include the time factor:  / ),,(),( tiE nlm nertr    Pb 4.10 Pb 4.11 docsity.com Quantum mechanics electron 7 7 ; Z proton...“ é er “=~"Hydrogen Atom FIGURE 4.5: Densit: plot to be rotated ab Box,” v1.0.8, by Dauger Rese. site http://dauger.com. y plots for the hydrogen wave functions (a, vertical) z axis. Printed by permissi atch. You can make your own plots by going to the Web The hydrogen atom Representation of “Prim (r 9, ~) (4,2,0) (4,3,0) , 1, m). Imagine each ion using “Atom in a docsity.com