Orthonormality-Quantum Physics and Mechanics-Lecture Slides, Slides of Quantum Mechanics

Download Orthonormality-Quantum Physics and Mechanics-Lecture Slides and more Slides Quantum Mechanics in PDF only on Docsity! Homework next week: • HW #17 Tuesday Nov 8 by 10pm • HW #18 Thursday Nov 10 by 10pm Announcements Phys 451 Homework tonight: • HW #16 Friday Nov 4 by 10pm docsity.com x y z   r Orthonormality    *i j ijr r dr   Phys 451 2 * ' ' ' ' 0 0 ( , ) ( , )sinm ml l ll mmY Y d d                     Angular part Spherical harmonics are orthogonal Also Pb 4.6 ' ' 2 ( ). ( ) 2 1 l l l ll l P x P x dx l     docsity.com The radial equation Example: infinite spherical well V=0 V  Phys 451 Change of variables ( )u rR r 2 2 2 2 ( 1) 2d u l l m E u dr r        Inside the well docsity.com The radial equation • For l = 0 2 2 2 2d u m E u dr        Infinite spherical well V=0 V  solution   sin( ) cos( )u r A kr B kr    sin( ) cos( )kr kr R r A B r r   Boundary condition 2 2 2 22 n n E ma   Phys 451 Three quantum numbers: (n, l, m) 00( , , ) ( , , )nlm nr r     here docsity.com The radial equation • If l ≠ 0 Infinite spherical well V=0 V  solution   ( ) ( )l lu r ArJ kr BrN kr  2 2 2 2 ( 1) 2d u l l m E u dr r        Spherical Bessel function Spherical Neumann function   1 sin ( ) l l l d x J x x x dx x           1 1 cos ( ) l l l d x N x x x dx x          Physical condition at r =0 0B  Phys 451 docsity.com Phys 451 Spherical well V=0 V  Pb 4.7: construct and 2 ( )n x1( )n x show that they blow up at zero Pb 4.8: case of l = 1 show that   2 2 2 1 2 1/ 2 2 nE n ma    Pb 4.9: Finite spherical well V=-V0 0V  Find the ground state (l =0) docsity.com The hydrogen atom What is the density of probability of the electron? Phys 451 docsity.com The hydrogen atom 2 2 2 1 1 1 sin ( 1) sin sin Y Y l l Y                       The angular equation (same)      , , ,r R r Y     Angular function (same)    , cosm im ml lY Ae P    Azimutal quantum number Magnetic quantum number l m 0l  m l Phys 451 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 The hydrogen atom 1 ( )lu e v   max 0 ( ) j j j j v c    0max 1 2 n j l      Quantization of the energy 2 2 2 2 0 1 2 4 n m e E n             Ground state: “binding energy” 2 2 1 2 0 13.6 2 4 m e E eV           Bohr 1913 Phys 451 docsity.com