Schrödinger Equation In 3 Dimensions-Quantum Physics and Mechanics-Lecture Slides, Slides of Quantum Mechanics

Download Schrödinger Equation In 3 Dimensions-Quantum Physics and Mechanics-Lecture Slides and more Slides Quantum Mechanics in PDF only on Docsity! Homework this week: • HW #15 Wednesday Nov 2 • HW #16 Friday Nov 4 Announcements Phys 451 Take the test by tonight! docsity.com Schrödinger equation in 3 dimensions x y z Phys 451 docsity.com Schrödinger equation in 3 dimensions i H t     2 2 ( ) 2 H V r m     2 2 2 2 2 2 2x y z           Laplacian p i  now Phys 451 docsity.com Schrödinger equation in 3 dimensions General solution   /( , ) niE tn nr t c r e     2 2 ( ) 2 n n n nV r E m       Each stationary state verifies Phys 451 docsity.com Spherical coordinates 2 2 2 2 2 2 2 2 1 1 1 sin sin sin r r r r r r                              Laplacian x y z   r r radius   polar angle azimutal angle Phys 451 docsity.com The angular equation 2 2 2 1 1 sin sin sink cst                           , , ,r R r Y     Further separation of variables:      , , ( )r R r       Azimutal equation: 2 2 2 1 d cst m d        ime   m integer  equation:  2 2sin sin sin 0 d d k m d d               x y z   r Phys 451 docsity.com x y z   r  2 2sin sin ( 1)sin 0 d d l l m d d                The angular equation Solution:           /2 2 cos 1 m l m m m l l AP d P x x P x dx            Legendre function Legendre polynomial ( 1)k l l   2 1 ( ) 1 2 ! l l l l d P x x l dx        Physical condition 0l  m ll,m integers Phys 451 docsity.com x y z   r The angular equation Solution:           /2 2 cos 1 m l m m m l l AP d P x x P x dx            Legendre function (cos )mlP  are polynoms in cos (multiplied by sin if m is odd) Phys 451 docsity.com