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Unformatted text preview: Chem 120A Review of hydrogen atom and orbitals 03/17/06 Spring 2006 Lecture 24 READING: Engel (CS): Chapter 10.1-10.2 (next week, read all of Ch. 10) Additional optional reading: McQuarrie and Simon, Physical Chemistry , Ch. 6 Review of the hydrogen atom The Schrodinger equation for the hydrogen atom in spherical polar coordinates is - h 2 2 m 2 r- Ze 2 4 r ( r , , ) = E ( r , , ) . (1) In order to solve this equation, we applied a separation of variables, ( r , , ) = R ( r ) Y ( , ) , yielding two separate differential equations: one for the angular part of the equation, L 2 Y ( , ) = h 2 Y ( , ) , (2) and one for the radial part, h 2 r r 2 r + 2 r 2 h 2 Ze 2 4 r + E R ( r ) = h 2 R ( r ) . Solutions to the angular equation have = l ( l + 1 ) , and the solutions of the radial equation, R nl ( r ) depend on two quantum numbers, l and n , with n = , 1 , 2 ,... . The angular momentum quantum number l is related to the principle quantum number n by 0 l n- 1. The solutions to the angular equation have depend on the angular momentum quantum number l , as well as the quantum number for the z-component of the angular momentum m . The allowed values of m are- l m l . The energies of the H atom depend only on the quantum number n , E =- Z 2 e 4 8 2 h 2 n 2 , or in atomic units, E =- Z 2 2 n 2 . The radial functions R nl ( r ) are Laguerre polynomials that are defined for 0 r < . The angular equations are the spherical harmonics, which depend on associated Legendre polynomials, P | m | l ( cos ) : Y ( , ) = Y m l ( , ) = N lm P | m | l ( cos ) e im , (3) Chem 120A, Spring 2006, Lecture 24 1 where N lm is a normalization constant. Notice that the Y lm can be factored into a function that depends only on and a function that depends only on : Y lm ( , ) = ( ) ( )...
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