Lecture24Notes - Chem 120A Spring 2006 READING Review of...

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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 Schr¨odinger equation for the hydrogen atom in spherical polar coordinates is - ¯ h 2 2 m 2 r - Ze 2 4 πε 0 r Ψ ( r , θ , φ ) = E Ψ ( r , θ , φ ) . (1) In order to solve this equation, we applied a separation of variables, Ψ ( r , θ , φ ) = R ( r ) Y ( θ , φ ) , yielding two separate differential equations: one for the angular part of the equation, ˆ L 2 Y ( θ , φ ) = ¯ h 2 β Y ( θ , φ ) , (2) and one for the radial part, ¯ h 2 r r 2 r + 2 μ r 2 ¯ h 2 Ze 2 4 πε 0 r + E ¶‚ R ( r ) = ¯ h 2 β R ( r ) . Solutions to the angular equation have β = l ( l + 1 ) , and the solutions of the radial equation, R nl ( r ) depend on two quantum numbers, l and n , with n = 0 , 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 ε 2 0 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 θ ) : Y ( θ , φ ) = Y m l ( θ , φ ) = N lm P | m | l ( cos θ ) e im φ , (3) Chem 120A, Spring 2006, Lecture 24 1
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where N lm is a normalization constant. Notice that the Y lm can be factored into a function that depends only on θ and a function that depends only on φ : Y lm ( θ , φ ) = Θ ( θ ) Φ ( φ ) , with Θ ( θ ) = P | m | l ( cos θ ) and Φ ( φ ) = e im φ . The solutions to the angular part of the equation are derived by using this separation of variables. From Eq. 2, sin θ Θ ( θ )
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