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Unformatted text preview: Lecture 14 October 12, 2009 The Hydrogen Atom Expression of the wave function Expression of the energy McQuarrie and Simon Chapter 6 pp 191231 The H atom A proton and an electron bound together. Quantum particles: they may not be still (that would violate the uncertainty principle); instead, they must be in motion . As for a rigid rotator : divide total motion into a center of mass motion and a rotational motion about the center of mass . The wave function associated with center of mass is that for a free particle having a mass equal to the sum of the masses of the proton and the electron. We will focus on the motion in the proton/electron system. The H amiltonian (141) Potential energy : Coulomb interaction between the proton and the electron (142) negative sign: attractive potential e: charge on an electron (on a proton with opposite sign) : dielectric permittivity of free space r : interparticle distance H = T + V V = " e 2 4 #$ r r Kinetic Energy Operator The radial form of the potential energy function: most convenient to work in spherical polar coordinates . Kinetic energy operator (143) reduced mass ( almost equal to the mass of the proton) r interparticle distance L 2 total angular momentum squared operator . Angular kinetic energy (rotation at a fixed value of r ) plus term associated with contraction/lengthening of r . T = " h 2 2 # 2 r , $ , % ( ) = " h 2 2 r 2 & & r r 2 & & r " L 2 h 2 ( ) * + , Time Independent Schrdinger Equation (144) Assume the wave function is a separable product of a radial function depending only on r and a remaining function depending on and (145) " h 2 2 r 2 # # r r 2 # # r " L 2 h 2 $ % & ( ) " e 2 4 *+ r , . / 1 2 r , 3 , 4 ( ) = E 2 r , 3 , 4 ( ) " h 2 2 r 2 # # r r 2 # # r " L 2 h 2 $ % & ( ) " e 2 4 *+ r , . / 1 R r ( ) Y 2 , 3 ( ) = E R r ( ) Y 2 , 3 ( ) Separation of Variables This can be rearranged to (146) Divide both sides by R ( r ) Y ( , ) (147) Usual separation of variables . Since the l.h.s. depends only on r and the r.h.s. depends only on and , both sides must be equal to a constant. We know the constant (lets temporarily call it K ) because we can solve for it from the r.h.s. Y " , # ( ) d dr r 2 d dr $ % & ( ) + 2 r 2 h 2 e 2 4 *+ r + E $ % & ( ) , ....
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This note was uploaded on 12/14/2010 for the course CHEM 3502 taught by Professor Staff during the Fall '08 term at Minnesota.
 Fall '08
 Staff

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