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# lecture14_umn - Lecture 14 The Hydrogen Atom Expression of...

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Lecture 14 October 12, 2009 The Hydrogen Atom Expression of the wave function Expression of the energy McQuarrie and Simon Chapter 6 pp 191-231

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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 (14-1) Potential energy : Coulomb interaction between the proton and the electron (14-2) negative sign: attractive potential e: charge on an electron (on a proton with opposite sign) ε 0 : dielectric permittivity of free space r : interparticle distance H = T + V V = " e 2 4 #\$ 0 r r

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Kinetic Energy Operator The radial form of the potential energy function: most convenient to work in spherical polar coordinates . Kinetic energy operator (14-3) μ 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 μ # 2 r , \$ , % ( ) = " h 2 2 μ r 2 & & r r 2 & & r " L 2 h 2 ( ) * + ,
Time Independent Schrödinger Equation (14-4) Assume the wave function is a separable product of a radial function depending only on r and a remaining function depending on θ and φ (14-5) " h 2 2 μ r 2 # # r r 2 # # r " L 2 h 2 \$ % & ( ) " e 2 4 *+ 0 r , - . / 0 1 2 r , 3 , 4 ( ) = E 2 r , 3 , 4 ( ) " h 2 2 μ r 2 # # r r 2 # # r " L 2 h 2 \$ % & ( ) " e 2 4 *+ 0 r , - . / 0 1 R r ( ) Y 2 , 3 ( ) = E R r ( ) Y 2 , 3 ( )

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Separation of Variables This can be rearranged to (14-6) Divide both sides by R ( r ) Y ( θ , φ ) (14-7) Usual separation of variables . Since the l.h.s. depends only on r and the r.h.s. depends only on θ and φ , both sides must be equal to a constant. We know the constant (let’s temporarily call it K ) because we can solve for it from the r.h.s.
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lecture14_umn - Lecture 14 The Hydrogen Atom Expression of...

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