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Unformatted text preview: The Hydrogen Atom The H atom is one of the problems we discussed two lectures ago. The Schrodinger equation is The angular terms are solved identically to last time. The separation constant β is so the radial equation becomes. ) , , ( ) , , ( ) ( ) , , ( ) ( sin 1 ) sin( ) sin( 1 1 2 2 2 2 2 2 2 2 2 φ θ ψ φ θ ψ φ θ ψ φ θ θ θ θ θ μ r E r r V r r r r r r r = + ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ − h ) 1 ( + l l ( ) r e r V r R E r R r V r dr r dR r dr r R d r E r V r dr r dR r dr d r R 2 2 2 2 2 2 2 2 2 2 4 ) ( where ) ( ) ( ) ( ) 1 ( 2 ) ( 2 ) ( 2 or ) 1 ( ) ( 2 ) ( ) ( 1 ε π μ μ μ − = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + − = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − l l h h l l h Since V(r) is a function only of r, the potential energy commutes with both L 2 and L z , and therefore so does the Hamiltonian operator itself. All have simultaneous eigenvalues and eigenfunctions. The constant ε = 8.854·10 12 just makes things work in the infuriating MKS electrical system. Note also that this equation will solve for any oneelectron ion if we replace e 2 by Ze 2 where Z is the atomic number of the nucleus, though answers become poor much above Kr because the electron motion starts becoming relativistic. This is an important equation, so lets choose it to discuss how these things are solved or rather were solved 160 years ago. As r becomes large, the terms in 1/r and 1/r 2 become progressively smaller, and at asymptotically large r we have whose general solution is If c is imaginary we find that E is positive. These are not the solutions we want, as they represent incoming and outgoing spherical wavefronts. If c is real and positive the term with A blows up as r gets large, which is wrong. So c is real and positive and As r goes to zero the term with 1/r 2 in it dominates (even the term in E becomes unimportant) and whose solution is of the form We then define a function v(r): and try to solve the equation for v(r) . large) r very (for ) ( ) ( 2 2 2 2 r u E dr r u d = − μ h imaginary or real be may c where ) ( cr cr Be Ae r u − + ≈ ∞ → ≈ − r Be r u cr as ) ( 2 2 2 2 2 2 4 where ) ( ) ( ) 1 ( 2 ) ( 2 becomes equation above the and define First we ε π μ μ e V r u E r u r V r dr r u d rR(r) u(r) − = ′ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ + + + − = l l h h ) ( ) 1 ( ) ( 2 2 2 r u r dr r u d + = l l r small for ) ( so r as up blows but ) ( 1 1 + + ≈ → + = l l l l Cr r u r D r D Cr r u ) ( ) ( 1 r v e r r u cr − + = l To solve for v(r) we write it as a power series expansion in powers r n : We then insert that in the differential equation and equate like powers of r . This results in what is called a recursion formula: For large n this becomes and hence which blows up at large r unless there is some n at which...
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This note was uploaded on 09/22/2011 for the course CHEM 222 taught by Professor Linda during the Spring '11 term at Edmonds Community College.
 Spring '11
 Linda
 Atom

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