Chapter4_10

# Chapter4_10 - determined by normalization and the recursion...

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PHY4604 R. D. Field Department of Physics Chapter4_10.doc University of Florida The Hydrogen Atom – Quantum Mechanics (1) For the hydrogen atom the radial equation is ) ( ) ( 2 ) 1 ( ) ( 2 2 2 2 2 2 2 r EU r U mr l l r Ke dr r U d m = + + + h h which becomes U l l d U d + + = 2 0 2 2 ) 1 ( 1 ρ where 2 / 2 h mE = κ and r and 2 2 0 2 h mKe = so that 2 0 2 2 2 0 2 2 2 2 2 2 2 2 α mc mc c Ke m E = = = h h . Now we let ) ( ) ( 1 V e U l + = . In terms of V( ρ ) the radial equation becomes 0 )] 1 ( 2 [ ) 1 ( 2 0 2 2 = + + + + V l d dV l d V d . Power Series Solution: We now look for a solution of the form = = 0 ) ( j j j c V . and hence = + = + = = 0 1 0 1 ) 1 ( j j j j j j c j jc d dV and = + + = 0 1 1 2 2 ) 1 ( j j j c j j d V d . Recursion Formula: Inserting these power series into the differential equation yields j j c l j j l j c + + + + + = + ) 2 2 )( 1 ( ) 1 ( 2 0 1 This recursion formula determines the coefficients and hence the function V( ρ ). We start with c 0 (which becomes the overall constant that will be
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Unformatted text preview: determined by normalization) and the recursion formula give us all the other constants c j . Truncation: If we keep and infinite number of terms the V( ρ ) diverges as r → ∞ hence we must find a way to terminate the series such that ∑ = − + = max 1 ) ( j j j j l c e U and 1 max = + j c . Hence require that ) 1 ( 2 max = − + + l j . If we define n = j max + l + 1, with j max = 0, 1, 2, … then ρ = 2n and 2 2 2 2 2 2 2 2 n mc mc E n − = − = , with n = 0, 1, 2 … and l max = n -1....
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