Chapter4_8 - H r |U n >=E n |U n > ,...

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PHY4604 R. D. Field Department of Physics Chapter4_8.doc University of Florida Same answer as Bohr! The Radial Equation Radial Equation: Setting 2 ) 1 ( h + = l l C gives 2 2 2 2 2 2 ) 1 ( ) ( ) ) ( ( 2 ) ( 2 ) ( ) ( h h + = + + l l r R E r V mr r r R r r r R r R r and hence 0 ) ( 2 ) 1 ( ) ( 2 ) ( 2 ) ( 2 2 2 2 2 = + + + r R E mr l l r V m r r R r r r R h h . We define the “effective potential” to be 2 2 2 ) 1 ( ) ( ) ( mr l l r V r V eff h + + = . Note that the potential V(r) is altered by the second term which corresponds to a repulsive “centrifugal barrier” . It is sometimes convenient to introduce the function U(r) = rR(r) which results in the following equation for U(r) ) ( ) ( ) ( ) ( 2 2 2 2 r EU r U r V dr r U d m eff = + h . This equation looks just like the one dimensional equation! The overall wavefunctions are given by Y nlm (r, θ , φ ) = U n (r)Y lm ( θ , φ )/r , where n is the quantum number that specifies the eigenstates of the eigenvalue equation
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Unformatted text preview: H r |U n >=E n |U n > , where ) ( 2 ) ( 2 2 2 r V dr d m H eff op r + = h Hydrogen Atom: The energy levels and wave functions of the hydrogen atom are calculated by setting V(r) equal to the Coulomb potential r Ke r V Coulomb 2 ) ( = . The resulting energy levels are ) ( 2 1 2 2 2 c m n E e n = , where n is the principle quantum number. The states are specified by the three quantum numbers n , l , and m . Quantum Number Allowed Values Specifies n 1, 2, 3, . .. Energy Level Radial Wavefunction l 0, 1, 2, . .., (n-1) L 2 (length of the angular momentum vector) m -l, -l+1,. .., l L z (component of the angular momentum in the z-direction)...
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