hw3_09 - EP 471 Homework Set#3 1 Legendres differential...

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EP 471 – Homework Set #3 1) Legendre’s differential equation, 0 , () 22 12 xy x y y λ ′′ −+= arises in numerous physical problems, particularly in BVPs involving spherical symmetry. In the quantum theory of the atom this is related to the polar part of the electron wave function. In these problems, the amplitude of the wave function is related to the probability of finding the electron. If the potential is symmetric (Coulomb attraction), the wave amplitude ( ) 2 x ψ has to be symmetric, leading to one of two kinds of solutions: Even: () ( ) x x =− , Odd: ( ) ( ) x x = −− In terms of the current problem, we wish to solve the above equation over the interval 1 x subject to one of two sets of boundary conditions: Even: 11 ; 1 −≤ ≤+ y −= ( ) y + = ; ( ) y ′ + = Odd : ; 1 y 1 ( ) y + = ; ( ) y ′ + = To solve this problem numerically, we should first write it as: 0 where ε is a small number (suggestion: use 6 10 y y ελ + = +− = ) so that we don’t get a divide-by- zero error at the ends of the interval. As a check on the result, you may be aware that
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This note was uploaded on 01/30/2012 for the course ENGINEERIN 471 taught by Professor Witt during the Spring '08 term at University of Wisconsin.

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hw3_09 - EP 471 Homework Set#3 1 Legendres differential...

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