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Question on orthogonal eigenfunctions

  1. Jun 15, 2010 #1
    in this book I have by G.L Squires. One of the questions is:

    if [tex]\phi[/tex]1 and [tex]\phi[/tex]2 are normalized eigenfunctions corresponding to the same eigenvalue. If:
    [tex]\int[/tex][tex]\phi[/tex]1*[tex]\phi[/tex]2 d[tex]\tau[/tex] = d

    where d is real, find normalized linear combinations of [tex]\phi[/tex]1 and
    [tex]\phi[/tex] 2 that are orthogonal to a) [tex]\phi[/tex] 1 b) [tex]\phi[/tex]1 + [tex]\phi[/tex]2

    the part I'm having trouble with is finding a linear combination. The book gives: c1phi1 + c2phi2 (to shorten it) as the linear combo but yeah, whats the deal with that. they give a solution for a but yeah what is going on there. whats the process involved. yes, I am basically asking for a "tutor" so to speak, so I am n0t asking for any old person to help me but someone who is willing to explain it without becoming angry that I know not of such things.
    thanks in advance!
    Last edited: Jun 15, 2010
  2. jcsd
  3. Jun 16, 2010 #2


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    Let L be the linear combination (depends on c1 and c2). In both cases you have one (of two) equation L.L=1.
    (. means dot product, which is the integral you described).

    For a) the other equation is L.φ1=0.
    For b) the other equation is L.(φ12)=0

    You wil have in both cases two equations in two unknowns (c1 and c2).
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