Question on orthogonal eigenfunctions

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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!
 
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Answers and Replies

  • #2
mathman
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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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