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There is nothing magical about the eigenfunctions being orthogonal

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There is nothing magical about the eigenfunctions being orthogonal. This can be shown by integration by parts twice. Consider two eigenfunctions Fn and Fm, both of which satisfy the following equations: d2Fn /dx2 = −λ2 nFn (11.79) and d2Fm /dx2 = −λ2 mFm (11.80) Obviously λn and λm are two distinct eigenvalues. Also assume that some homogeneous boundary conditions apply for both Fn and Fm at x = 0 and x = 1 (the end points assuming dimensionless variables). Now consider the integral 0∫ 1 [ Fm d2Fn dx2 ] dx = −λ2 n 0∫ 1 FmFn dx (11.81) Show that this follows from Eq. (11.79).

Integrate this by parts twice. Then use the homogeneous boundary condition. Then use Eq. (11.80) to get rid of the second-derivative term for Fm. This involves a lot of algebra, but all of it is elementary. Finally show that the resulting expression is (λ2 m − λ2 n) 0∫1 FmFn dx = 0 (11.82) Since λm is not equal to λn (the eigenvalues are distinct), we conclude that the functions Fm and Fn are orthogonal to each other. Generalization of this forms the basis of the Sturm– Liouville theory for eigenfunctions indicated in Problem 2 above.