Show eigenfunctions are orthogonal

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Two eigenfunctions, A and B, of an operator O with distinct eigenvalues a and b are orthogonal if their eigenvalues are real. The integral relationship provided, ∫ A*OB dx = ∫ (OA)*B dx, serves as a basis for demonstrating this orthogonality. To establish the eigenvalue equations, one must express them as OA = aA and OB = bB. The discussion emphasizes the importance of correctly formatting LaTeX equations for clarity. Ultimately, the orthogonality of eigenfunctions corresponding to different eigenvalues is a fundamental property in linear algebra and quantum mechanics.
indie452
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hi

one of my past papers needs me to show that if 2 eigenfunctions, A and B, of an operator O possesses different eigenvalues, a and b, they must be orthogonal. assume eigenvalues are real.

we are given

\int A<sup>*</sup>OB dx = \int(OA)<sup>*</sup>B dx

* indicates conjugate
 
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Can you write two eigenvalue equations, one for each eigenfunction? Also, please preview and write correctly your LateX equations before posting them, they will be easier to read.
 
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Thread 'Magnitude of buoyant force in fluids of different densities'

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