Show eigenfunctions are orthogonal

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SUMMARY

The discussion focuses on proving the orthogonality of two eigenfunctions, A and B, corresponding to distinct eigenvalues a and b of an operator O. It is established that if the eigenvalues are real, the integral condition ∫ A*OB dx = ∫(OA)*B dx holds true. This leads to the conclusion that eigenfunctions associated with different eigenvalues are orthogonal, reinforcing the fundamental properties of linear operators in quantum mechanics.

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  • Understanding of linear operators in functional analysis
  • Familiarity with eigenvalues and eigenfunctions
  • Knowledge of integral calculus and complex conjugates
  • Basic proficiency in LaTeX for mathematical notation
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  • Study the properties of linear operators in quantum mechanics
  • Learn about the spectral theorem and its implications for eigenfunctions
  • Explore the concept of orthogonality in Hilbert spaces
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Mathematicians, physicists, and students studying quantum mechanics or functional analysis who seek to deepen their understanding of eigenfunction properties and orthogonality.

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

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

* 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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