Show eigenfunctions are orthogonal

In summary, eigenfunctions are special functions that are associated with a particular operator in mathematics and are important for solving differential equations and representing physical systems. They are also orthogonal, meaning their inner product is equal to zero, and this property is crucial in many mathematical and physical applications. Eigenfunctions can be proven to be orthogonal by calculating their inner product and showing it is equal to zero. They are used in various fields, including quantum mechanics, signal processing, and differential equations, to represent physical systems such as vibrating strings and electromagnetic fields.
  • #1
indie452
124
0
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*OB dx[/tex] = [tex]\int(OA)*B dx[/tex]

* indicates conjugate
 
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  • #2
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.
 
Last edited:

Related to Show eigenfunctions are orthogonal

1. What are eigenfunctions?

Eigenfunctions are special functions that are associated with a particular operator in mathematics. These functions are important because they can be used to solve differential equations and represent physical systems.

2. How are eigenfunctions related to orthogonality?

In mathematics, two functions are considered orthogonal if their inner product is equal to zero. Eigenfunctions are orthogonal because they are associated with different eigenvalues, and the inner product of two different eigenfunctions is equal to zero.

3. Why is it important to show that eigenfunctions are orthogonal?

The orthogonality of eigenfunctions is crucial in many mathematical and physical applications. It allows us to use the eigenfunctions as a basis for representing more complex functions, making calculations and solutions easier and more efficient.

4. How can you prove that eigenfunctions are orthogonal?

To show that eigenfunctions are orthogonal, we need to calculate the inner product of two different eigenfunctions and show that it is equal to zero. This can be done using integration techniques and the properties of the operator associated with the eigenfunctions.

5. What are some examples of systems where eigenfunctions are used?

Eigenfunctions are used in many areas of mathematics and science, including quantum mechanics, signal processing, and differential equations. They are also commonly used to represent physical systems such as vibrating strings, quantum harmonic oscillators, and electromagnetic fields.

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