Proving Orthogonality of Eigenfunctions for Hermitian Operators

In summary, the conversation discusses how to show that the eigenfunctions of a Hermitian operator, with different eigenvalues, are orthogonal. This is done by using the definitions and properties of Hermitian operators and eigenfunctions. It is also noted that the eigenvalues of Hermitian operators are real.
  • #1
leila
19
0
Hi there,

Was wondering if anyone could point me in the right direction for this one?

Show that the eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal?

Thanks
 
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  • #2
Alright, say you have an hermitian operator, O with eigenfunctions |a1> and |a2>, with eigenvalues of a1 and a2 respectively. Then:
O|a1>=a1|a1> (1)
<a1|O=a1*<a1| (2)
O|a2>=a2|a2> (3)
and
<a2|O=a2*<a2| (4)
Now right multiply |a1> in equation (4) and left multiply by <a2| in equation (1) to get two expressions for <a2|O|a1>. Subtract the two equations and observe.
-edit: keep in mind that the eigenvalues of hermitian operators are real. You can prove this by letting a1=a2
 
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FAQ: Proving Orthogonality of Eigenfunctions for Hermitian Operators

What is the definition of "orthogonality" in the context of eigenfunctions for Hermitian operators?

In mathematics, orthogonality refers to the property of two functions being perpendicular to each other, meaning that their inner product (integral) is equal to zero. In the context of eigenfunctions for Hermitian operators, it means that the eigenfunctions are independent of each other and do not overlap in their function space.

Why is it important to prove orthogonality of eigenfunctions for Hermitian operators?

Proving orthogonality of eigenfunctions is essential in the study of Hermitian operators because it allows us to decompose a function into a linear combination of eigenfunctions. This decomposition simplifies the problem and makes it easier to solve, as well as providing important insights into the behavior of the operator.

What is the method for proving orthogonality of eigenfunctions for Hermitian operators?

The standard method for proving orthogonality of eigenfunctions for Hermitian operators involves showing that the inner product of two eigenfunctions is equal to zero. This can be done by integrating the product of the two functions over the entire function space and using properties of Hermitian operators, such as self-adjointness and eigenvalue equations.

Are all eigenfunctions for Hermitian operators orthogonal?

No, not all eigenfunctions for Hermitian operators are orthogonal. It depends on the specific operator and the basis of eigenfunctions being used. However, it is always possible to construct a set of orthogonal eigenfunctions for any Hermitian operator.

What are the practical applications of proving orthogonality of eigenfunctions for Hermitian operators?

The concept of orthogonality of eigenfunctions for Hermitian operators has many practical applications in physics, engineering, and other fields. For example, it is used in quantum mechanics to solve for the energy levels of a system, in signal processing to analyze signals, and in numerical methods for solving differential equations.

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