Hermitian Operators and Non-Orthogonal Bases: Exploring Infinite Spaces

In summary, the conversation discusses the difference between hermitian operators acting on finite and infinite spaces. The rule involving eigenvectors is not valid in the infinite space, and it is unclear if other rules are affected as well. Additionally, it is mentioned that the basis in question is not orthonormal. A detail is also mentioned to be forgotten, with thanks.
  • #1
LCSphysicist
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TL;DR Summary
It is not necessary.
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The basis he is talking about: {1,x,x²,x³,...}
I don't know how to answer this question, the only difference i can see between this hermitians and the others we normally see, it is that X is acting on an infinite space, and, since one of the rules involving Hermitian fell into decline in the infinity space (e.g Eigenvectors span the space),i am not sure if another rules changes in the infinity space too..
 
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  • #2
The matrix of a self-adjoint operator with respect to an orthonormal basis is equal to its conjugate transpose. Your basis is not orthonormal.
 
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Likes Abhishek11235, Delta2, LCSphysicist and 3 others
  • #3
Infrared said:
The matrix of a self-adjoint operator with respect to an orthonormal basis is equal to its conjugate transpose. Your basis is not orthonormal.
Forgot this detail, thank you.
 

1. What is a Hermitian operator matrix?

A Hermitian operator matrix is a square matrix that is equal to its own conjugate transpose. In other words, the matrix is equal to its complex conjugate reflected across the main diagonal.

2. What are the properties of a Hermitian operator matrix?

Some key properties of a Hermitian operator matrix include: all eigenvalues are real numbers, the matrix is diagonalizable, and the eigenvectors corresponding to distinct eigenvalues are orthogonal.

3. How is a Hermitian operator matrix related to quantum mechanics?

In quantum mechanics, Hermitian operator matrices are used to represent physical observables, such as energy or momentum. The eigenvalues of these matrices correspond to the possible measurement outcomes of the observable.

4. Can a non-square matrix be a Hermitian operator?

No, a non-square matrix cannot be a Hermitian operator. The definition of a Hermitian operator requires the matrix to be square and equal to its own conjugate transpose.

5. How is a Hermitian operator matrix different from a unitary matrix?

While both Hermitian operator matrices and unitary matrices have real eigenvalues, a unitary matrix is also orthogonal, meaning its columns and rows are all orthogonal to each other. Additionally, the inverse of a unitary matrix is equal to its conjugate transpose, while the inverse of a Hermitian operator matrix is equal to its transpose.

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