Prove that eigenstates of hermitian operator form a complete set

In summary, the conversation is about the proof of a property regarding hermitian operators and eigenvectors. The speaker is unsure how to approach it and is asking for help. They mention that the lecture only briefly mentioned it and the proof is long. Additionally, they note that they have not studied Hilbert spaces, which may be relevant to the proof. Another person suggests starting with the property that a hermitian operator has at least one eigenvector.
  • #1
fa2209
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Not really sure how to go about this. Our lecture said "it can be shown" but didn't go into any detail as apparently the proof is quite long. I'd really appreciate it if someone could show me how this is done. Thanks. (Not sure if this is relevant but I have not yet studied Hilbert spaces).
 
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  • #2
The proof is complicated and it appears in dozens of books on functional analysis. If you're not acquainted to HS and measure theory, then it basically makes little sense to read it.
 
  • #3
fa2209 said:
Not really sure how to go about this. Our lecture said "it can be shown" but didn't go into any detail as apparently the proof is quite long. I'd really appreciate it if someone could show me how this is done. Thanks. (Not sure if this is relevant but I have not yet studied Hilbert spaces).

Can you prove that a hermitian operator has at least one eigenvector? You should really start with this property.
 

1. What is a hermitian operator?

A hermitian operator is a linear operator in quantum mechanics that corresponds to an observable physical quantity. This means that it has real eigenvalues and its corresponding eigenvectors are orthogonal.

2. What does it mean for eigenstates of a hermitian operator to form a complete set?

It means that any state of a quantum system can be expressed as a linear combination of the eigenstates of the hermitian operator. This is known as the spectral theorem.

3. How can we prove that eigenstates of a hermitian operator form a complete set?

We can prove this by showing that the eigenstates of a hermitian operator are orthogonal to each other and that they span the entire vector space. This can be done using mathematical proofs and the properties of hermitian operators.

4. Why is it important for eigenstates of a hermitian operator to form a complete set?

This is important because it allows us to fully describe the state of a quantum system, and make predictions about the possible outcomes of measurements. It also helps us understand the mathematical structure of quantum mechanics.

5. Are there any real-world applications of the concept of eigenstates of hermitian operators forming a complete set?

Yes, the concept of eigenstates of hermitian operators is used in many areas of physics, such as quantum mechanics, quantum computing, and spectroscopy. It is also used in engineering fields, such as signal processing and control theory.

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