Discussion Overview
The discussion revolves around the eigenkets of a function of a hermitian operator A, particularly in the context of a problem from Sakurai's textbook. Participants explore the implications of defining functions of operators and the relationship between the eigenkets of A and f(A).
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant questions whether the function f(A) has the same eigenkets as the hermitian operator A, expressing concern that adding anything to A could make it non-hermitian.
- Another participant seeks clarification on how a function of an operator A is defined, referencing a specific page from a lecture note.
- Further inquiries are made regarding Definition 27 from the referenced material, specifically about the nature of the Ai and the implications of the power series expansion in relation to the continuous spectrum.
- There is a discussion about the completeness of eigenkets of a hermitian operator in the discrete spectrum and how this relates to the power series expansion.
- Clarification is provided that the Ai refers to powers of A, such as 1, A, A^2, A^3, etc.
- Participants discuss the notation of sums and integrals in the context of the spectrum, indicating a nuanced understanding of the mathematical framework involved.
Areas of Agreement / Disagreement
Participants express differing views on whether f(A) shares eigenkets with A, indicating a lack of consensus. There are also multiple questions and clarifications regarding definitions and mathematical implications, suggesting ongoing exploration rather than agreement.
Contextual Notes
Participants highlight potential limitations in understanding the definitions and the implications of the continuous versus discrete spectrum, as well as the completeness of eigenkets, which remain unresolved.