Discussion Overview
The discussion centers on the properties of evanescent waves in relation to vector spaces, particularly in the context of wave equations such as Poisson, Laplace, and Schrödinger. Participants explore the completeness and orthogonality of evanescent waves compared to propagating waves, examining definitions and mathematical implications.
Discussion Character
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant questions whether evanescent waves can satisfy the completeness condition, suggesting it seems unlikely.
- Another participant notes that the definition of evanescent waves and the scalar product used will significantly affect the outcome, proposing that a pre-Hilbert space may arise from these definitions.
- A specific form of evanescent waves is introduced as Exp[-k x], with both k and x being real.
- There is a query about whether "real" refers to all real numbers or specifically positive values.
- A participant raises a concern regarding the lack of symmetry between propagating and evanescent waves, questioning the ability to expand functions in the evanescent wave basis in infinite space.
Areas of Agreement / Disagreement
Participants express differing views on the definitions and properties of evanescent waves, with no consensus reached on their ability to form a vector space or satisfy completeness conditions.
Contextual Notes
Discussions involve assumptions about the definitions of evanescent waves and the scalar product, as well as the implications of working in finite versus infinite spaces. The mathematical treatment of orthogonality and completeness remains unresolved.