Discussion Overview
The discussion revolves around the calculation of the determinant of a continuous matrix, specifically in the context of a Hamiltonian in momentum representation. Participants explore the implications of having infinite-dimensional matrices and the challenges associated with finding determinants and eigenvectors in this framework.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- One participant questions how to calculate the determinant of a continuous matrix where the elements are defined as a function of real numbers.
- Another participant notes that if the matrix elements are real, the matrix is symmetric, and suggests that an orthogonal matrix can be found to diagonalize the Hamiltonian matrix.
- A formula for the determinant involving the trace and logarithm of the diagonalized matrix is proposed, but the participant expresses uncertainty about how to find the orthogonal matrix needed for this process.
- There is a suggestion to search for a new basis made up of eigenvectors of the Hamiltonian to simplify the problem, although a challenge is raised regarding how to find these eigenvectors if only the matrix elements are known.
Areas of Agreement / Disagreement
Participants express multiple competing views on how to approach the problem, particularly regarding the methods for finding the orthogonal matrix and the eigenvectors. The discussion remains unresolved.
Contextual Notes
Participants highlight the complexity of working with infinite-dimensional matrices and the dependence on the specific definitions of matrix elements and eigenvectors. There are unresolved steps regarding the determination of eigenvectors from given matrix elements.