Homework Help Overview
The discussion revolves around the properties of matrix representations of the momentum operator \( p \) and the position operator \( x \) in quantum mechanics, specifically addressing the commutation relation \([p,x] = -ih/2\pi\). The original poster seeks to prove that no finite dimensional representations satisfy this relation and questions how the argument changes in the context of infinite dimensional matrices.
Discussion Character
- Conceptual clarification, Assumption checking, Mathematical reasoning
Approaches and Questions Raised
- Participants explore the implications of taking the trace of both sides of the commutation relation and question the validity of the argument in finite versus infinite dimensions. They discuss the trace of the identity matrix and its implications for dimensionality, as well as the definition of the trace operation in infinite dimensions.
Discussion Status
Participants are actively engaging with the problem, raising questions about the mathematical definitions involved, particularly regarding the trace operation in infinite dimensions. There is an exploration of the consequences of the trace being zero in finite dimensions and how this might differ when considering infinite dimensional representations.
Contextual Notes
There is an ongoing discussion about the assumptions related to the dimensionality of the matrices and the implications of the trace operation being well-defined in infinite dimensions. Participants are questioning whether the trace of the identity matrix remains meaningful in this context.