Discussion Overview
The discussion revolves around the nature of the commutator of two operators in quantum mechanics, specifically whether it is always a scalar or can be a matrix. Participants explore the implications of the commutator in the context of quantum mechanics and its representation in mathematical terms.
Discussion Character
- Technical explanation, Conceptual clarification, Debate/contested
Main Points Raised
- One participant states that the commutator [A,B] is generally a matrix, but notes that the specific case of the position and momentum operators results in a scalar, i*hbar.
- Another participant clarifies that in this context, i*hbar is shorthand for i*hbar times the identity operator.
- A later reply expresses surprise that this point was not made clear in Griffiths' text, suggesting a gap in the explanation of the mathematics involved.
- Another participant points out that the commutator of two scalars is a scalar, while the commutator of two vectors results in a matrix, providing an example involving the Kronecker delta.
- This participant also mentions that the trace of the commutator in four dimensions would yield a specific numerical result.
Areas of Agreement / Disagreement
Participants express differing views on the nature of the commutator, with some asserting it is a scalar in specific cases while others argue it is generally a matrix. The discussion remains unresolved regarding the broader implications of these interpretations.
Contextual Notes
There is a lack of clarity regarding the definitions and assumptions surrounding the commutator, particularly in relation to the identity operator and the dimensionality of the operators involved.