Discussion Overview
The discussion centers around the nature of the commutator of position and momentum operators in quantum mechanics, specifically whether it should be considered a postulate. Participants explore the implications of this commutator, its derivation, and its relationship to classical mechanics through the lens of canonical quantization.
Discussion Character
- Debate/contested
- Conceptual clarification
- Mathematical reasoning
Main Points Raised
- Some participants question whether the commutator [x,p]=i$\hbar$ is a postulate, noting that no standard texts explicitly state it as such.
- Others argue that the momentum operator can be derived from the principle of translation invariance, suggesting that the momentum operator is fundamentally linked to this symmetry.
- A participant introduces the concept of the Poisson bracket from classical mechanics, proposing that its quantum counterpart, involving the commutator, provides a deeper understanding of the relationship between classical and quantum mechanics.
- Some participants suggest that taking the commutator as a postulate allows for the derivation of both the momentum operator in position representation and the position operator in momentum representation.
- There is mention of canonical quantization, where conjugate variables are related through the Poisson bracket and swapped with the commutator, reinforcing the idea that this commutator can be viewed as a postulate in this formalism.
Areas of Agreement / Disagreement
Participants express differing views on whether the commutator should be treated as a postulate. While some see it as a foundational aspect of quantum mechanics, others emphasize its derivation from broader principles, indicating that the discussion remains unresolved.
Contextual Notes
Participants note that the numerical value of $\hbar$ needs to be measured or postulated, which adds complexity to the discussion about the foundational nature of the commutator.