Uncertain principle and Noether theorem?

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Discussion Overview

The discussion revolves around the relationship between Noether's theorem, symmetries, and conservation laws in classical and quantum mechanics. Participants explore how these concepts interact, particularly in the context of the uncertainty principle and observable quantities in quantum mechanics.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses confusion about how Noether's theorem, which relates symmetries to conservation laws, applies in the quantum framework where the uncertainty principle is also relevant.
  • Another participant states that if an observable A is conserved, its commutator with the Hamiltonian is zero, leading to a specific relationship involving the uncertainties of A and the Hamiltonian.
  • A question is raised regarding the relationship between symmetry and conservation observations in quantum mechanics, beyond the established commutation relationship.
  • Another participant suggests that if an observable A implements a symmetry, then A is conserved, provided it commutes with the Hamiltonian under certain assumptions about time independence in the Schrödinger picture.

Areas of Agreement / Disagreement

The discussion contains multiple viewpoints regarding the relationship between symmetries and conservation laws in quantum mechanics, and no consensus is reached on the implications of these relationships.

Contextual Notes

Participants assume certain conditions regarding the time independence of observables and the common domain of operators, which may not be explicitly defined in the discussion.

ndung200790
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Please teach me this:
It seem Noether theorem say that a symmetry corespondant a conservation observation at classical level, and at quantum framework the uncertain principle works.So I don't understand why at quantum level,there still exist conservation law.Example momentum conservation at vertex of Feymann diagram.
Thank you very much in advance.
 
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Well, if an observable A is conserved, then its commutator with the Hamiltonian is zero, thus we can say that [itex]\Delta A \dot \Delta H \geq 0[/itex] on all the common domain. That's all to it.
 
So,are there any relation between symmetry and conservation observations in quantum mechanics(beside the claim: if A is conserved , A must be to commute with H)?
 
If A implements a symmetry in the sense of Wigner and Weyl, then A is conserved which means that A commutes with H on all the common domain, of course, assuming the explicit time independence of A in the Schroedinger picture,
 

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