Observables commute and time operator

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Discussion Overview

The discussion revolves around the commutation relations of operators in quantum mechanics, specifically focusing on the total energy and momentum operators versus kinetic energy and momentum operators. It also explores the implications of the Heisenberg Uncertainty Principle (HUP) regarding time and energy, including the existence of a time operator and its potential consequences.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that the total energy and momentum operators do not commute, while kinetic energy and momentum operators do, seeking an explanation for this distinction.
  • Another participant asserts that momentum determines kinetic energy, but this does not hold for total energy, and states that there is no time operator in quantum mechanics.
  • A different participant argues that whether the Hamiltonian commutes with the momentum operator depends on the specific Hamiltonian used, mentioning that for a free particle, it may commute.
  • This participant also claims that the origin of the HUP related to time and energy is different from the usual interpretations and suggests that the delta-t in the HUP refers to lifetimes of states rather than measurement times.
  • One participant posits that the existence of a time operator with standard commutation relations would imply the absence of a lower energy bound, leading to no ground state.
  • Another participant references Pauli's argument regarding the time operator and suggests that a careful analysis contradicts this view, inviting further discussion on the implications of a time operator.
  • A follow-up question is raised about how to address the implications of a time operator leading to no ground state.

Areas of Agreement / Disagreement

Participants express differing views on the existence of a time operator and its implications, with no consensus reached on the validity of the arguments presented regarding the commutation relations and the HUP.

Contextual Notes

There are unresolved assumptions regarding the definitions of operators and the specific conditions under which they commute. The discussion also highlights the complexity of the HUP and its interpretations in quantum mechanics.

CAF123
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I just have two questions relating to what I have been studying recently.
1) I know that the total energy and momentum operators don't commute, while the kinetic energy and momentum operators do. Why is this the case? (explanation rather than mathematically).
2) One form of the HUP says that we can't measure position and momentum of a particle simultaneously and when I evaluate the commuator , it gives a non zero operator. The other form of the HUP says that ## ΔEΔt ≥\frac{\hbar}{2}.##Is there a way to evaluate the commutator here - to similarly show that a non zero commutator between time and energy (if it exists) is in agreement with the HUP? (I.e do we define a time operator)?
Many thanks.
 
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Momentum determines the kinetic energy. This is not true for the total energy.
## ΔEΔt ≥\frac{\hbar}{2}.## is an "effective" rule - there is no time operator in quantum mechanics.
 
1) Who says the hamiltonian doesn't commute with the momentum operator? That really depends on the hamiltonian. For example, a hamiltonian for a free particle trivially commutes with p. Most of the time, this won't happen because H will contain both position and momentum operators, which means that it won't commute with either (because p and q don't commute: HUP).2) As for the second question, no there's no time operator in QM and the origin of that HUP is different than the usual ones (and I think an explanation must involve QED), so you can't really evaluate a commutator for it. And keep in mind that the delta-t in that expression refers to lifetimes of certain states (and not the time you take to make a measurement). Someone might be able to elaborate further on this point.
 
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An existence of time operator with the usual commutation rule with hamiltonian implies no bound of lower energy(no ground state)
 
Last edited:
andrien said:
An existence of time operator with the usual commutation rule with hamiltonian implies no bound of lower energy(no ground state)

This is essentially Pauli's argument in the article on wave mechanics in the <Enzyklopädie der Physik>. A careful analysis (Eric Galapon in Proc.Roy.Soc.London) shows he's quite wrong. See my blog article on this.
 
So you mean a time operator exist.so can you tell how to get rid of the condition it implies, i mean no ground state and it's significance.
 

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