Observables commute and time operator

In summary, the conversation discusses the commutation of operators in quantum mechanics, specifically the differences between total energy and momentum operators and kinetic energy and momentum operators. The concept of the Heisenberg Uncertainty Principle is also brought up, with one form stating that position and momentum cannot be measured simultaneously. There is no time operator in quantum mechanics and the origin of the HUP in this context is different. The existence of a time operator with the usual commutation rule with the hamiltonian would imply no bound of lower energy, but this has been proven to be incorrect.
  • #1
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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  • #2
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.
 
  • #3
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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  • #4
An existence of time operator with the usual commutation rule with hamiltonian implies no bound of lower energy(no ground state)
 
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  • #5
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.
 
  • #6
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.
 

1. What is the concept of "observables commute"?

The concept of "observables commute" refers to the property of two observables in quantum mechanics being able to be measured simultaneously without affecting each other's outcome. This means that the order in which the observables are measured does not affect the final result.

2. What is the significance of observables commuting in quantum mechanics?

The significance of observables commuting is that it allows for the prediction of measurement outcomes in quantum systems. It also allows for the mathematical description of quantum mechanics to be simpler and more intuitive.

3. What is the time operator in quantum mechanics?

The time operator in quantum mechanics is a mathematical representation of time in a quantum system. It is used to describe the evolution of a quantum state over time and is an essential component in calculating the probability of a specific outcome in a measurement.

4. How does the time operator relate to observables commuting?

The time operator and the concept of observables commuting are closely related. If two observables commute, it means that they can be measured simultaneously at any given time. This is equivalent to saying that the time operator and the two observables commute.

5. Can all observables commute in a quantum system?

No, not all observables can commute in a quantum system. The commutativity of observables depends on the specific properties of the system and the observables being measured. In some cases, observables may partially commute, but there will always be some that do not commute.

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