Time-energy uncertainty relation

In summary: Eq. (55).In summary, the time-energy uncertainty principle relates the uncertainty in energy and the characteristic time it takes for a system to change by one standard deviation. It sets a lower limit for the product of these two quantities and can be derived from Heisenberg's inequality. The interpretation of this principle can vary, but it is based on the idea that accurately measuring one quantity will decrease the accuracy of the measurement of the other quantity.
  • #1
quasar987
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How should [itex]\Delta E \Delta t \geq \hbar/2[/itex] be interpreted? When does it apply? Delta t refers to the time of what? etc.

Thx.
 
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  • #2
For a given time independent observable A that doesn't commute with the Hamiltonian and a state |psi>, interpret [itex]\Delta E = \Delta H [/itex] (H is the hamiltonian and delta means standard deviation) and define:

[tex]\Delta t := \frac{\Delta A}{|d\langle A \rangle/dt|}[/tex]

So [tex]\Delta t[/tex] is the characteristic time it takes for the observable to change by one standard deviation.
I think this is the best interpretation of the inequality. No mystic mojo is involved.

Then by Heisenberg's inequality:

[tex]\Delta H \Delta A \geq \frac{1}{2}|\langle [H,A] \rangle|[/tex]

And using:
[tex]\frac{d\langle A\rangle}{dt}=\frac{i}{\hbar}\langle [H,A]\rangle[/tex]

you can rewrite it as [itex]\Delta E \Delta t \geq \frac{\hbar}{2}[/itex]
 
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  • #3
It sets a lower limit to the products of the uncertainty in those two quantities, ie for heisenburgs uncertainty principle its displacement and velocity
 
  • #4
See the explanation in Sakurai's book.

Daniel.
 
  • #5
quasar987 said:
How should [itex]\Delta E \Delta t \geq \hbar/2[/itex] be interpreted? When does it apply? Delta t refers to the time of what? etc.

Thx.
To be precise there is no such thing as a time-energy uncertainty principle. Uncertainty principles are the relationship between two operators and while there is an energy operator there is no such thing as a time operator. Best to call it the time-energy uncertainty relation. The meaning of this relation is dt is the amount of time it takes for a system to evolve and dE represents the average change in the amount of energy during this time of evolution.

Pete
 
  • #6
But, I heard somewhere that in one case dW dt>=hbar/2 is correct. What is this example? Maybe for photons?
 
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  • #7
I have two questions:

1) what's the motivations for this expression?

Galileo said:
And using:
[tex]\frac{d\langle A\rangle}{dt}=\frac{i}{\hbar}\langle [H,A]\rangle[/tex]

2) why is 1 standarad deviation chosen? Wouldn't the inequality change for any other choice?
 
  • #8
The time-energy uncertainty principle is a little bit different from the position-momentum one. This is perhaps best illustrated by Landau's quote, "To violate the time-energy uncertainty relation all I have to do is measure the energy very precisely and then look at my watch!"
 
  • #9
I'd say the physical meaning behind the time-energy uncertainty relation has to do with the fact that a state of definite energy is characterized (physically) by having a definite frequency of change of its phase. To decide what that frequency is, you need to watch many cycles of time, and the more cycles you follow, the more precisely you know that frequency. But the more cycles you watch, the less you can say about the actual time at which you "looked", for you looked over a range of times. Conversely, if you look at "your watch" at a very specific time, then you cannot say what is the frequency at which the phase of the state is changing. Note that if you divide through by h, the expression becomes uncertainty in frequency times uncertainty in time exceeds 1 cycle.
 
  • #10
All I know about it is that you have to be carefull with the interpretation. It seems that there is not a single version of it and not even experts agree on it.
There is however a simple version that can be derived clearly from first principles an is the one Galileo expleined, but that is not the only interpretation.
 
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  • #11
Galileo said:
For a given time independent observable A that doesn't commute with the Hamiltonian and a state |psi>, interpret [itex]\Delta E = \Delta H [/itex] (H is the hamiltonian and delta means standard deviation) and define:

[tex]\Delta t := \frac{\Delta A}{|d\langle A \rangle/dt|}[/tex]

So [tex]\Delta t[/tex] is the characteristic time it takes for the observable to change by one standard deviation.
I think this is the best interpretation of the inequality. No mystic mojo is involved.

Then by Heisenberg's inequality:

[tex]\Delta H \Delta A \geq \frac{1}{2}|\langle [H,A] \rangle|[/tex]

And using:
[tex]\frac{d\langle A\rangle}{dt}=\frac{i}{\hbar}\langle [H,A]\rangle[/tex]

you can rewrite it as [itex]\Delta E \Delta t \geq \frac{\hbar}{2}[/itex]
In this approach, Delta t is NOT UNCERTAINTY of time, but a time DURATION of a physical process. There is however a similar way to introduce UNCERTAINTY of time measured by a clock, as explained, e.g., in
http://xxx.lanl.gov/abs/1203.1139 (v3)
Eqs. (19)-(23)
 

What is the time-energy uncertainty relation?

The time-energy uncertainty relation, also known as the Heisenberg uncertainty principle, is a fundamental concept in quantum mechanics that states that it is impossible to know both the exact position and momentum of a particle at the same time. This means that there is a limit to how precisely we can measure the energy and time of a quantum system.

Why is the time-energy uncertainty relation important?

The time-energy uncertainty relation is important because it sets a fundamental limit on our ability to make precise measurements in the quantum world. It also has implications for the behavior of particles at the subatomic level and is a key principle in understanding the uncertainty and randomness of quantum mechanics.

How does the time-energy uncertainty relation relate to the uncertainty principle?

The time-energy uncertainty relation is a specific case of the uncertainty principle, which states that there is a limit to how precisely we can measure certain pairs of physical properties of a quantum system. The time-energy uncertainty relation specifically deals with the uncertainty between time and energy measurements.

Can the time-energy uncertainty relation be violated?

No, the time-energy uncertainty relation is a fundamental principle of quantum mechanics and cannot be violated. It is a consequence of the wave-particle duality of particles at the subatomic level and is supported by numerous experimental evidence.

What are some practical applications of the time-energy uncertainty relation?

The time-energy uncertainty relation has many practical applications, such as in nuclear physics where it helps to explain the behavior of subatomic particles, and in quantum computing where it is used to understand the limitations of measuring and manipulating quantum states. It also has implications in fields such as optics, chemistry, and astronomy.

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