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