In all the textbooks that I have seen, energy-time relation is written in the following way:$$\Delta E \cdot \Delta t \geqslant \frac{\hbar}{2}$$ Here is my interpretation of this principle: The energy of a system that exists for a finite time $\Delta t$ has an energy uncertainty of at least $\frac{\hbar}{2\Delta t}$.

So here is where I get really confused.. This relation then suggests that if $\Delta t$ and $\Delta E$ can both be very large, which means that huge fluctuations in energy can occur for a very long period of time. How is that possible? Doesn't that violate the conservation of energy? My thought was maybe the relation also suggests that:$$\Delta E \cdot \Delta t \sim \frac{\hbar}{2} $$ That would resolve the issue here. But I am not sure if this speculation is actually correct.


The energy-time uncertainty principle tells us only that $\Delta E \Delta t$ must be greater than $\hbar/2$ but it does not specify how large $\Delta E$ and $\Delta t$ are. That will be determined by other properties of the system.

Put another way, if I have some system with a very large timescale $\Delta t$ then my system could also have a large energy uncertainty or it could have a small energy uncertainty. That depends on the nature of my system. All the energy-time uncertainty principle tells us is that $\Delta E$ has a lower limit.

I suspect you are worrying about the often repeated claim that the energy-time uncertainty principle allows a virtual particle with an energy $E$ to flash into existance for a time $t$ as long as $Et \approx \hbar/2$. That's because this claim is complete rubbish, or at best a gross misrepresentation of reality. See for example the answers to Creation of particle anti-particle pairs or search this site for more related questions.


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