Energy-time uncertainty relation for a volume and macroscopic objects?

[Aidyan]

As I understand it, the energy-time uncertainty relation $$\triangle E \triangle t \geq \hbar /2$$ expresses a trade-off between the precision with which energy and time can be simultaneously measured. I understand that, unlike position and momentum, time is not an operator, so the relation is not a formal uncertainty between two observables. Instead, it reflects the idea that the shorter the duration $$\triangle t$$ over which a quantum state exists, the more uncertain is its energy $$\triangle E$$. For example, it underlies the phenomenon of the broadening of spectral lines.
I'm wondering whether it can be applied to a volume of space, where the time interval is related to a length L divided to the speed of light c as: $$\triangle t=\frac{L}{c}$$. Then $$\triangle E \geq \frac{\hbar}{2 \triangle t}=\frac{\hbar c}{2 L}=\frac{\hbar c}{2 V^{1/3}}$$. If so, can this be applied to macroscopic objects occupying a volume V? Does this make sense? If not, why not?

 

[PeroK]

As I understand it, the energy-time uncertainty relation $$\triangle E \triangle t \geq \hbar /2$$ expresses a trade-off between the precision with which energy and time can be simultaneously measured.

It's nothing like that. Time is not an observable in QM; it's an independent variable, unrelated to the particle itself (unlike the position of the particle). In any case, Griffiths deals with this common misunderstanding in his Introduction to Quantum Mechanics. See section 3.5.3:

$\Delta t$ in the energy-time uncertainty principle is not the standard deviation of a collection of time measurements. Roughly speaking, it's the time it takes the system to change substantially.

He then goes on to make this more precise.

 

[otennert]

I think we have covered the "time-energy uncertainty relation" a couple of times already, see e.g. here: https://www.physicsforums.com/threads/time-energy-uncertainty-relation.1046138/post-6807748