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I can understand the Heisenberg Uncertainty Principle which states that $$\Delta{x}\Delta{p_x}\ge \hbar/2$$

I also understand that this can be extended to any canonical conjugates.

But I can't seem to understand the Time-Energy Uncertainty Principle. Where does it have its origin? Does the Heisenberg Uncertainty principle support it (in which case I would like to see an elegant derivation) or is it an independent principle in itself?

Also, how do we interpret this?

In measurement theory people seem to say that any energy measurement has an uncertainty that has to do with the duration of measurement,
Atomic/Molecular physicists say that spectral energy has an error that has to do with the lifetime of the excited states and so on.

It seems to me that people are randomly interpreting the Time-Energy Uncertainty anywhere time and energy come together. What is the mathematical basis of such an interpretation? And can it be literally used anywhere where energy and time come together - which is everywhere?

DanielC
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TheImperfectCrazy
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  • Time and energy are the first components of the position and momentum four-vectors in relativity.. – Deschele Schilder Apr 01 '21 at 07:51
  • Okay I had realized that, but then the dot product of these four vectors (say $\Delta x . \Delta P$) gives a sum of two terms (corresponding to zeroth component and the x-axis component) which should satisfy $\ge \hbar /2$ but then how can we say that this is same as the first term being $\ge \hbar /2$? – TheImperfectCrazy Apr 01 '21 at 08:04
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    Have you tried reading other discussions here?https://physics.stackexchange.com/questions/501145/validity-of-the-derivation-of-time-energy-uncertainty-principle?rq=1 – DanielC Apr 01 '21 at 08:05
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    Also this one: https://physics.stackexchange.com/questions/53802/what-is-delta-t-in-the-time-energy-uncertainty-principle?noredirect=1&lq=1 – DanielC Apr 01 '21 at 08:08

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