Beside momentum and position I read that Heisenberg Uncertainty can also apply to energy and time so for instance virtual particle (energy) could pop into existence but for a very brief moment (time) unlike say an electron (if we know the energy level) it will lasts for a very very long period of time, so I just wondering if there are any other more empirical examples.


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The fist thing to mention is that the uncertainty relation between energy and time is in this sense different from position and momentum that in quantum mechanics time is a parameter and not a dynamical variable (like position and momentum, which depend on this parameter). Beside that, there is no intuitive hermitian time-operator $\hat T$ which is canonical conjugate to the energy-operator (the Hamiltonian, $\hat H$) such that the commutator is $[\hat T,\hat H] = i \hbar$ from which we can obtain the uncertainty relation from the expectation value of the commutator, i.e. $\Delta A \Delta B \ge \frac{1}{2} |<[\hat A, \hat B]>|$.

Since the uncertainty relation make statements about the dispersion of observables of experimental outcomes we can construct measurement schemes where the energy-time uncertainty relation come into play.

Therefore, let us consider the Einstein ''clock-in-the-box'' thought-experiment: Suppose we have a box which emits a photon through a slit and that we measure the weight of the box and can measure the time when the photon was released with the clock inside the box. Since the weight is proportional to the rest mass $m$ and by Einstein's famous fomula proportional to its energy $E = mc^2$. Here, I just want to mention that in the original discussion between Einstein and Bohr this scheme was proposed by Einstein to show that he can violate the energy-time uncertainty relation, but, however, Bohr was able to defeat Einsteins argumentation by considering the uncertainty of the pointer in the weight measuring device. I want to point out, that a consistent description of measurments in quantum mechanics has to include the measuring device as a quantum system and thus the pointer has to obey the uncertainty relations of position and momentum as well (the generel measurement interactions were already described by von-Neumann in his famous book).

The argument from Bohr was like this:

The position and momentum of the pointer are denoted as $x$ and $p$ with its corresponding uncertainties $\Delta x $ and $\Delta p$. Once wehave choosen our degree of certainty in the pointer poisition the uncertainty in the pointer momentum is bounded from below by $\frac{\hbar}{2\Delta x}$. We now want to measure the weight of the emitted photon with the following procedure: after the photon is emitted, the pointer for th weight measurement points at a different position and by adding some new weight to the box we can bring the pointer back to its original position. But the accuracy of this weight measurement can notbe better then the smallest adde weight $g \Delta m$ and if we add the mass $\Delta m $ and wait a time $t$, the momentum delivered to the box can not be greater than $g \Delta m t$, and this mustbe greater than $\Delta p$ to have an observable effect. Thus we have $\Delta p < gt \Delta m$. Putting all together and using $\Delta E = c^2 \Delta m$, we have

$\frac{\hbar}{2} < \frac{gt \Delta x \Delta E}{c^2}$.

And now comes the funny part, which I personally like the most in the old Einstein-Bohr debates, namely that Bohr uses the theory of general relativity from Einstein to defeat him. Since a clock in the gravitational field ticks more slowly than a clock which is in free fall. And two clockes at different heights will run at different rates, because of their gravitational potential difference.If the difference in height is $\Delta x$, the fractional difference $\Delta t / t$ in their measured times will be $\Delta t / t = g \Delta x / c^2$. Thus over a measuring period $t$, the uncertainty in the time of the clock is $\Delta t = t g \Delta x / c^2$ and combining with the previous results we have

$\Delta t \Delta E \ge \frac{\hbar}{2}$,

your energy-time uncertainty relation.

I hope this is example is empirical enough despite beeing just a thought-experiment.

A formal understanding of the uncertainty relation between energy and time can be seen, like for position and momentum, from the Fourier Transformation between time and frequency domain (frequency and energy are essentially the same despite a proportionality factor). Since every wave-mechanics has the property of dispersion relations between the two spaces, and since the Schrödinger wave-mechanics is such a wave-theory we have this uncertainty, but which is not purely quantum mechanical since the same appears in classical wave mechanics.

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    $\begingroup$ While this explanation might have defeated Einstein, it seems rather odd that one should have to invoke general relativity to save Heisenberg time-energy uncertainty principle that is assumed to be true in non-relativistic QM. Is there a very precise understanding of what $\Delta E$ and $\Delta t$ means in this case? $\endgroup$
    – Vivek
    Commented Nov 23, 2019 at 0:25
  • $\begingroup$ Thank you for your comment, it is indeed a very surprising resolution of the 'paradox' posed by Einstein! From my point of view $\Delta E$ and $\Delta t$ can be understood in how precise you can determine the photons energy (by measuring its mass) during a time interval with accury $\Delta t$. I think, Einstein introduced this thought-experiment in order to decouple a measurement of time from a measurement of time by inferring the energy of the photon from a position measurement which has no uncertainty relation with time. $\endgroup$
    – MST
    Commented Nov 23, 2019 at 16:43

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