# Energy time complementarity from unitary evolution

I am looking for a well posed experimental situation that illustrates energy time complementarity. I know of Einsteins box, which is discussed quite nicely in Bohr's article Discussions with Einstein on Epistemological Problems in Atomic Physics. He used time dilation due to gravity, I am looking for examples that do not involve gravitational physics.

Bohr in his article never applies unitary evolution $e^{-iHt}$ directly to the problem. This is analogous to $e^{ipx}$ which is at the heart of momentum position uncertainty . It would be nice to see Energy time Complementarity directly from $e^{-iHt}$. What does energy time complementarity imply for energy conservation?

Does such an experiment exist? If not, is there some fundamental impedance preventing it from being done in a gravity-free context?

Please take care to pose the experimental situation completely, as these are subtle issues. I would prefer examples that do not involve QFT, as it would unnecessarily distract us from the main question.

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Since a state which exists only for a finite duration can't have a precisely defined energy, you can note that the shorter the lifetime of an excited state of an atom, the greater will be the spread of energies assigned to that state. So when it transitions to the groundstate, this spread is transferred to the emitted photon, hence the spectral line has an unavoidable broadening from the ideal infinitely thin case.

(Incidentally, not related to examples but there is a good discussion of the general meaning of the energy time uncertainty principle here).

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I have heard about this situation, What does this imply for energy conservation? The only time that one seems to be able to measure is when the state got excited. we must be able to account for energy conservation if we look closely at how the atom was excited. – Prathyush Jun 15 '13 at 7:46
I think we have to be very precise about what we mean by energy conservation. Here, the system isn't in an energy eigenstate - it just doesn't have a definite energy - so energy conservation can only be phrased in terms of the energy expectation value, i.e the average over a number of trials. Energy conservation is not violated in this sense. – twistor59 Jun 15 '13 at 8:18
I think there is a well defined meaning for energy atleast at large times. When the atom and the photon are sufficiently far apart we can say that interaction between the atom and photon is zero, the total energy of the atom +photon is a well defined quantity. However as you said we will observe distributions. The final energy is a well defined measurable object, however the initial energy(of state in an atomic superposition) is not, however if we look at how the state was prepared then i think it is meaningful to study energy conservation. – Prathyush Jun 15 '13 at 9:42
Energy conservation has a well defined meaning. That is you measure the energy of the system at an instant(in above case system + environment) then the laws of quantum mechanics guarantee that time evolution will conserve energy or in other words you will always be in the same eigenstate. This picture doesn't change even if you have two or more systems talking to each other. The total energy is always conserved and energy-time complimentary or any interpretation of it thereof is meaningless. – Noob Rev B Jun 23 '13 at 22:57
As far as I understand there is heuristic classical picture for energy-time uncertainty. Imagine bringing a system about which you know nothing(say you don't know the energy levels or for that matter the no. of particles in it) to a thermal environment. If you give the system long enough time to relax in the thermal environment which is much larger than the system size you expect thermal distribution of energy of the system even after it has been isolated from the environment. Thus having large $\delta t$ implies more certain energy distribution without invoking any quantum mechanics. – Noob Rev B Jun 23 '13 at 23:09

In my opinion energy-time uncertainty relation is a historical baggage and has no meaning whatsoever in the sense of position-momentum uncertainty. This is due to the fact that time is not measurable attribute of a particle, correspondingly you don't have a hermitian operator for it. You can't have "wavepacket" in time. Time, like position is a coordinate. However a particle can have a spread in position coordinate and a corresponding spread in momentum at a given time ($[x,p]$ commutator is $i$ when both operators are taken at the same time) . These two quantities are enough to determine the Hamitonian at that time (hence the energy) and unique time evolution of the wave function for infinitesimally later time. Forget everything that is written in QFT textbooks about virtual particles and energy-time uncertainty! Time is certain and energy is conserved.

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This answer is opinion-based and controversial, at best. – Wouter Jun 23 '13 at 22:53
@Wouter: very well sir, could you please point out which part of my answer is scientifically wrong or unfalsifiable. When you do that my answer will qualify as "opinion" then we can talk about bias and controversy. – Noob Rev B Jun 23 '13 at 23:14
One directly observable effect of the energy-time uncertainty relation is the natural linewidth of spectral lines. – Wouter Jun 23 '13 at 23:23
my arguments do not preclude natural linewidth at all. Linewidth is natural consequence of deterministic schrodinger equation (or master equation for density matrix). Just because one is interested(& observes energy loss) in a subsystem doesn't mean the energy conservation for the universe is violated. See my comments below the previous answer for more details. I would again renew my request to find flaw in my answer(the one which you termed as opinion, biased and controversial) above rather than linking wikipedia pages. I have already stated that energy-time complimentary is heuristic tool – Noob Rev B Jun 23 '13 at 23:36
My first comment was directed at the fact that your answer is not based on mainstream physics, which is what this site is about. You'll notice I didn't downvote (I rarely do). Perhaps I did blaze through to my second comment too quickly. The point is that the energy-time uncertainty relation is part of mainstream physics (again, the subject of this site), so your answer is opinion-based and controversial in that sense. In any case, I'll mention this link which I found interesting. – Wouter Jun 24 '13 at 0:38