# Meaning of the Energy-Time Uncertainty Relation

1. May 30, 2014

### MrRobotoToo

I'd like to know what exactly it's telling us. Does it mean that the more accurately we measure the energy of a system the less accurately we know for how long the system has been in that range of energies? Or does it mean that the more accurately energy is measured the less accurately we know how long the measurement took? Or something else entirely? Some clarification will be much appreciated.

2. May 30, 2014

### The_Duck

The time-energy uncertainty relation gives a minimum value for the product of two numbers:

1) the uncertainty in the energy of a system
2) the typical time interval over which the state of the system changes appreciably

So systems with sharply defined energy change only slowly. Conversely fast-changing systems have poorly defined energies. Perhaps the best example is atomic energy levels. Electrons in excited states will decay back to the ground state by emitting a photon after some typical lifetime. As a result the excited energy levels of an atom have slightly uncertain energies, with uncertainty inversely proportional to the lifetime. This is observed as a slight broadening of spectral lines, because photons from this transition can actually be emitted with a range of energies instead of one sharply defined energy (note that there are also other effects that broaden spectral lines).

How do we define "the typical time interval over which the state of the system changes appreciably?" It's something like this. Pick any observable X. X has some expectation value and some uncertainty. The time interval of interest is the time it takes for X's expectation value to change by more than its uncertainty.

I think that Griffiths, for one, has a somewhat careful discussion of the time-energy uncertainty relation.

3. May 30, 2014

### MrRobotoToo

Thanks, The_Duck. That was quite useful.

4. Jun 15, 2014

### QuasiParticle

I myself am still somewhat puzzled with the time-energy uncertainty relation. It doesn't seem as fundamental as, for example, the position-momentum relation, because it cannot be derived as straightforwardly. Moreover, there have been claims that for some systems the time-energy uncertainty relation can be violated*. As posted by The_Duck, the correct way to think about the time-energy uncertainty relation seems to be in terms of the lifetime of the system, not as the inability to measure its energy with infinite accuracy in a finite time.

*Aharonov and Bohm, “Time in quantum theory and the uncertainty relation for time and energy,” Phys. Rev. 122, 1649 (1961).

http://journals.aps.org/pr/abstract/10.1103/PhysRev.122.1649