A “free” particle not bound by any sort of potential well or inverse-square force can theoretically have any energy (until those Planck energy guys argue that there is an ultimate quantization even for free particles – but those would be incredibly tiny divisions, so we’ll ignore them), so in those cases you can have any uncertainty in position or momentum, or ebergy and time or whatever.

I’ve always been a bit annoyed by the description that the uncertainty principle is a case of our inability to *measure* the energy levels, because it is arguably an even deeper problem. The uncertainty relationship exists even if you *don’t* try to measure anything. As long as you buy the de Broglie hypothesis that matter has wave nature, you can represent you electron, photon, whatever as a wave, and you can break that wave up into its Fourier components. To create a wave localized in space requires a *lot* of frequency components. The tighter you wish to localize it, the more frequency components you need. To absolutely localize the particle at one point by a “delta function” requires an infinite range of frequencies. Similarly, to determine a particular range of frequencies requires an uncertainty of position, and requiring a single frequency requires an infinite range in the position uncertainty. Since the momentum of a particle is proportional to the frequency, there is an inverse relationship between momentum and position. The more accurately you know one, the less accurately you know the other. This holds true for photons (which have no mass, but do have momentum) just as much as for particles such as electrons that have a rest mass.

In other words, if you “buy” modern quantum mechanics, with the Schroedinger equation and Planck’s constant, there doesn’t exist any theoretical situation in which you can simultaneously specify both position and momentum. The same reasoning doesn’t let you know both Energy and Time to arbitrary accuracy, or total angular momentum and its projection along onees, or any other set of operators which do not commute.

Of course, in the real world we still have to measure things, and QM is still a theory that might still have its shortcomings disproven by experiment, but QM is remarkably consistent and accurate in its predictive capability (shut up – I don’t want to talk about EPR or Maxwell’s Demon), so it’s not a problem of us not having sufficiently accurate equipment or sufficiently devious experimenters. If QM is correct, it’s not possible to come up with a measurement scheme sufficiently subtle to measure both position and momentum accuirately, because there’s no wave packet that exists with such a combination that it can collapse to.