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I was also thinking about the uncertainty principle in regards with energy & time. The question of something like:

Alpha tunneling out of the nucleus is where this can be invoked, but having an uncertainty in energy does not imply that energy is not conserved - it just means that energy got borrowed from somewhere and then can be returned such that the product of uncertainties remains above the minimum and it looks like a fluctuation. So I got to wondering where the energy is borrowed from and the only thing I can really come up with right now is that it gets borrowed from another field that it interacts with. Perhaps because of Coulomb repulsion in the nucleus, whose interaction field would extend far outside the nucleus, there is a temporary drop in the Coulomb field energy? That was my guess, because I don't consider borrowing from other matter fields very realistic, since several would all have to contribute something, rather than just one. Anyways, just thinking "out loud" here.

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Have you read the alpha decay article in wikipedia? . I believe your question is based on a misconception of what tunneling is. . No extra energy is needed. The wave function has a probability for the alpha particle to be outside the nucleus with the energy level it had when within it. – anna v Oct 27 '12 at 11:03

Classically the answer on this question was depending on your favorite interpretation of quantum mechanics, anyway now in the framework of Quantum field theory and quantum electrodynamics, we can say that this "extra energy" comes from vacuum, because we now know (but to be very precises not 100% sure yet) that vacuum not that empty thing, and it is fluctuate continuously, and this fluctuations makes very small input on various process that happens in quantum word, anyway this input makes such a very interesting phenomenons as this energy borrowing you mentioned.

Anyway one should emphasize that the energy is really "borrowed" in sense that it should be "returned back", for this we say that average vacuum expectation value is always zero.

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It's usually explained with respect to the wave functions of the nucleus and alpha particle. It's been a while so someone else might be able to explain it better!

I'm not sure what you are asking, if it is where does the energy come from for the alpha particle to tunnel out of the nucleus...this is how I would describe the set of events.

I think that the alpha particle and nucleus can be in different states (with the same energy). Normally they are in one of the possible states where the nucleus is stable. However there are states with the same energy where the nucleus is not stable enough and the nucleus will decay. Obviously the less likely that the nucleus appears in this unstable state the longer the half life.

It's hard to think of a good analogy, perhaps a mixture of two gas molecules in a box. Normally (on average) they are evenly distributed through out the box. However sometimes due to random motion of all the particles the density of one of the species increases in one area. This could trigger a reaction like combustion or a nuclear chain reaction if you like, which is equivalent to the decay.

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