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This is a question about the negative energy solutions to the free particle Dirac Equation in the first quantized picture. We need both the positive and negative energy solutions to have a complete set of states.

For example, the bound states of the hydrogen atom consists of superpositions of positive and negative energy free particle solutions. So then, what is the physical interpretation of a superposition of positive and negative energy states?

Suppose we had a hydrogen atom, and we immediately removed the nucleus. If we measured the energy of the resulting free electron, what would we see?

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A free electron? –  Alfred Centauri Jul 1 '12 at 2:28
    
We would see a free particle, but I guess the question I'm trying to ask is: what is it's energy? –  ChickenGod Jul 1 '12 at 6:29
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I don't understand your claim "the bound states of the hydrogen atom consists of superpositions of positive and negative energy free particle solutions". Can you explain what you mean by this. The last time I learned the Dirac solution of the H atom (many many years ago) I don't recall anything resembling your statement. –  John Rennie Jul 1 '12 at 9:16
    
Perhaps I'm mistaken, but I recall that if you expand the ground state of the hydrogen atom in the basis of free particle plane waves, you get small but nonzero coefficients for the negative energy waves. Then, if you make a measurement of the energy, what are the possible outcomes? –  ChickenGod Jul 2 '12 at 10:45
    
At least heuristically, I would have to agree with @ChickenGod that the hydrogen atom has negative energy electron states in it. Since its a bound state the energy itself is negative you won't be able to produce this with just linear combos of positive energy solutions. –  DJBunk Jul 11 '12 at 17:14
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