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"Negative Energy'" Solutions: Hole Theory, Dirac's goal had been to find a relativistic equation for electrons which was free of the, negative probabilities and the "negative energy" states of the Klein-Gordon equation. By, developing an equation that was first order in the time derivative, he hoped to have an, , equation that behaved like the Schrödinger equation, an equation for a single particle. The Dirac, equation also has "negative energy" solutions. While the probability is positive, the flux that, we have derived is in the opposite direction of the momentum vector for the "negative, , energy" solutions., We cannot leave the "negative energy" solutions since the positive energy solutions, alone do not form a complete set. An electron which is localized in space, will have, components of its wave function which are "negative energy"., , E, positive-energy contiuum, +mc, , 0, , -, , -mc, , Figure, , -, , bound states, negative-energy continuum, , 1:, , Spectrum of energy eigenvalues, , of the free, , particle Dirac equation., , One problem of the "negative energy" states is that an electron in a positive energy, , (bound or free) state should be able to emit a photon and make a transition to a "negative, , of, , Dirac, , energy., energy" state. The process could continue giving off an infinite, are, postulated a solution to this problem by assuming that all of the ""negative energy states, all filled and the Pauli Exclusion Principle keeps positive energy electrons from making, amount, , transitions to them. It is possible to imagine an infinite "sea" of negative energy'ëlectrons. This, sea is referred to as the Dirac sea., , E, , +mc2, , 0, -mc, , Figure 2: Negative-energy states are ocupied (Dirac Sea). This represents the vacuum state.

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The, , positive energies, , must, , must all be, , bigger than mc and the negative energies, energy", would be possible for a *"negative, gap of 2mc. It, minimum, , all be, , less than-mes. There is an energy, state. The, make a transition to a positive energy, electron to absorb a photon and, sea is, 2me. If a member of this negative energy, cause this would be, could, that, photon energy, state., A resulting hole, for it to rise into a positive energy, is, it, possible, given enough energy, which is truly an absence of negative, sea. This hole,, would be created in the negative energy, to be a positively, The hole, in fact, would appear, would appear to be positively charged., , charge,, charged particle., , the only, were assumed to be, his equation the electron and proton, At time Dirac invented, by an, assumed that this positive particle, produced, was therefore, It, two subatomic particles., There was a problem with this theory, proton., state,, a, was, energy, a, negative, electron leaving, have the same, sea would have to, how the hole in the Dirac, Several scientists showed, however,, that an electron. Is it possible, is much more massive, obviously, A, electron., proton, an, as, mass, another positively charged, then that there could be, , particle?, , a new particle.., results were found to support, these, In 1932, Anderson's experimental, electron. Anderson called, same mass as an, the, had, these particles, discovered antimatter., The positron was the first, , He found that, new, , particles, , "positrons.", , creation; it, , pair, positive energy state is, The process of moving, The discovery of the, which we interpret as a positron., hole, a, and, electron, produces both an, to the hole theory., of, positron gave a great deal support, the electron and his, relativistic quantum theory of, famous for his 1928, an, , electron to a, , like, , Dirac is most, Prize for Physics with, 1933 he shared the Nobel, existence of antiparticles. In, the, of, prediction, , Erwin Schrödinger..