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MARKUS KLUTE: Welcome back to 8.701. So well continue the discussion on QED. In the last video, we looked at wave equations and we discussed Dirac equations. Now we want to look at solutions to the Dirac equations. All right, so remember that the overall goal now is to find a description of spin-half particles, which we can then use in our matrix element calculation in order to get to cross sections or decay rates of particles. If you just ad hoc or natural choice for a solution would be a wave equation, which is a product of a spinor, which depends on energy and momentum, and an exponent. So we have a free plane wave as a solution to our free particle base waveform. We have to show or we have to make sure that this wave equation satisfies the Dirac equation as shown here. Since the spinor depends only on energy and momentum here, its rather simple to write down the derivatives, because they only depend on the exponent. So we can do this here. And we find those solutions here for th