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# Plane-Wave Solutions

We have already solved the Dirac equation for free particles at rest. The Lorentz transformation may be used to construct the free-particle solutions for an arbitrary velocity.

It is easy to see the invariant form of the exponential by writing its form in the rest frame of the particle

 (5.179)

where , and for and for . The positive-energy solutions () and negative-energy solutions () transform among themselves separately and do not mix with each other under proper Lorentz transformations, as well as, under spatial inversions.

For a Lorentz transformation of a particle from rest ( )

 (5.180)

Using (3.17) and (3.18) we can thus write

 (5.181) (5.182)

To calculate the Lorentz transformation matrix, , we will need

 (5.183)

Using this results with (5.147) gives

 (5.184)

where . From (5.18) and (5.19), we see that each of the four general solutions to the Dirac equation correspond to one of the columns of the above transformation matrix.

Let us now look at some of the properties of the general solution to the Dirac equation. The general form of the wave function is

 (5.185)

Substitution into the Dirac equation gives

 (5.186)

 (5.187)

which is the Dirac equation in momentum space. The adjoint equation is

 (5.188)

Since the covariant normalization statement

 (5.189)

is a Lorentz scalar we may evaluate it in the rest frame for simplicity.

 (5.190)

The completeness relationship is

 (5.191)

We first evaluate it in the rest frame and then boost to the moving frame. For a particle at rest

Therefore

 (5.192)

For a particle in motion

 (5.193)

The probability density

 (5.194)

will not be an invariant but transforms as the zeroth component of a vector. Boosting the particle at rest we have

 (5.195)

Defining and noticing that , we calculate

 (5.196)

 (5.197)

Subsections

Next: Spin Up: Dirac Equation Previous: Bilinear Covariants
Douglas M. Gingrich (gingrich@ ualberta.ca)
2004-03-18