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# Free Motion of a Dirac Particle

Consider the Dirac equation without potentials (free equation)

 (5.59)

Its stationary states are found with the ansatz

 (5.60)

where , and

 (5.61)

We split up the 4-component spinor into two 2-component spinors and to give

 (5.62)

where

 (5.63)

Substituting our definitions into the Dirac equation we have

 (5.64)

 (5.65) (5.66)

States with definite momentum can be written as

 (5.67)

We recognize that these states are eigenfunctions of the momentum operator:

 (5.68)

Substitution into equations 5.65 and 5.66 gives

 (5.69) (5.70)

Rearranging we have

 (5.71) (5.72)

For a solution we require

 (5.73)

Using identity 2.24 we have

 (5.74)

where we have been careful with the sign of when solving for both cases at ones.

We also have

 (5.75)

Let us denote the two-component spinor by

 (5.76)

where and are complex and is normalized according to .

The complete set of positive- and negative-energy free solutions is

 (5.77)

The normalization is determined from

 (5.78)

 (5.79)

or

 (5.80)

We now show that another quantum number, helicity, can be used to classify free one-particle states. Define

 (5.81)

This is the four dimensional generalization of the spin vector operator.

We form

 (5.82)

and show that it commutes with the free Dirac Hamiltonian operator.

 (5.83)

Since , we need only consider

 (5.84)

Also , therefore can all be diagonalized together.

can be rewritten as the helicity operator

 (5.85)

Helicity is the projection of the spin onto the direction of the momentum.

For an electron propagating in the -direction ,

 (5.86)

with eigenvalues . The eigenvectors of are

 (5.87)

with

 (5.88)

We can write , where

 (5.89) (5.90)

and

 (5.91)

We thus see that we have two independent states for a positive-energy electron:

 (5.92) (5.93)

We expect the absence of spin-up'' to be equivalent to the presence of spin-down''. This implies that a negative-energy solution with spin down is equivalent to a positive-energy solution with spin up. We choose

 (5.94) (5.95)

Why dose the Dirac spinor need four components to describe a spin-1/2 particle? The answer has to do with parity. in a relativistic theory, we would not be able to represent the effect of parity satisfactorily with only two components.

Next: Covariant Form of the Up: Dirac Equation Previous: Constants of the Motion
Douglas M. Gingrich (gingrich@ ualberta.ca)
2004-03-18