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) |

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.

2004-03-18