Consider the Coulomb interaction

(4.156) |

and the substitution

(4.157) |

The Klein-Gordon equation with a Coulomb potential is

(4.158) |

For stationary states we can write

(4.159) |

For spherical coordinates

(4.160) |

(4.161) |

(4.162) |

where . The Klein-Gordon equation in spherical coordinates is

(4.163) |

(4.164) |

Defining we can write

(4.165) |

Defining we can write

(4.166) |

Defining we can write

(4.167) |

Defining and

(4.168) |

This radial equation is the same as in the nonrelativistic case if . Solving for we have

(4.169) |

The parameter is determined by the boundary condition on when .

The remainder of this problem is left as an exersize for the student.

2004-03-18