# 11.7: Fine Structure of Hydrogen Energy Levels

For the case of a hydrogen atom,

(1260) |

Hence, Equations (1258) and (1259) yield

(1261) | ||

(1262) |

where , and

(1263) | ||

(1264) |

with . Here, is the Bohr radius, and the fine structure constant. Writing

where

(1267) |

we obtain

Let us search for power law solutions of the form

where successive values of differ by unity. Substitution of these solutions into Equations (1268) and (1269) leads to the recursion relations

Multiplying the first of these equations by , and the second by , and then subtracting, we eliminate both and , since . We are left with

The physical boundary conditions at require that and as . Thus, it follows from (1265) and (1266) that and as . Consequently, the series (1270) and (1271) must terminate at small positive . If is the minimum value of for which and do not both vanish then it follows from (1272) and (1273), putting and , that

(1275) | ||

(1276) |

which implies that

(1277) |

Since the boundary condition requires that the minimum value of be greater than zero, we must take

(1278) |

To investigate the convergence of the series (1270) and (1271) at large , we shall determine the ratio for large . In the limit of large , Equations (1273) and (1274) yield

(1279) | ||

(1280) |

since . Thus,

(1281) |

However, this is the ratio of coefficients in the series expansion of . Hence, we deduce that the series (1270) and (1271) diverge unphysically at large unless they terminate at large .

Suppose that the series (1270) and (1271) terminate with the terms and , so that . It follows from (1272) and (1273), with substituted for , that

(1282) | ||

(1283) |

These two expressions are equivalent, because . When combined with (1274) they give

(1284) |

which reduces to

(1285) |

or

(1286) |

Here, , which specifies the last term in the series, must be greater than by some non-negative integer . Thus,

(1287) |

where is the eigenvalue of . Hence, the energy eigenvalues of the hydrogen atom become

(1288) |

Given that , we can expand the above expression in to give

(1289) |

where is a positive integer. Of course, the first term in the above expression corresponds to the electron's rest mass energy. The second term corresponds to the standard non-relativistic expression for the hydrogen energy levels, with playing the role of the radial quantum number (see Section 4.6). Finally, the third term corresponds to the fine structure correction to these energy levels (see Exercise 4). Note that this correction only depends on the quantum numbers and . Now, we showed in Section 7.7 that the fine structure correction to the energy levels of the hydrogen atom is a combined effect of spin-orbit coupling and the electron's relativistic mass increase. Hence, it is evident that both of these effects are automatically taken into account in the Dirac equation.

### Contributors

- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)