it is convenient to transform to polar coordinates. Let

(1219) |

and

It is easily demonstrated that

which implies that in the Schrödinger representation

Now, by symmetry, an energy eigenstate in a central field is a simultaneous eigenstate of the total angular momentum

(1223) |

Furthermore, we know from general principles that the eigenvalues of are , where is a positive half-integer (since , where is the standard non-negative integer quantum number associated with orbital angular momentum.)

It follows from Equation (1192) that

(1224) |

However, because is an angular momentum, its components satisfy the standard commutation relations

(1225) |

Thus, we obtain

(1226) |

However, , so

Further application of (1192) yields

(1228) | ||

(1229) |

However, it is easily demonstrated from the fundamental commutation relations between position and momentum operators that

(1230) |

Thus,

(1231) |

which implies that

(1232) |

Now,

(1233) |

Finally, since commutes with and , but anti-commutes with the components of

where

If we repeat the above analysis, starting at Equation (1228), but substituting for , and making use of the easily demonstrated result

(1236) |

we find that

(1237) |

Now,
commutes with
, as well as the components of
**
**
and
. Hence,

Moreover, commutes with the components of , and can easily be shown to commute with all components of

Hence, Equations (1218), (1234), (1238), and (1239) imply that

(1240) |

In other words, an eigenstate of the Hamiltonian is a simultaneous eigenstate of . Now,

(1241) |

where use has been made of Equation (1227), as well as . It follows that the eigenvalues of are . Thus, the eigenvalues of can be written , where is a non-zero integer.

Equation (1192) implies that

where use has been made of (1220) and (1235).

It is helpful to define the new operator , where

Moreover, it is evident that

Hence,

(1245) |

where use has been made of (1117). It follows that

We have already seen that commutes with

(1247) |

Equation (1192) gives

(1248) |

where use has been made of the fundamental commutation relations for position and momentum operators. However, and

(1249) |

Equation (1221) then yields

(1250) |

Equation (1242) implies that

(1251) |

Making use of Equations (1238), (1243), (1244), and (1246), we get

(1252) |

Hence, the Hamiltonian (1218) becomes

(1253) |

Now, we wish to solve the energy eigenvalue problem

(1254) |

where is the energy eigenvalue. However, we have already shown that an eigenstate of the Hamiltonian is a simultaneous eigenstate of the operator belonging to the eigenvalue , where is a non-zero integer. Hence, the eigenvalue problem reduces to

(1255) |

which only involves the radial coordinate . It is easily demonstrated that anti-commutes with . Hence, given that takes the form (1125), and that , we can represent as the matrix

(1256) |

Thus, writing in the spinor form

(1257) |

and making use of (1222), the energy eigenvalue problem for an electron in a central field reduces to the following two coupled radial differential equations: