it is convenient to transform to polar coordinates. Let

(11.129) |

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

(11.133) |

Furthermore, we know from general principles that the eigenvalues of are , where is a positive half-integer (because , where is the standard non-negative integer quantum number associated with orbital angular momentum.) (See Chapter 6.)

It follows from Equation (11.101) that

(11.134) |

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

(11.135) |

(See Section 4.1.) Thus, we obtain

(11.136) |

However, , so

Further application of Equation (11.101) yields

(11.138) | ||

(11.139) |

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

(11.140) |

(See Section 2.2.) Thus,

(11.141) |

which implies that

(11.142) |

Now,

(11.143) |

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

where

If we repeat the previous analysis, starting at Equation (11.138), but substituting for , and making use of the easily demonstrated result

(11.146) |

we find that

(11.147) |

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 of the components of

Hence, Equations (11.128), (11.144), (11.148), and (11.149) imply that

(11.150) |

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

(11.151) |

where use has been made of Equation (11.137), as well as . It follows that the eigenvalues of are . Thus, the eigenvalues of can be written , where is a non-zero integer.

Equation (11.101) implies that

where use has been made of Equations (11.130) and (11.145).

It is helpful to define the dimensionless operator , where

Moreover, it is evident that

Hence,

(11.155) |

where use has been made of Equation (11.24). It follows that

We have already seen that commutes with

(11.157) |

Because
**
**
commutes with
and
, and
, as well
as
, we obtain

(11.158) |

However, and

(11.159) |

Equation (11.131) then yields

(11.160) |

Equation (11.152) implies that

(11.161) |

Making use of Equations (11.148), (11.153), (11.154), and (11.156), we get

(11.162) |

Hence, the Hamiltonian (11.128) becomes

(11.163) |

Now, we wish to solve the energy eigenvalue problem

(11.164) |

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

(11.165) |

which only involves the radial coordinate, . It is easily demonstrated that anti-commutes with . Hence, given that takes the form (11.32), and that , we can represent as the matrix

(11.166) |

Thus, writing in the spinor form

(11.167) |

and making use of Equation (11.132), the energy eigenvalue problem for an electron in a central field reduces to the following two coupled radial differential equations: