Derivation of Radial Equation

(624) |

Recall that the angular momentum vector, , is defined [see Eq. (526)]

Here, the (where all run from 1 to 3) are elements of the so-called

Thus, , , and ,

Let us calculate the value of using Eq. (627). According
to our new notation, is the same as . Thus, we
obtain

Here is the usual

(631) |

Here, we have made use of the fairly self-evident result that . We have also been careful to preserve the order of the various terms on the right-hand side of the above expression, since the and the do not necessarily commute with one another.

We now need to rearrange the order of the terms on the right-hand
side of Eq. (632). We can achieve this by making use of
the fundamental commutation relation for the and the [see Eq. (483)]:

(634) |

Here, we have made use of the fact that , since the commute with one another [see Eq. (482)]. Next,

(635) |

(636) |

(637) |

Note that if we had attempted to derive the above expression directly from Eq. (626), using standard vector identities, then we would have missed the final term on the right-hand side. This term originates from the lack of commutation between the and operators in quantum mechanics. Of course, standard vector analysis assumes that all terms commute with one another.

Equation (638) can be rearranged to give

(639) |

(640) |

(641) |

Let us now consider whether the above Hamiltonian commutes with the angular momentum operators and . Recall, from Sect. 8.3, that and are represented as differential operators which depend solely on the angular spherical polar coordinates, and , and do not contain the radial polar coordinate, . Thus, any function of , or any differential operator involving (but not and ), will automatically commute with and . Moreover, commutes both with itself, and with (see Sect. 8.2). It is, therefore, clear that the above Hamiltonian commutes with both and .

Now, according to Sect. 4.10, if two operators commute with
one another then they possess simultaneous eigenstates. We thus conclude
that *for a particle moving in a central potential the eigenstates of the
Hamiltonian are simultaneous eigenstates of and *.
Now, we have already found the simultaneous eigenstates of
and --they are the spherical harmonics,
,
discussed in Sect. 8.7. It follows that the spherical
harmonics are also eigenstates of the Hamiltonian. This observation leads
us to try the following separable form for the stationary
wavefunction:

Recall that the quantum numbers and are restricted to take certain integer values, as explained in Sect. 8.6.

Finally, making use of Eqs. (622), (642), and (645),
we obtain the following differential equation which determines the radial variation of the stationary wavefunction:

(646) |