Motion in Central Field

(387) |

Adopting Schrödinger's representation, we can write . Hence,

(388) |

When written in spherical polar coordinates, the above equation becomes

(389) |

Comparing this equation with Equation (374), we find that

Now, we know that the three components of angular momentum commute with
(see Section 4.1). We also know, from Equations (369)-(371), that
,
, and
take the
form of partial derivative operators involving only *angular* coordinates,
when written in terms of spherical polar coordinates using the Schrödinger representation. It follows from Equation (390) that all three components of the angular
momentum commute with the Hamiltonian:

(391) |

It is also easily seen that (which can be expressed as a purely angular differential operator) commutes with the Hamiltonian:

(392) |

According to Section 3.2, the previous two equations ensure that the angular momentum and its magnitude squared are both constants of the motion. This is as expected for a spherically symmetric potential.

Consider the energy eigenvalue problem

(393) |

where is a number. Since and commute with each other and the Hamiltonian, it is always possible to represent the state of the system in terms of the simultaneous eigenstates of , , and . But, we already know that the most general form for the wavefunction of a simultaneous eigenstate of and is (see previous section)

Substituting Equation (394) into Equation (390), and making use of Equation (382), we obtain

This is a