Motion in Central Field

(4.110) |

Adopting the Schrödinger representation, we can write . Hence,

(4.111) |

When written in spherical coordinates, the previous equation becomes [92]

(4.112) |

Comparing this equation with Equation (4.85), we find that

Now, we know that the three components of angular momentum commute with . (See Section 4.1.) We also know, from Equations (4.80)-(4.82), that , , and take the form of partial derivative operators involving only angular coordinates, when written in terms of spherical coordinates using the Schrödinger representation. It follows from Equation (4.113) that all three components of the angular momentum commute with the Hamiltonian: that is,

(4.114) |

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

(4.115) |

According to Section 3.3, 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

(4.116) |

where is a number (with the dimensions of energy). Because 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 the previous section.) Substituting Equation (4.117) into Equation (4.113), and making use of Equation (4.105), we obtain

This is a