# 4.5: Motion in Central Field

Consider a particle of mass \(M\) moving in a spherically symmetric potential. The Hamiltonian takes the form

\[H = \frac{{p}^2}{2\,M} + V(r). \tag{387}\]

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 *Sturm-Liouville equation* for the function . We know, from the general properties of this type of equation, that if is required to be well-behaved at and as then solutions only exist for a discrete set of values of . These are the energy eigenvalues. In general, the energy eigenvalues depend on the quantum number , but are independent of the quantum number .

### Contributors

- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)