The One-Particle Central-Force Problem

Before studying the hydrogen atom, we shall consider the more general problem of a single particle moving under a central force. The results of this section will apply to any central-force problem, for example, the hydrogen atom (Section 6.5), the isotropic three-dimensional harmonic oscillator (Problem 6.1). 

A central force is one derived from a potential-energy function that is spherically symmetric, which means that it is a function only of the distance of the particle from the origin V = V r). The relation between force and potential energy is given by (5.32) as 

Now we consider the quantum mechanics of a single particle subject to a central force. The Hamiltonian operator is 

Looking back to (5.68), which gives the operator for the square of the magnitude of the orbital angular momentum of a single particle, L, we see that 

In classical mechanics a particle subject to a central force has its angular momentum conserved (Section 5.3). In quantum mechanics we might ask whether we can have states with definite values for both the energy and the angular momentum. To have the set of eigenfunctions of H also be eigenfunctions of V-, the commutator [H, D] must vanish. We have 

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