If the potential only depends on the distance between two particles,
If a the Hamiltonian has rotational symmetry, we can
show that the Angular Momentum operators commute with the Hamiltonian.
We will not be able to label our states with the quantum numbers for the three components of angular momentum. Recall that we are looking for a set of mutually commuting operators to label our energy eigenstates. We actually want two operators plus to give us three quantum numbers for states in three dimensions.
components of angular momentum do not commute
with each other
These commutators lead us to choose the mutually commuting set of operators to be , , and . We could have chosen any component, however, it is most convenient to choose given the standard definition of spherical coordinates.
The Schrödinger equation now can be rewritten with only radial derivatives and .
It is possible to separate the Schrödinger equation since and appear separately. Write the solution as a product
We get the three equations.
We will use the algebra of the angular momentum operators to help us solve the angular part of the problem in general.
For any given problem with rotational symmetry, we will need to solve a particular differential equation in one variable . This radial equation can be simplified a bit.
Jim Branson 2013-04-22