##

Rotational Symmetry

If the potential only depends on the distance between two particles,

the Hamiltonian has **Rotational Symmetry**.
This is true for the coulomb (and gravitational) potential as well as many others.
We know from classical mechanics that these are important problems.
If a the Hamiltonian has rotational symmetry, we can
show that the Angular Momentum operators commute with the Hamiltonian.

We therefore expect each component of
to be conserved.
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.

The
components of angular momentum do not commute
with each other

but the **square of the angular momentum commutes** with any of the components
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
.

This leads to a great simplification of the 3D problem.
It is possible to **separate the Schrödinger equation** since
and
appear separately.
Write the solution as a product

where
labels the eigenvalue of the
operator
and
labels the eigenvalue of the
operator.
Since
does not appear in the Schrödinger equation,
we only label the radial solutions with the energy and the eigenvalues of
.
We get the three equations.

By assuming the eigenvalues of
have the form
,
we have **anticipated the solution but not constrained it**, since the units of
angular momentum are those of
and since we expect
to have positive
eigenvalues.

The assumption that the eigenvalues of
are some (dimensionless) number times
does not constrain our solutions at all.
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.

We have grouped the term due to angular momentum with the potential.
It is often called a pseudo-potential.
For
, it is like a repulsive potential.

Jim Branson
2013-04-22