We can also look at the **eigenfunctions of the momentum operator**.

The **eigenstates** are

with allowed to be positive or negative.

These solutions do not go to zero at infinity so they are not normalizable to one particle.

This is a common problem for this type of state.

We will use a
**different type of normalization for the momentum eigenstates**
(and the position eigenstates).

Instead of the Kronecker delta, we use the Dirac delta function. The momentum eigenstates have a continuous range of eigenvalues so that they cannot be indexed like the energy eigenstates of a bound system. This means the Kronecker delta could not work anyway.

These are the **momentum eigenstates**

For a **free particle Hamiltonian**,
both momentum and parity commute with
.
So we can make simultaneous eigenfunctions.

We

So we have the

Jim Branson 2013-04-22