We can also look at the eigenfunctions of the momentum operator.
The eigenstates are
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
satisfying the normalization condition
For a free particle Hamiltonian,
both momentum and parity commute with
So we can make simultaneous eigenfunctions.
We cannot make eigenfunctions of all three operators since
So we have the choice of the
which are eigenfunctions of
but contain positive and negative parity components.
or we have the
states which contain two momenta but are
These are just different linear combinations of the same solutions.