A.40 Deuteron wave function

This addendum examines the form of the wave function of the deuteron. It assumes that the deuteron can be described as a two particle system; a proton and a neutron. In reality, both the proton and the neutron consist of three quarks. So the deuteron is really a six particle system. That will be ignored here.

Then the deuteron wave function is a function of the positions and
spin angular momenta of the proton and neutron. That however can be
simplified considerably. First of all, it helps if the center of
gravity of the deuteron is taken as origin of the coordinate system.
In that coordinate system, the individual positions of proton and
neutron are no longer important. The only quantity that is important
is the position vector going from neutron to proton,
{A.5}:

That represents the relative position of the proton relative to the neutron.

Consider now the spin angular momenta of proton and neutron. The two
have spin angular momenta of the same magnitude. The corresponding
quantum number, called the spin

for short, equals

All together it means that the deuteron wave function depends
nontrivially on both the nucleon spacing and the spin components in
the

The square magnitude of this wave function gives the probability density to find the nucleons at a given spacing

It is solidly established by experiments that the wave function of the
deuteron has net nuclear spin

The conditions for a state to have definite orbital angular momentum
were discussed in chapter 4.2. The angular dependence of
the state must be given by a spherical harmonic

As far as the combined spin angular momentum is concerned, the
possibilities were discussed in chapter 5.5.6 and in
more detail in chapter 12. First, the proton and
neutron spins can cancel each other perfectly, producing a state of
zero net spin. This state is called the singlet

state. Zero net spin has a corresponding quantum number

The second possibility is that the proton and neutron align their
spins in parallel, crudely speaking. More precisely, the combined
spin has a magnitude given by quantum number

The wave function of the deuteron can be written as a combination of
the above states of orbital and spin angular momentum. It then takes
the generic form:

The above expression for the wave function is quite generally valid for a system of two fermions. But it can be made much more specific based on the mentioned known properties of the deuteron.

The simplest is the fact that the parity of the deuteron is even.
Spherical harmonics have odd parity if

Physically, that means that the spatial wave function is symmetric
with respect to replacing

The spatial symmetry also means that the wave function is symmetric
with respect to swapping the two nucleons. That is because

The condition that the nuclear spin

The key is now that unless a state

Using the above criterion, consider which states cannot appear in the
deuteron wave function. First of all, states with

Secondly, states with

Now states with

States with

The bottom line is that the deuteron wave function can have
uncertainty in the orbital angular momentum. In particular, both
orbital angular momentum numbers