The previous section has taught us that in general the orbital
angular momentum
of strongly interacting
states is not a conserved quantum number due to the presence of
non-central forces. These forces commute
with the total spin of the system and hence we can classify the
states by the
and
quantum numbers.
For example, the deuteron is a
object, with orbital angular
momentum being a mixture of
and
.
It is therefore convenient to define states of good
and
which are linear combinations of the spatial and spin wavefunctions
with different .
The wavefunction for a given system is then a sum
over these states of different .
For the deuteron, we need
and
for each .
The
by

where is the spin wavefunction for the state. For the deuteron in the states the relevant configurations (orthonormal) are

A formula that you have seen before, but will be very useful to us is the triple integration formula

We can now compute some angular matrix elements relevant to the deuteron.
In order to make progress we rewrite the
operator in terms of
the total spin operator, ,

(4) |

where we have used the fact that is symmetric under interchange , and are the spin raisong and lowering operators. Acting on it is easy to show that

(5) |

We have used

(6) |

and the standard representation of the 's.

Similar calculations lead to

(7) |

The deuteron wavefunction
(where
is the magnetic substate)
is a linear combination of
and
components

where this relation defines the mixing angle between the orthonormal states. The wavefunction satisfies the schrodinger equation for the nucleons moving in the nuclear potential

where is the reduced mass of the N-N system and writing this form out explicitly

This represents a set of couple equations, which we can obtain by projecting with and to obtain

These equations have exactly the form we described above. If there are non-zero matrix elements between states of different , as induced by the tensor force, then the and states mix with each other as expected.

We recall that from our naive guestimates made previously,
the deuteron is a very extended object compared to
the range of the nuclear interaction. In the region outside
the potential, the coupled equations become

which have solutions of the form

and by direct substitution we find that

where the are normalization constants determined solving the equation directly including the potentials.

It is clear that solving the coupled equations is not going to be so easy. In particular, we see that there is the mixing angle that is apriori undetermined. We see that for , the Swave wavefunction is uncoupled from the D-wave admixture, but the D-wave is infinitely coupled to the S-wave component. We see that in order to solve these equations, we must compare with data, so that the solution reproduce the binding energy, magnetic moment and quadrupole moment of the deuteron.

In order for us to proceed, we therefore need to know how to relate these wavefunctions to the magnetic and quadrupole moments that are measured.