Using the Lowering Operator to Find Total Spin States
The total spin lowering operator is
First lets remind ourselves of what the individual lowering operators do.
Now we want to identify
Lets operate on this equation with
. First the RHS gives
Operating on the LHS gives
So equating the two we have
Now we can lower this state.
Lowering the LHS, we get
Lowering the RHS, gives
Therefore we have found 3 s=1 states that work together.
They are all symmetric under interchange of the two particles.
There is one state left over which is orthogonal to the three states we identified.
We have guessed that this is an
state since there is only one state and it has m=0.
We could verify this by using the