• View
• Edit
• History
• Print

2bChannels

Back up to 3b

Allowed two-body channels and ordering

Note that we have to fulfill the selection rules:

• Triangular condition: {$ |l - s| \le j \le |l+s| $}
• Antisymmetry of two-nucleon state: {$ (-1)^{l+s+t} = -1 $}

Two-body channels are constructed in the following order:

1. Start with j =0
2. Start with the s=0 state (there will only be one for each j)
   • with s=0 you must have l=j
   • furthermore t will be given by the antisymmetry condition l+s+t = odd.
3. Continue with the s=1 states (unless j=0, there will be three s=1 states)
   • start by keeping t from the single state created in step 2 (s=0) and continue in increasing l order.
   • finally create the state with s=1 and the other value of t
4. increase j by 1
   • If {$ j > j_\mathrm{max} $}: quit
   • Else: go back to step 2.

As an example, let us generate the j=2 states:

• Starting with s=0 will give us l=2 (to give j=2) and t=1 (from antisymmetry)
• We continue with s=1 and start with t=1 states (i.e. same t as for the s=0 state)
  • Possible values for l are given by two constraints: 1) j=2 (combined with s=1 gives l=1,2,3) and 2) t=1 (combined with s=1 gives l odd).
  • I.e. l=1,3 in increasing order
• Finally, we create the s=1 and t=0 state. I.e. l=2

So we have:

      l   s   j   t
     --  --  --  --
      2   0   2   1
      1   1   2   1
      3   1   2   1
      2   1   2   0