Example case: pairing Hamiltonian

Another example Using the \( (1/2)^4 \) single-particle space, resulting in eight single-particle states

Index \( n \) \( l \) \( s \) \( m_s \)
1 0 0 1/2 -1/2
2 0 0 1/2 1/2
3 1 0 1/2 -1/2
4 1 0 1/2 1/2
5 2 0 1/2 -1/2
6 2 0 1/2 1/2
7 3 0 1/2 -1/2
8 3 0 1/2 1/2

and then taking only 4-particle, \( M=0 \) states that have no `broken pairs', there are six basis Slater determinants:

  • \( | 1, 2 , 3 , 4 \rangle , \)
  • \( | 1 , 2 , 5 , 6 \rangle , \)
  • \( | 1 , 2 , 7 , 8 \rangle , \)
  • \( | 3 , 4 , 5 , 6 \rangle , \)
  • \( | 3 , 4 , 7 , 8 \rangle , \)
  • \( | 5 , 6 , 7 , 8 \rangle \)