Another, slightly more challenging space is the \( (1/2)^4 \) space, that is, with eight single-particle states we have
| 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 |
For \( N=2 \) there are 16 states with \( M=0 \); for \( N=3 \) there are 24 states with \( M=1/2 \), and for \( N=4 \) there are 36 states with \( M=0 \).