| Index | \( n \) | \( l \) | \( 2j \) | \( 2m_j \) |
| 1 | 1 | 0 | 1 | -1 |
| 2 | 1 | 0 | 1 | 1 |
| 3 | 0 | 2 | 3 | -3 |
| 4 | 0 | 2 | 3 | -1 |
| 5 | 0 | 2 | 3 | 1 |
| 6 | 0 | 2 | 3 | 3 |
| 7 | 0 | 2 | 5 | -5 |
| 8 | 0 | 2 | 5 | -3 |
| 9 | 0 | 2 | 5 | -1 |
| 10 | 0 | 2 | 5 | 1 |
| 11 | 0 | 2 | 5 | 3 |
| 12 | 0 | 2 | 5 | 5 |
This represents the \( 1s_{1/2}0d_{3/2}0d_{5/2} \) valence space, or just the \( sd \)-space. There are twelve single-particle states, labeled by an overall index, and which have associated quantum numbers the number of radial nodes, the orbital angular momentum \( l \), and the angular momentum \( j \) and third component \( j_z \). To keep everything as integers, we could store \( 2 \times j \) and \( 2 \times j_z \).