Question

Which of the following are invalid sets of quantum numbers?

`3, 4, 0, 1/2`
`2, 1, 3, 1/2`
`-2, 1, 0, -1/2`
`3, 2, 2, -1/2`
`3, 2, 0, -1/2`

Answer 1

Several rules apply here:

1. `n >= 1` in integer increments.
2. `l >= 0` in integer increments.
3. `l_max = n-1`.
4. `m_l = {-l, -l+1, . . . , -1, 0, +1, . . . , l - 1, l}`
5. `m_s = pm"1/2"`

For the format `(n,l,m_l,m_s)`:

• `(3, 4, 0, 1/2)` is invalid because `l > n`.
• `(2, 1, 3, 1/2)` is invalid because `m_l` is outside the range of `l`.
• `(-2, 1, 0, -1/2)` is invalid because `n < 1`.
• `(3, 2, 2, -1/2)` is valid because `n > 1`, `l < n`, `m_l` is within the range of `l`, and `m_s` is correctly either `+"1/2"` or `-"1/2"`.
• `(3, 2, 0, -1/2)` is valid for the same reason as the fourth choice.