# Which of the following are invalid sets of quantum numbers?

## $3 , 4 , 0 , \frac{1}{2}$ $2 , 1 , 3 , \frac{1}{2}$ $- 2 , 1 , 0 , - \frac{1}{2}$ $3 , 2 , 2 , - \frac{1}{2}$ $3 , 2 , 0 , - \frac{1}{2}$

Nov 9, 2016

Several rules apply here:

1. $n \ge 1$ in integer increments.
2. $l \ge 0$ in integer increments.
3. ${l}_{\max} = n - 1$.
4. ${m}_{l} = \left\{- l , - l + 1 , . . . , - 1 , 0 , + 1 , . . . , l - 1 , l\right\}$
5. ${m}_{s} = \pm \text{1/2}$

For the format $\left(n , l , {m}_{l} , {m}_{s}\right)$:

• $\left(3 , 4 , 0 , \frac{1}{2}\right)$ is invalid because $l > n$.
• $\left(2 , 1 , 3 , \frac{1}{2}\right)$ is invalid because ${m}_{l}$ is outside the range of $l$.
• $\left(- 2 , 1 , 0 , - \frac{1}{2}\right)$ is invalid because $n < 1$.
• $\left(3 , 2 , 2 , - \frac{1}{2}\right)$ is valid because $n > 1$, $l < n$, ${m}_{l}$ is within the range of $l$, and ${m}_{s}$ is correctly either $+ \text{1/2}$ or $- \text{1/2}$.
• $\left(3 , 2 , 0 , - \frac{1}{2}\right)$ is valid for the same reason as the fourth choice.