Question

1. which of the following combinations of quantum numbers is permissible? a. n=3, l=3, m_l=1, m_s=-½ b. n=4, l=3, m_l=4, m_s=-½ c. n=3, l=2, m_l=1, m_s=+½ d. n=1, l=2, m_l=0, m_s=-½ e. n=2, l=1, m_l=-1, m_s=0

Explanation:

• For quantum numbers: \( n \geq 1 \), \( 0 \leq l < n \), \( -l \leq m_l \leq l \), \( m_s = \pm\frac{1}{2} \).
• Option A: \( l = 3 \) but \( n = 3 \), \( l \) must be \( < n \) (invalid).
• Option B: \( m_l = 4 \) but \( l = 3 \), \( m_l \) must be \( \leq l \) (invalid).
• Option C: \( n = 3 \), \( l = 2 \) (\( 2 < 3 \)), \( m_l = 1 \) (\( -2 \leq 1 \leq 2 \)), \( m_s = +\frac{1}{2} \) (valid).
• Option D: \( l = 2 \) but \( n = 1 \), \( l \) must be \( < n \) (invalid).
• Option E: \( m_s = 0 \) (must be \( \pm\frac{1}{2} \), invalid).

Answer:

C. \( n = 3, l = 2, m_l = 1, m_s = +\frac{1}{2} \)