Question

which of the following is an acceptable set of quantum numbers ?

○ n = 2, l = 0, m_l = 0, m_s = +1

● n = 2, l = 1, m_l = 2, m_s = -1/2

○ n = 0, l = 0, m_l = 0, m_s = +1/2

○ n = 1, l = 1, m_l = 0, m_s = -1/2

○ n = 3, l = 1, m_l = -1, m_s = -1/2

Explanation:

1. Recall quantum number rules:

• Principal quantum number \( n \geq 1 \) (so \( n = 0 \) is invalid).
• Azimuthal quantum number \( l \) satisfies \( 0 \leq l < n \) (e.g., \( n = 1 \) implies \( l = 0 \); \( n = 2 \) implies \( l = 0, 1 \)).
• Magnetic quantum number \( m_l \) satisfies \( -l \leq m_l \leq l \).
• Spin quantum number \( m_s = \pm \frac{1}{2} \) (so \( m_s = +1 \) is invalid).

1. Analyze each option:

• \( n = 2, l = 0, m_l = 0, m_s = +1 \): \( m_s \) is invalid (must be \( \pm \frac{1}{2} \)).
• \( n = 2, l = 1, m_l = 2 \): \( m_l \) must satisfy \( -1 \leq m_l \leq 1 \) (invalid, since \( l = 1 \)).
• \( n = 0, l = 0, m_l = 0, m_s = +\frac{1}{2} \): \( n = 0 \) is invalid (\( n \geq 1 \)).
• \( n = 1, l = 1, m_l = 0, m_s = -\frac{1}{2} \): \( l \) must be \( < n \) ( \( n = 1 \) implies \( l = 0 \), invalid).
• \( n = 3, l = 1, m_l = -1, m_s = -\frac{1}{2} \):
• \( n = 3 \geq 1 \), \( l = 1 < 3 \), \( m_l = -1 \) (within \( -1 \leq m_l \leq 1 \) for \( l = 1 \)), \( m_s = -\frac{1}{2} \) (valid).

Answer:

\( \boldsymbol{n = 3, l = 1, m_l = -1, m_s = -1/2} \) (the last option)