Official Correct Choice:
Option A: \( n = 3, l = 2, m = 0, s = +1/2 \)
Concept: The validity of a set of quantum numbers depends on strict boundary rules connecting \( n, l \), and \( m \).
Formula: $$ l = 0 \text{ to } (n-1) $$ $$ m = -l \text{ to } +l $$
Solution: - Let's evaluate each option to see if it breaks any rules:
- Option A: \( n=3 \). Max \( l = 2 \). So \( l=2 \) is allowed. If \( l=2 \), \( m \) can be from -2 to +2. So \( m=0 \) is allowed. Spin is +1/2. This set is perfectly valid.
- Option B: \( n=3 \). Max \( l = 3-1 = 2 \). Here \( l=3 \), which violates the rule (\( l \) cannot equal \( n \)). Invalid.
- Option C: \( n=4 \), \( l=3 \) is allowed. However, if \( l=3 \), the max value of \( m \) is +3. Here \( m=4 \) is given, which violates the rule. Invalid.
- Option D: \( n=4 \), \( l=2 \) is allowed. If \( l=2 \), max \( m \) is +2. Here \( m=4 \) is given, which violates the rule. Invalid.
Why other options are incorrect: Options B, C, and D mathematically violate the restrictive boundary rules of quantum mechanics.
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