Sure, let's analyze each set of quantum numbers and determine their validity.
Quantum numbers in an atom are characterized by four different numbers:
1. [tex]\( n \)[/tex] (Principal quantum number) - Indicates the main energy level and must be a positive integer ([tex]\( n > 0 \)[/tex]).
2. [tex]\( l \)[/tex] (Angular momentum quantum number) - Must be an integer in the range [tex]\( 0 \leq l < n \)[/tex].
3. [tex]\( m_l \)[/tex] (Magnetic quantum number) - Must be an integer in the range [tex]\( -l \leq m_l \leq l \)[/tex].
4. [tex]\( m_s \)[/tex] (Spin quantum number) - Must be either [tex]\( -\frac{1}{2} \)[/tex] or [tex]\( +\frac{1}{2} \)[/tex].
Now, let's examine each set:
Set (A) [tex]\( (1,0,0,-\frac{1}{2}) \)[/tex]:
- [tex]\( n = 1 \)[/tex] (positive integer, valid)
- [tex]\( l = 0 \)[/tex] (falls in range [tex]\( 0 \leq l < 1 \)[/tex], valid)
- [tex]\( m_l = 0 \)[/tex] (falls in range [tex]\( -0 \leq 0 \leq 0 \)[/tex], valid)
- [tex]\( m_s = -\frac{1}{2} \)[/tex] (valid value)
- Result: All parameters are valid for set (A).
Set (B) [tex]\( (3,0,0,+\frac{1}{2}) \)[/tex]:
- [tex]\( n = 3 \)[/tex] (positive integer, valid)
- [tex]\( l = 0 \)[/tex] (falls in range [tex]\( 0 \leq l < 3 \)[/tex], valid)
- [tex]\( m_l = 0 \)[/tex] (falls in range [tex]\( -0 \leq 0 \leq 0 \)[/tex], valid)
- [tex]\( m_s = +\frac{1}{2} \)[/tex] (valid value)
- Result: All parameters are valid for set (B).
Set (C) [tex]\( (2,1,1,+\frac{1}{2}) \)[/tex]:
- [tex]\( n = 2 \)[/tex] (positive integer, valid)
- [tex]\( l = 1 \)[/tex] (falls in range [tex]\( 0 \leq l < 2 \)[/tex], valid)
- [tex]\( m_l = 1 \)[/tex] (falls in range [tex]\( -1 \leq 1 \leq 1 \)[/tex], valid)
- [tex]\( m_s = +\frac{1}{2} \)[/tex] (valid value)
- Result: All parameters are valid for set (C).
Set (D) [tex]\( (4,4,-2,+\frac{1}{2}) \)[/tex]:
- [tex]\( n = 4 \)[/tex] (positive integer, valid)
- [tex]\( l = 4 \)[/tex] (not valid, should be in the range [tex]\( 0 \leq l < 4 \)[/tex])
- [tex]\( m_l = -2 \)[/tex] (out of the permissible range for [tex]\( l \)[/tex], hence irrelevant since [tex]\( l \)[/tex] itself is invalid)
- [tex]\( m_s = +\frac{1}{2} \)[/tex] (valid value)
- Result: Set (D) is invalid.
Therefore, the unacceptable set of quantum numbers is (D) [tex]\( (4,4,-2,+\frac{1}{2}) \)[/tex].
