From the given sets of quantum numbers, that one that is inconsistent with the theory is

Options

A. `n=3, l=2, m=-3, s=+1/2`
B. `n=4, l=3, m=3, s=+1/2`
C. `n=2, l=1, m=0, s=-1/2`
D. `n=4, l=3, m=2, s=+1/2`

Answer

To determine which set of quantum numbers is inconsistent with the theory, we will analyze each option based on the rules of quantum mechanics.

### Step-by-step Solution:

1. **Understand Quantum Numbers**:
- The principal quantum number (n) can take positive integer values (1, 2, 3, ...).
- The azimuthal quantum number (l) can take values from 0 to (n-1).
- The magnetic quantum number (m_l) can take values from -l to +l.
- The spin quantum number (s) can be either +1/2 or -1/2.

2. **Evaluate Option 1**:
- Given: n = 3, l = 2, m_l = -3, s = +1/2.
- For n = 3, l can be 0, 1, or 2 (valid).
- For l = 2, m_l can be -2, -1, 0, +1, or +2 (invalid since m_l = -3 is not allowed).
- **Conclusion**: This option is inconsistent.

3. **Evaluate Option 2**:
- Given: n = 4, l = 3, m_l = 3, s = +1/2.
- For n = 4, l can be 0, 1, 2, or 3 (valid).
- For l = 3, m_l can be -3, -2, -1, 0, +1, +2, or +3 (valid).
- **Conclusion**: This option is consistent.

4. **Evaluate Option 3**:
- Given: n = 2, l = 1, m_l = 0, s = -1/2.
- For n = 2, l can be 0 or 1 (valid).
- For l = 1, m_l can be -1, 0, or +1 (valid).
- **Conclusion**: This option is consistent.

5. **Evaluate Option 4**:
- Given: n = 4, l = 3, m_l = 2, s = unspecified.
- For n = 4, l can be 0, 1, 2, or 3 (valid).
- For l = 3, m_l can be -3, -2, -1, 0, +1, +2, or +3 (valid).
- **Conclusion**: This option is consistent.

### Final Conclusion:
The only inconsistent set of quantum numbers is from **Option 1** (n = 3, l = 2, m_l = -3, s = +1/2).

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