Which of the following sets of quantum number is not possible

Options

A. `n =4,l = 1,m = 0,s = +1//2`
B. `n =4,l = 3,m = -3,s = -1//2`
C. `n =4,l = -1,m = +2,s = _1//2`
D. `n =4,l = 1,m = 0,s = -1//2`

Answer

To determine which set of quantum numbers is not possible, we need to analyze each set based on the rules governing quantum numbers. The four quantum numbers are:

1. Principal quantum number (n)
2. Azimuthal quantum number (l)
3. Magnetic quantum number (m)
4. Spin quantum number (s)

### Step-by-Step Solution:

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

2. **Analyzing Each Set of Quantum Numbers:**
- **Set 1:** (n = 4, l = 1, m = 0, s = +1/2)
- n = 4 (valid)
- l = 1 (valid since l can be 0 to 3)
- m = 0 (valid since m can be -1, 0, +1)
- s = +1/2 (valid)
- **Conclusion:** This set is possible.

- **Set 2:** (n = 4, l = 3, m = -3, s = -1/2)
- n = 4 (valid)
- l = 3 (valid since l can be 0 to 3)
- m = -3 (valid since m can be -3, -2, -1, 0, +1, +2, +3)
- s = -1/2 (valid)
- **Conclusion:** This set is possible.

- **Set 3:** (n = 4, l = -1, m = +2, s = 1/2)
- n = 4 (valid)
- l = -1 (invalid since l must be between 0 and 3)
- m = +2 (cannot be determined since l is invalid)
- s = 1/2 (valid, but irrelevant due to invalid l)
- **Conclusion:** This set is not possible.

- **Set 4:** (n = 4, l = 1, m = 0, s = -1/2)
- n = 4 (valid)
- l = 1 (valid since l can be 0 to 3)
- m = 0 (valid since m can be -1, 0, +1)
- s = -1/2 (valid)
- **Conclusion:** This set is possible.

3. **Final Answer:**
- The set of quantum numbers that is not possible is **(n = 4, l = -1, m = +2, s = 1/2)**.