Which of the following sets of quantum number is //are not perrmitted ?

Options

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

Answer

To determine which sets of quantum numbers are not permitted, we need to understand the rules governing quantum numbers:

1. **Principal Quantum Number (n)**: This number can take any positive integer value (n = 1, 2, 3,...).
2. **Azimuthal Quantum Number (l)**: This number can take integer values from 0 to (n-1). Therefore, for a given n, l must be less than n.
3. **Magnetic Quantum Number (m)**: This number can take integer values from -l to +l, including 0.
4. **Spin Quantum Number (s)**: This number can be either +1/2 or -1/2.

Now, let's analyze the given sets of quantum numbers one by one:

### Option 1: n = 3, l = 3, m = 1, s = +1/2
- **n = 3**: Valid
- **l = 3**: Not valid because l must be less than n (l can be 0, 1, or 2 for n = 3).
- **m = 1**: Valid (but irrelevant since l is invalid).
- **s = +1/2**: Valid.

**Conclusion**: This set is **not permitted** due to the invalid l value.

### Option 2: n = 3, l = 2, m = 2, s = -1/2
- **n = 3**: Valid
- **l = 2**: Valid (l can be 0, 1, or 2 for n = 3).
- **m = 2**: Valid (m can be -2, -1, 0, 1, 2 for l = 2).
- **s = -1/2**: Valid.

**Conclusion**: This set is **permitted**.

### Option 3: n = 3, l = 1, m = 2, s = -1/2
- **n = 3**: Valid
- **l = 1**: Valid (l can be 0, 1, or 2 for n = 3).
- **m = 2**: Not valid because m must be in the range of -l to +l (for l = 1, m can be -1, 0, or 1).
- **s = -1/2**: Valid.

**Conclusion**: This set is **not permitted** due to the invalid m value.

### Option 4: n = 3, l = 0, m = 0, s = +1/2
- **n = 3**: Valid
- **l = 0**: Valid (l can be 0, 1, or 2 for n = 3).
- **m = 0**: Valid (for l = 0, m can only be 0).
- **s = +1/2**: Valid.

**Conclusion**: This set is **permitted**.

### Final Summary:
- **Not Permitted**: Option 1 (l value is invalid) and Option 3 (m value is invalid).
- **Permitted**: Option 2 and Option 4.