Which combinations of quantum number `n,l,m,s` for the electron in an atom does not provide a permissible solution of the wave equation ?

Options

A. `3,2,-2 ,(1)/(2)`
B. `3,3,1,-(1)/(2)`
C. `3,2,1(1)/(2)`
D. `3,1,1,-(1)/(2)`

Answer

To determine which combinations of quantum numbers \( n, l, m, s \) for an electron in an atom do not provide a permissible solution of the wave equation, we need to understand the rules governing these quantum numbers.

### Step-by-Step Solution:

1. **Understanding Quantum Numbers**:
- **Principal Quantum Number (\( n \))**: This number can take positive integer values \( n = 1, 2, 3, \ldots \).
- **Azimuthal Quantum Number (\( l \))**: This number can take values from \( 0 \) to \( n-1 \). Therefore, for a given \( n \), \( l \) must satisfy \( 0 \leq l < n \).
- **Magnetic Quantum Number (\( m \))**: This number can take values from \( -l \) to \( +l \). Thus, for a given \( l \), \( m \) must satisfy \( -l \leq m \leq l \).
- **Spin Quantum Number (\( s \))**: This number can take values of \( +\frac{1}{2} \) or \( -\frac{1}{2} \).

2. **Checking the Options**:
- **Option 1**: \( n = 3, l = 2, m = -2, s = \frac{1}{2} \)
- Here, \( l = 2 \) is valid since \( 0 \leq 2 < 3 \).
- \( m = -2 \) is also valid since \( -2 \leq -2 \leq 2 \).
- \( s = \frac{1}{2} \) is valid.
- **This option is permissible.**

- **Option 2**: \( n = 3, l = 3, m = 1, s = -\frac{1}{2} \)
- Here, \( l = 3 \) is **not valid** since \( l \) must be less than \( n \) (i.e., \( 0 \leq l < 3 \)).
- **This option is not permissible.**

- **Option 3**: \( n = 3, l = 2, m = 1, s = \frac{1}{2} \)
- Here, \( l = 2 \) is valid since \( 0 \leq 2 < 3 \).
- \( m = 1 \) is valid since \( -2 \leq 1 \leq 2 \).
- \( s = \frac{1}{2} \) is valid.
- **This option is permissible.**

- **Option 4**: \( n = 3, l = 1, m = 1, s = -\frac{1}{2} \)
- Here, \( l = 1 \) is valid since \( 0 \leq 1 < 3 \).
- \( m = 1 \) is valid since \( -1 \leq 1 \leq 1 \).
- \( s = -\frac{1}{2} \) is valid.
- **This option is permissible.**

3. **Conclusion**:
- The combination of quantum numbers that does not provide a permissible solution of the wave equation is from **Option 2** where \( n = 3 \) and \( l = 3 \).

### Final Answer:
**Option 2**: \( n = 3, l = 3, m = 1, s = -\frac{1}{2} \) is not permissible.