Which of the following sets of quantum numbers are correct ?

Options

A. `{:(n,l,m_(1)),(1,1,+2):}`
B. `{:(n,l,m_(1)),(2,1,+1):}`
C. `{:(n,l,m_(1)),(3,2,-2):}`
D. `{:(n,l,m_(1)),(3,4,-2):}`

Explanation

To determine which sets of quantum numbers are correct, we need to analyze each set based on the rules governing quantum numbers. Quantum numbers consist of four values: principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (m), and spin quantum number (s).

### Step-by-Step Solution:

1. **Understanding Quantum Numbers**:
- The principal quantum number (n) can take positive integer values (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 take values of +1/2 or -1/2.

2. **Analyzing Each Option**:
- **Option A**: Let's say n = 1 and l = 1.
- Here, l cannot be equal to 1 when n = 1 because l must be in the range of 0 to (n-1). Thus, this option is **incorrect**.

- **Option B**: Let's assume n = 2, l = 1, and m = 0.
- Here, n = 2 is valid, l = 1 is valid (since l can be 0 or 1 when n = 2), and m = 0 is valid (since m can range from -l to +l, which is -1 to +1). Thus, this option is **correct**.

- **Option C**: Let's assume n = 3, l = 2, and m = -1.
- Here, n = 3 is valid, l = 2 is valid (since l can be 0, 1, or 2 when n = 3), and m = -1 is valid (since m can range from -2 to +2). Thus, this option is **correct**.

- **Option D**: Let's assume n = 3 and l = 3.
- Here, l cannot be equal to 3 when n = 3 because l must be in the range of 0 to (n-1), which is 0 to 2. Thus, this option is **incorrect**.

3. **Conclusion**:
- The correct options based on the analysis are **Option B** and **Option C**.

### Final Answer:
The correct sets of quantum numbers are **Option B and Option C**.