Which one of the following sets of quantum numbers represents an impossible arrangement ?

Options

A. `{:(n,l,m,s),(3,2,-2,1//2):}`
B. `{:(n,l,m,s),(4,0,0,1//2):}`
C. `{:(n,l,m,s),(3,2,-3,1//2):}`
D. `{:(n,l,m,s),(5,3,0,-1//2):}`

Explanation

To determine which set of quantum numbers represents an impossible arrangement, we need to understand the rules governing the quantum numbers:

1. **Principal Quantum Number (n)**: This number can take any positive integer value (1, 2, 3, ...).
2. **Azimuthal Quantum Number (l)**: This number can take values from 0 to (n-1). For example, if n=3, then l can be 0, 1, or 2.
3. **Magnetic Quantum Number (m)**: This number can take values from -l to +l. For example, if l=2, then m can be -2, -1, 0, 1, or 2.
4. **Spin Quantum Number (s)**: This number can be either -1/2 or +1/2.

Now, let’s analyze the options one by one:

### Option 1:
- n = 3
- l = 2 (valid since l can be 0, 1, or 2)
- m = 1 (valid since m can be -2, -1, 0, 1, 2)
- s = +1/2 (valid since s can be -1/2 or +1/2)

**Conclusion**: This option is valid.

### Option 2:
- n = 4
- l = 0 (valid since l can be 0 to 3)
- m = 0 (valid since m can be -0 to +0)
- s = +1/2 (valid since s can be -1/2 or +1/2)

**Conclusion**: This option is valid.

### Option 3:
- n = 3
- l = 2 (valid since l can be 0, 1, or 2)
- m = -3 (invalid since m must be between -l and +l, which means m can only be -2, -1, 0, 1, or 2)
- s = +1/2 (valid)

**Conclusion**: This option is invalid due to the value of m.

### Option 4:
- n = 5
- l = 3 (valid since l can be 0 to 4)
- m = 0 (valid since m can be -3, -2, -1, 0, 1, 2, 3)
- s = -1/2 (valid since s can be -1/2 or +1/2)

**Conclusion**: This option is valid.

### Final Answer:
The impossible arrangement of quantum numbers is **Option 3**.

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