Which one of the following sets of quantum numbers is possible ?

Options

A. `n = 3, l = 3, m_l = -3, m_s = +1//2`
B. `n = 2, l = 1, m_l = 2, m_s = -1//2`
C. `n = 2, l = 0, m_l = 0, m_s = 0`
D. `n = 3, l = 2, m_l = 3, m_s = -1//2`

Explanation

To determine which set of quantum numbers is possible, we need to understand the rules governing the quantum numbers. The quantum numbers are:

1. Principal quantum number (n)
2. Azimuthal quantum number (l)
3. Magnetic quantum number (m_l)
4. Spin quantum number (m_s)

Let's analyze the quantum numbers step by step.

### Step 1: Principal Quantum Number (n)
The principal quantum number (n) can take any positive integer value: n = 1, 2, 3, ...

### Step 2: Azimuthal Quantum Number (l)
The azimuthal quantum number (l) can take integer values from 0 to (n-1).
- For n = 3, the possible values of l are 0, 1, and 2.

### Step 3: Magnetic Quantum Number (m_l)
The magnetic quantum number (m_l) can take integer values from -l to +l.
- If l = 0, then m_l = 0.
- If l = 1, then m_l can be -1, 0, or +1.
- If l = 2, then m_l can be -2, -1, 0, +1, or +2.

### Step 4: Spin Quantum Number (m_s)
The spin quantum number (m_s) can take values of +1/2 or -1/2. It cannot be 0.

### Analyzing the Options
Now, let's analyze the given options based on the above rules:

1. **Option 1: n = 3, l = 3, m_l = 2, m_s = +1/2**
- Here, l = 3 is not possible since l must be less than n (l < n). This option is invalid.

2. **Option 2: n = 3, l = 2, m_l = 2, m_s = +1/2**
- Here, l = 2 is valid. However, m_l = 2 is not valid because for l = 2, m_l can be -2, -1, 0, +1, or +2. This option is invalid.

3. **Option 3: n = 2, l = 1, m_l = 0, m_s = 0**
- Here, l = 1 is valid. m_l = 0 is also valid. However, m_s = 0 is not valid since m_s can only be +1/2 or -1/2. This option is invalid.

4. **Option 4: n = 2, l = 0, m_l = 0, m_s = +1/2**
- Here, l = 0 is valid. m_l = 0 is valid. m_s = +1/2 is also valid. This option is valid.

### Conclusion
Out of the given options, only **Option 4: n = 2, l = 0, m_l = 0, m_s = +1/2** is a possible set of quantum numbers.

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