Give all possible values of `l, m_l and m_s` for electrons when `n = 3`.

Answer

To find all possible values of the quantum numbers \( l \), \( m_l \), and \( m_s \) for electrons when \( n = 3 \), we can follow these steps:

### Step 1: Determine the values of \( l \)
The azimuthal quantum number \( l \) can take values from \( 0 \) to \( n-1 \). Therefore, for \( n = 3 \):
- Possible values of \( l \) are:
- \( l = 0 \) (s orbital)
- \( l = 1 \) (p orbital)
- \( l = 2 \) (d orbital)

### Step 2: Determine the values of \( m_l \)
The magnetic quantum number \( m_l \) can take values from \( -l \) to \( +l \), including zero. We will calculate \( m_l \) for each value of \( l \):

- For \( l = 0 \):
- \( m_l = 0 \)

- For \( l = 1 \):
- \( m_l = -1, 0, +1 \)

- For \( l = 2 \):
- \( m_l = -2, -1, 0, +1, +2 \)

### Step 3: Determine the values of \( m_s \)
The spin quantum number \( m_s \) can take values of \( +\frac{1}{2} \) or \( -\frac{1}{2} \) for each electron. Therefore, for any electron, the possible values of \( m_s \) are:
- \( m_s = +\frac{1}{2} \)
- \( m_s = -\frac{1}{2} \)

### Summary of Values
- For \( n = 3 \):
- Possible values of \( l \): \( 0, 1, 2 \)
- Possible values of \( m_l \):
- For \( l = 0 \): \( 0 \)
- For \( l = 1 \): \( -1, 0, +1 \)
- For \( l = 2 \): \( -2, -1, 0, +1, +2 \)
- Possible values of \( m_s \): \( +\frac{1}{2}, -\frac{1}{2} \)