The azimuthal quantum number `l` of an orbital is `3`. What are the possible value of `m`?

Answer

To determine the possible values of the magnetic quantum number \( m \) when the azimuthal quantum number \( l \) is given as \( 3 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Quantum Numbers**: The azimuthal quantum number \( l \) determines the shape of the orbital and can take on integer values from \( 0 \) to \( n-1 \), where \( n \) is the principal quantum number. The magnetic quantum number \( m \) describes the orientation of the orbital and can take values from \( -l \) to \( +l \).

2. **Identify the Range of \( m \)**: For a given \( l \), the possible values of \( m \) are:
\[ m = -l, -l + 1, -l + 2, \ldots, 0, \ldots, l - 1, l \]
This means \( m \) can take on any integer value between \( -l \) and \( +l \).

3. **Substituting the Value of \( l \)**: Here, \( l = 3 \). Therefore, the possible values of \( m \) will be:
\[ m = -3, -2, -1, 0, 1, 2, 3 \]

4. **Counting the Values**: The total number of possible values for \( m \) can be calculated using the formula:
\[ \text{Number of values} = 2l + 1 \]
Substituting \( l = 3 \):
\[ \text{Number of values} = 2 \times 3 + 1 = 7 \]

5. **List the Values**: The complete set of possible values for \( m \) when \( l = 3 \) is:
\[ m = -3, -2, -1, 0, 1, 2, 3 \]

### Conclusion:
The possible values of \( m \) when the azimuthal quantum number \( l \) is \( 3 \) are:
\[ -3, -2, -1, 0, 1, 2, 3 \]