Final answer:
The value of the angular momentum quantum number l when n is not a multiple of 3 is not given in the provided options. The correct answer is that l can be any integer less than n. Examples are provided for specific values of n with the corresponding allowed values for l and m_l.
Step-by-step explanation:
The question posed by the student pertains to quantum numbers and their allowed values, particularly focusing on the angular momentum quantum number l in relation to the principal quantum number n. According to quantum mechanics, for a given principal quantum number n, the possible values of the angular momentum quantum number l range from 0 up to n-1. Therefore, when n is not a multiple of 3, or for any other integers, l can be any integer such that l < n. For example, if n = 4, then the allowed values for l are 0, 1, 2, and 3.
So, when the student asks what the value of l is when n is not a multiple of 3, none of the given options (l=0, l=n, l=3n, l=an) are correct by default. The value of l depends on the specific value of n and should be less than n.
To answer the other parts:
- For n = 4, the allowed values of l are 0, 1, 2, 3.
- For n = 9, there are 9 possible values of l (0 through 8).
- For l = 3, the allowed values of ml (the magnetic quantum number) are -3, -2, -1, 0, 1, 2, 3.
Lastly, it's important to remember that in any set of quantum numbers, the quantum number l must be a non-negative integer that is less than n, and the magnetic quantum number ml must have values ranging from -l to +l, including zero.