Question

What values of `m_l` are permitted for an electron with `l=3` ?

Answer 1

Seven.

Explanation:

The number of values that the magnetic quantum number, `m_l`, can take tells you the number of orbitals present in an energy subshell described by the angular momentum quantum number `l`.

The values that the magnetic quantum number can take depend on the value of the angular momentum quantum number as described by the relation

`m_l = {-l, -(l-1), ..., -1, 0, 1, ..., (l-1), l}`

In your case, you have

`l = 3`

This value of the angular momentum quantum number describes the `f` subshell. Consequently, you can say that magnetic quantum number can take the following values

`m_l = {-3, -2, -1, 0, 1, 2, 3}`

The fact that the magnetic quantum number can take `7` possible values for an `f` subshell tells you that this subshell holds a total of `7` orbitals, each described by a value of the magnetic quantum number.

`underbrace(l = 3)_ (color(blue)("the f subshell")) => m_l = underbrace( {-3, -2, -1, 0, 1, 2, 3} )_ (color(blue)("7 values for m"_l quad = quad "7 distinct f orbitals"))`