This is College Physics Answers with Shaun Dychko. We are told that the principal quantum number for an electron around an atom is 4 and we are asked what are the possible values of the angular momentum projection quantum number, <i>m l</i>. Now, we don't know what <i>l</i> is because we are not told but we can choose a maximum <i>l</i> such that we get the maximum number of possibilities for <i>m l</i> because <i>m l</i> is possibilities are constrained by whatever the value of <i>l</i> is. And so the maximum possible <i>l</i> is 3 because it follows the sequence starting from 0, 1 to up to a maximum of principal quantum number minus 1 so that is 4 minus 1. So the maximum possible angular momentum quantum number is 3. So with that being 3, the angular momentum projection quantum number can start from negative 3 which is negative <i>l</i> and increasing by 1's to negative 2, negative 1, 0, 1, 2 or a maximum of <i>l</i> so a maximum of 3. So these are the possible values for <i>m l</i> given a principal quantum number of 4.
What are the possible values of $m_l$ for an electron in the $n = 4$ state?
$m_l = -3, -2, -1, 0, 1, 2, 3$