For the principal quantum number n = 4, how many possible combinations are there for the values of the angular momentum quantum number l and the magnetic quantum number ml?

• A. 12
• B. 16
• C. 20
• D. 10

Answer: (C)

Verified step by step guidance

1. Recall that the principal quantum number \(n\) determines the possible values of the angular momentum quantum number \(l\), which range from \(0\) to \(n-1\). For \(n=4\), \(l\) can be \$0, 1, 2,$ or \(3\).
2. For each value of \(l\), the magnetic quantum number \(m_l\) can take integer values from \(-l\) to \(+l\), inclusive. This means the number of possible \(m_l\) values for each \(l\) is \(2l + 1\).
3. Calculate the number of \(m_l\) values for each \(l\): - For \(l=0\), number of \(m_l\) values = \(2(0) + 1 = 1\) - For \(l=1\), number of \(m_l\) values = \(2(1) + 1 = 3\) - For \(l=2\), number of \(m_l\) values = \(2(2) + 1 = 5\) - For \(l=3\), number of \(m_l\) values = \(2(3) + 1 = 7\).
4. Add all the possible combinations of \(l\) and \(m_l\) values together: \(1 + 3 + 5 + 7\).
5. The sum gives the total number of possible \((l, m_l)\) combinations for \(n=4\). This total represents all the unique pairs of angular momentum and magnetic quantum numbers for that principal quantum number.