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.