Which of the following sets of quantum numbers for an electron in a hydrogen atom is valid?

• A. n = 2, l = 1, m_l = 0, m_s = +1/2
• B. n = 3, l = 3, m_l = 2, m_s = +1/2
• C. n = 1, l = 1, m_l = 0, m_s = -1/2
• D. n = 2, l = 0, m_l = 2, m_s = -1/2

Answer: (A)

Verified step by step guidance

1. Recall the allowed ranges for each quantum number in a hydrogen atom: the principal quantum number \(n\) must be a positive integer (\(n = 1, 2, 3, \ldots\)), the azimuthal quantum number \(l\) can take integer values from \$0\( up to \)n-1\(, the magnetic quantum number \)m_l\( ranges from \)-l\( to \)+l\( in integer steps, and the spin quantum number \)m_s\( can be either \)+\frac{1}{2}\( or \)-\frac{1}{2}$.
2. Check the first set: \(n = 2\), \(l = 1\), \(m_l = 0\), \(m_s = +\frac{1}{2}\). Since \(l\) must be less than \(n\), and \(m_l\) must be between \(-l\) and \(+l\), verify if these conditions hold.
3. Check the second set: \(n = 3\), \(l = 3\), \(m_l = 2\), \(m_s = +\frac{1}{2}\). Verify if \(l\) is less than \(n\) and if \(m_l\) is within the allowed range for \(l\).
4. Check the third set: \(n = 1\), \(l = 1\), \(m_l = 0\), \(m_s = -\frac{1}{2}\). Confirm if \(l\) is less than \(n\) and if \(m_l\) is valid.
5. Check the fourth set: \(n = 2\), \(l = 0\), \(m_l = 2\), \(m_s = -\frac{1}{2}\). Verify if \(m_l\) lies between \(-l\) and \(+l\) for the given \(l\).