Question

an electron cannot have the quantum numbers n = _______, l = _____, ml = _______.
2, 1, -1
3, 1, -1
1, 1, 1
3, 2, 1
2, 0, 0

Explanation:

Step1: Check the relationship between \(n\) and \(l\)

The rule for quantum numbers is \(0\leq l\leq n - 1\).
For \(n = 1\), \(l\) can only be \(0\) (since \(l\) must satisfy \(l\leq n-1\), when \(n = 1\), \(n - 1=0\)).

Step2: Analyze each option

• Option \(n = 2\), \(l = 1\): Since \(n=2\), \(n - 1=1\), so \(l = 1\) is valid. And \(m_l=-1\) (where \(m_l\) ranges from \(-l\) to \(+l\), here \(l = 1\), so \(m_l=-1,0,1\)) is valid.
• Option \(n = 3\), \(l = 1\): Since \(n = 3\), \(n-1 = 2\), \(l = 1\) (because \(1\leq2\)) is valid. And \(m_l=-1\) (since \(l = 1\), \(m_l=-1,0,1\)) is valid.
• Option \(n = 1\), \(l = 1\): Since \(n = 1\), \(n - 1=0\), but \(l = 1>0\), so this set of quantum numbers is invalid.
• Option \(n = 3\), \(l = 2\): Since \(n = 3\), \(n - 1=2\), \(l = 2\) is valid. And \(m_l = 1\) (since \(l = 2\), \(m_l=-2,-1,0,1,2\)) is valid.
• Option \(n = 2\), \(l = 0\): Since \(n = 2\), \(n - 1=1\), \(l = 0\) (because \(0\leq1\)) is valid. And \(m_l = 0\) (since \(l = 0\), \(m_l = 0\)) is valid.

Answer:

\(1,1,1\)