Question

which of the following sets of quantum numbers are not allowed. explain your answer.

1. n = 4, l = 3, m_l = -4, m_s = +1/2
2. n = 2, l = 3, m_l = 0, m_s = +1/2
3. n = 3, l = 2, m_l = -1, m_s = +1/2

3 only
1 only
2 only
2 and 3
1 and 2

Explanation:

• For set 1:
• The principal quantum number \(n = 4\). The angular - momentum quantum number \(l\) can range from \(0\) to \(n - 1\), so \(l=3\) is allowed (\(0\leqslant l\leqslant3\)).
• The magnetic quantum number \(m_l\) should satisfy \(-l\leqslant m_l\leqslant l\). When \(l = 3\), \(m_l\) can be \(- 3,-2,-1,0,1,2,3\). Here \(m_l=-4\) is not allowed.
• For set 2:
• The principal quantum number \(n = 2\). The angular - momentum quantum number \(l\) should satisfy \(0\leqslant l\leqslant n - 1\). So \(l\) can be \(0\) or \(1\). Here \(l = 3\) is not allowed.
• For set 3:
• The principal quantum number \(n = 3\). The angular - momentum quantum number \(l = 2\) (\(0\leqslant l\leqslant2\) is allowed).
• The magnetic quantum number \(m_l=-1\) satisfies \(-l=- 2\leqslant m_l=-1\leqslant l = 2\). The spin quantum number \(m_s=+\frac{1}{2}\) is allowed (\(m_s=\pm\frac{1}{2}\)).

Answer:

1 and 2