Question

which of the following sets of quantum numbers describe valid orbitals? check all that apply.
□ n = 1, l = 0, m = 0
□ n = 2, l = 1, m = 3
□ n = 2, l = 2, m = 2
□ n = 3, l = 0, m = 0
□ n = 5, l = 4, m = - 3
□ n = 4, l = - 2, m = 2

Explanation:

• For quantum numbers:
• The principal quantum number \(n\) can take positive integer values (\(n = 1,2,3,\cdots\)).
• The angular - momentum quantum number \(l\) satisfies \(0\leq l\leq n - 1\).
• The magnetic quantum number \(m\) satisfies \(-l\leq m\leq l\).
• For \(n = 1,l = 0,m = 0\):
• \(n=1\) (valid as \(n\geq1\)).
• \(l = 0\) (since \(l\leq n - 1=1 - 1 = 0\)).
• \(m = 0\) (since \(-l=- 0\leq m\leq l = 0\)).
• For \(n = 2,l = 1,m = 3\):
• \(n = 2\) (valid).
• \(l = 1\) (since \(l\leq n - 1=2 - 1=1\)).
• But \(m = 3\) is invalid because for \(l = 1\), \(m\) should satisfy \(-1\leq m\leq1\).
• For \(n = 2,l = 2,m = 2\):
• \(n = 2\) (valid).
• \(l = 2\) is invalid because \(l\leq n - 1=2 - 1 = 1\).
• For \(n = 3,l = 0,m = 0\):
• \(n = 3\) (valid).
• \(l = 0\) (since \(l\leq n - 1=3 - 1 = 2\), and \(l = 0\) is within the range \(0\leq l\leq2\)).
• \(m = 0\) (since \(-l=-0\leq m\leq l = 0\)).
• For \(n = 5,l = 4,m=-3\):
• \(n = 5\) (valid).
• \(l = 4\) (since \(l\leq n - 1=5 - 1 = 4\)).
• \(m=-3\) (since for \(l = 4\), \(-4\leq m\leq4\)).
• For \(n = 4,l=-2,m = 2\):
• \(l=-2\) is invalid because \(l\geq0\).

Answer:

\(\boldsymbol{n = 1,l = 0,m = 0}\), \(\boldsymbol{n = 3,l = 0,m = 0}\), \(\boldsymbol{n = 5,l = 4,m=-3}\)