Question

which is a set of valid quantum numbers?
○ n = 4, l = 3, m = 3
○ n = -1, l = 0, m = 0
○ n = 1, l = -2, m = 0
○ n = 4, l = 4, m = 4

Explanation:

To determine valid quantum numbers, we use the rules for principal (\(n\)), azimuthal (\(l\)), and magnetic (\(m\)) quantum numbers:

1. Principal quantum number (\(n\)): Must be a positive integer (\(n = 1, 2, 3, \dots\)).
2. Azimuthal quantum number (\(l\)): Must satisfy \(0 \leq l \leq n - 1\).
3. Magnetic quantum number (\(m\)): Must satisfy \(-l \leq m \leq l\).

Analyze each option:

• Option 1: \(n = 4, l = 4, m = 4\)

For \(n = 4\), \(l\) must satisfy \(0 \leq l \leq 3\) (since \(l \leq n - 1 = 3\)). Here, \(l = 4\) violates \(l \leq n - 1\). Invalid.

• Option 2: \(n = 1, l = -2, m = 0\)

\(l\) must be non - negative (since \(l \geq 0\)). Here, \(l = -2\) is negative. Invalid.

• Option 3: \(n = -1, l = 0, m = 0\)

\(n\) must be a positive integer. Here, \(n = -1\) is negative. Invalid.

• Option 4: \(n = 4, l = 3, m = 3\)
• \(n = 4\) (positive integer, valid).
• \(l = 3\): For \(n = 4\), \(l\) ranges from \(0\) to \(3\) (since \(n - 1 = 3\)), so \(l = 3\) is valid.
• \(m = 3\): For \(l = 3\), \(m\) ranges from \(-3\) to \(3\), so \(m = 3\) is valid.

Answer:

\(n = 4, l = 3, m = 3\) (the last option in the given choices)