Multiple Choice
For an electron with principal quantum number n = 5, how many different values of the angular momentum quantum number l are possible?
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9. Quantum Mechanics
Quantum Numbers: Angular Momentum Quantum Number
Multiple Choice
For the principal quantum number n = 3, how many possible combinations of the angular momentum quantum number l and the magnetic quantum number m_l are there?
A
15
B
6
C
9
D
12
Verified step by step guidance1
Recall that the principal quantum number \(n\) determines the possible values of the angular momentum quantum number \(l\), where \(l\) can take integer values from \$0\( up to \)n-1\(. For \)n=3\(, list all possible \)l\( values: \)l = 0, 1, 2$.
For each value of \(l\), the magnetic quantum number \(m_l\) can take integer values from \(-l\) to \(+l\), including zero. This means the number of possible \(m_l\) values for each \(l\) is \$2l + 1$.
Calculate the number of \(m_l\) values for each \(l\):
- For \(l=0\), number of \(m_l\) values = \$2(0) + 1 = 1\(
- For \)l=1\(, number of \)m_l\( values = \)2(1) + 1 = 3\(
- For \)l=2\(, number of \)m_l\( values = \)2(2) + 1 = 5$.
Add the number of \(m_l\) values for all \(l\) values to find the total number of possible \((l, m_l)\) combinations: \$1 + 3 + 5$.
The sum from the previous step gives the total number of possible combinations of \(l\) and \(m_l\) for \(n=3\). This total represents all the unique pairs of angular momentum and magnetic quantum numbers for that principal quantum number.
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