객관식
Which of the following correctly describes the allowed values for the four quantum numbers n, l, m_l, and m_s for an electron in an atom?
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9. Quantum Mechanics
Quantum Numbers: Principal Quantum Number
객관식
Which of the following sets of quantum numbers is acceptable for an electron in an atom?
A
n = 2, l = 0, m_l = 2, m_s = -1/2
B
n = 1, l = 1, m_l = 0, m_s = +1/2
C
n = 3, l = 3, m_l = 2, m_s = -1/2
D
n = 2, l = 1, m_l = 0, m_s = +1/2
검증된 단계별 안내1
Recall the allowed ranges for each quantum number: the principal quantum number \(n\) must be a positive integer (\(n = 1, 2, 3, \ldots\)), the azimuthal quantum number \(l\) can take integer values from \(0\) to \(n-1\), the magnetic quantum number \(m_l\) ranges from \(-l\) to \(+l\), and the spin quantum number \(m_s\) can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
Check the first set: \(n = 2\), \(l = 0\), \(m_l = 2\), \(m_s = -\frac{1}{2}\). Since \(l = 0\), \(m_l\) must be between \(-0\) and \(0\), so \(m_l = 2\) is not allowed. Therefore, this set is invalid.
Check the second set: \(n = 1\), \(l = 1\), \(m_l = 0\), \(m_s = +\frac{1}{2}\). For \(n = 1\), \(l\) must be between \(0\) and \(n-1 = 0\), so \(l = 1\) is not allowed. This set is invalid.
Check the third set: \(n = 3\), \(l = 3\), \(m_l = 2\), \(m_s = -\frac{1}{2}\). For \(n = 3\), \(l\) must be between \(0\) and \(2\), so \(l = 3\) is not allowed. This set is invalid.
The correct set must satisfy all quantum number rules. For example, \(n = 2\), \(l = 1\), \(m_l = 0\), \(m_s = +\frac{1}{2}\) is valid because \(l\) is between \(0\) and \(1\), \(m_l\) is between \(-1\) and \(1\), and \(m_s\) is either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
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