Question

Explain, giving reasons, which of the following sets of quantum numbers are not possible.
(a) n = 0, l = 0, ml = 0, ms = +\(\frac{1}{2}\) (b) n = 1, l = 0, ml = 0, ms = –\(\frac{1}{2}\) (c) n = 1, l = 1, ml = 0, ms = + \(\frac{1}{2}\) (d) n = 2, l = 1, ml = 0, ms = – \(\frac{1}{2}\) (e) n = 3, l = 3, ml = –3, ms = + \(\frac{1}{2}\) (f) n = 3, l = 1, ml = 0, ms = + \(\frac{1}{2}\)

Answer 1

Rules for quantum numbers:

• Principal quantum number, n = 1, 2, 3, … (n ≠ 0)
• Azimuthal quantum number, l = 0 to (n − 1)
• Magnetic quantum number, m_l = −l to +l
• Spin quantum number, m_s = +½ or −½

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(a) n = 0, l = 0, m_l = 0, m_s = +½

Not possible, because principal quantum number n cannot be zero.

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(b) n = 1, l = 0, m_l = 0, m_s = −½

Possible, because all quantum numbers satisfy the allowed rules.

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(c) n = 1, l = 1, m_l = 0, m_s = +½

Not possible, because for n = 1, l can only be 0 (l = 0 to n − 1).

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(d) n = 2, l = 1, m_l = 0, m_s = −½

Possible, because all quantum numbers are within the allowed limits.

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(e) n = 3, l = 3, m_l = −3, m_s = +½

Not possible, because for n = 3, l can have values only 0, 1, or 2.

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(f) n = 3, l = 1, m_l = 0, m_s = +½

Possible, because all quantum numbers obey the required conditions.

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Final Answer:

Not possible sets: (a), (c), (e)
Possible sets: (b), (d), (f)