Question

1. label each of the following sets of quantum numbers as valid or invalid.

n\tl\tm₁\tmₛ\tvalid or invalid?

1\t0\t0\t½

1\t1\t+1\t-½

3\t0\t1\t½

Explanation:

Step1: Check the rules for quantum numbers

• Principal quantum number (\(n\)): \(n = 1,2,3,\cdots\)
• Azimuthal quantum number (\(l\)): \(l=0,1,\cdots,n - 1\)
• Magnetic quantum number (\(m_l\)): \(m_l=-l,-l + 1,\cdots,0,\cdots,l-1,l\)
• Spin quantum number (\(m_s\)): \(m_s=\pm\frac{1}{2}\)

Step2: Analyze the first set \((n = 1,l = 0,m_l=0,m_s=\frac{1}{2})\)

• \(n = 1\) (valid as \(n\geq1\))
• \(l=0\) (since \(n = 1\), \(l\) can be \(0\) (\(l=n - 1\) when \(n = 1\))
• \(m_l = 0\) (since \(l = 0\), \(m_l\) can only be \(0\))
• \(m_s=\frac{1}{2}\) (valid as \(m_s=\pm\frac{1}{2}\))

Step3: Analyze the second set \((n = 1,l = 1,m_l=+1,m_s=-\frac{1}{2})\)

• \(n = 1\) (valid)
• \(l = 1\) (invalid because when \(n = 1\), \(l\) can only be \(0\) (\(l\leq n-1\))

Step4: Analyze the third set \((n = 3,l = 0,m_l=1,m_s=\frac{1}{2})\)

• \(n = 3\) (valid)
• \(l = 0\) (valid as \(l\leq n - 1=2\) is not violated here)
• \(m_l=1\) (invalid because when \(l = 0\), \(m_l\) can only be \(0\) (\(m_l\) ranges from \(-l\) to \(l\))

Answer:

\(n\)	\(l\)	\(m_l\)	\(m_s\)	valid or invalid?
\(1\)	\(1\)	\(+1\)	\(-\frac{1}{2}\)	invalid
\(3\)	\(0\)	\(1\)	\(\frac{1}{2}\)	invalid