Question

which of the following combinations of quantum numbers are allowed for an electron in a one - electron atom?
□ ( n = 6, l = 3, m = 4, m _ { s } = + \frac { 1 } { 2 } )
□ ( n = 3, l = 1, m = 0, m _ { s } = - \frac { 1 } { 2 } )
□ ( n = 3, l = 0, m = - 1, m _ { s } = 0 )
□ ( n = 4, l = 0, m = 0, m _ { s } = - \frac { 1 } { 2 } )

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 \(m\): \(m=-l,-l + 1,\cdots,0,\cdots,l-1,l\)
• Spin quantum number \(m_s=\pm\frac{1}{2}\)

Step2: Analyze each option

• For \(n = 4,l = 0,m = 0,m_s=-\frac{1}{2}\)
• \(n = 4\) (valid as \(n\geq1\))
• \(l=0\) (since \(l\leq n - 1=3\), valid)
• \(m = 0\) (since \(m\) ranges from \(-l\) to \(l\) and \(l = 0\), valid)
• \(m_s=-\frac{1}{2}\) (valid as \(m_s=\pm\frac{1}{2}\))
• For \(n = 3,l = 0,m=-1,m_s = 0\)
• \(n = 3\) (valid)
• \(l = 0\) (valid as \(l\leq n-1 = 2\))
• \(m=-1\) (invalid as when \(l = 0\), \(m\) can only be \(0\))
• \(m_s = 0\) (invalid as \(m_s=\pm\frac{1}{2}\))
• For \(n = 3,l = 1,m = 0,m_s=-\frac{1}{2}\)
• \(n = 3\) (valid)
• \(l = 1\) (valid as \(l\leq n-1=2\))
• \(m = 0\) (valid as \(m\) ranges from \(-l=-1\) to \(l = 1\))
• \(m_s=-\frac{1}{2}\) (valid)
• For \(n = 6,l = 3,m = 4,m_s=+\frac{1}{2}\)
• \(n = 6\) (valid)
• \(l = 3\) (valid as \(l\leq n-1 = 5\))
• \(m = 4\) (invalid as when \(l = 3\), \(m\) ranges from \(-3\) to \(3\))
• \(m_s=+\frac{1}{2}\) (valid)

Answer:

\(n = 4,l = 0,m = 0,m_s=-\frac{1}{2}\) and \(n = 3,l = 1,m = 0,m_s=-\frac{1}{2}\)