Question

given ( n = 2 ), identify which of the quantum numbers is not possible. answer: a ( l = 0 ) b ( l = 1 ) c ( m_l=-2 ) d ( m_s=+\frac{1}{2} ) e ( m_s=-\frac{1}{2} )

Explanation:

Step1: Determine the range of \( l \)

For a given principal quantum number \( n \), the angular - momentum quantum number \( l \) can take values \( l = 0,1,\cdots,n - 1 \). Given \( n=2 \), then \( l = 0,1 \). So options A (\( l = 0 \)) and B (\( l = 1 \)) are possible.

Step2: Determine the range of \( m_l \)

For a given \( l \), the magnetic quantum number \( m_l \) can take values \( m_l=-l,-l + 1,\cdots,0,\cdots,l-1,l \). When \( n = 2 \), the maximum value of \( l \) is \( 1 \). So \( m_l=-1,0,1 \). Option C (\( m_l=-2 \)) is not possible.

Step3: Determine the range of \( m_s \)

The spin - quantum number \( m_s=\pm\frac{1}{2} \). So options D (\( m_s =+\frac{1}{2} \)) and E (\( m_s=-\frac{1}{2} \)) are possible.

Answer:

C. \( m_l=-2 \)