Question

an electron in an atom is known to be in a state with magnetic quantum number ( m_{l}=0 ). what is the smallest possible value of the principal quantum number ( n ) of the state?

Explanation:

Step1: Recall the relationship between quantum numbers

The principal quantum number \(n\), the orbital angular momentum quantum number \(l\), and the magnetic quantum number \(m_l\) are related. The possible values of \(l\) range from \(0\) to \(n - 1\). The possible values of \(m_l\) range from \(-l\) to \(+l\).

Step2: Determine the smallest \(n\)

Since \(m_l = 0\) is allowed for any \(l\) (because \(m_l\) values include \(0\) when \(l\geq0\)), the smallest non - zero value of \(l\) is \(l = 0\). For \(l = 0\), the smallest value of \(n\) (since \(n>l\) and \(n\) is a positive integer) is \(n = 1\).

Answer:

\(n = 1\)