Question

An electron in an atom is in the \(n=3\) quantum level. List the possible values of \(\ell\) and \(m_{\ell}\) that it can have.

Short answer

For the third quantum level (\(n = 3\)), the possible combinations of \(\ell\) and \(m_{\ell}\) are: (\(0, 0\)), (\(1, -1\)), (\(1, 0\)), (\(1, 1\)), (\(2, -2\)), (\(2, -1\)), (\(2, 0\)), (\(2, 1\)), (\(2, 2\)).

Step-by-step solution

Determine the possible values of \(\ell\)

The azimuthal quantum number \(\ell\) can have integer values ranging from 0 to \(n-1\). Here, with \(n = 3\), the possible values of \(\ell\) can be 0, 1, or 2.

Determine the possible values of \(m_{\ell}\) for each \(\ell\)

For each value of \(\ell\), \(m_{\ell}\) can have integer values ranging from \(-\ell\) to \(\ell\). Consequently, when \(\ell = 0\), \(m_{\ell} = 0\); when \(\ell = 1\), \(m_{\ell}\) can be \(-1, 0, 1\); and when \(\ell = 2\), \(m_{\ell}\) can be \(-2, -1, 0, 1, 2\).