Question

How many electron states of the H atom have the quantum numbers \(n=3\) and \(\ell=1 ?\) Identify each state by listing its quantum numbers.

Short answer

Answer: There are 3 electron states with the quantum numbers \(n=3\) and \(\ell=1\) for hydrogen atom. These states are: \((3, 1, -1)\), \((3, 1, 0)\), and \((3, 1, 1)\).

Step-by-step solution

Find the possible values of the magnetic quantum number \(m_\ell\)

Given that \(\ell = 1\), we can find the possible values of the magnetic quantum number \(m_\ell\) using the range \((-\ell, -\ell+1, ..., 0, ..., \ell-1, \ell)\). So, for \(\ell = 1\), the possible values of \(m_\ell\) will be \((-1, 0, 1)\).

Count the number of electron states and identify them

Now that we know the possible values of \(m_\ell\), we can count the number of electron states and identify each state by listing its quantum numbers \((n, \ell, m_\ell)\). There are in total 3 values of \(m_\ell = -1, 0, 1\). The electron states with quantum numbers \((n, \ell, m_\ell)\) are as follows: 1. State 1: \((3, 1, -1)\) 2. State 2: \((3, 1, 0)\) 3. State 3: \((3, 1, 1)\) There are 3 electron states of the hydrogen atom with the given quantum numbers \(n = 3\) and \(\ell = 1\).