Question

What are the possible values of \(L_{z}\) (the component of angular momentum along the \(z\) -axis) for the electron in the second excited state \((n=3)\) of the hydrogen atom?

Short answer

Answer: The possible values of the electron's angular momentum component along the z-axis in the second excited state of the hydrogen atom are -2ħ, -ħ, 0, ħ, and 2ħ.

Step-by-step solution

Determine values of 'l'

Recall that the angular momentum quantum number (\(l\)) can have integer values ranging from 0 to (n-1) where n is the principal quantum number. Since \(n=3\), the possible values of \(l\) are 0, 1, and 2.

Determine values of 'm_l' for each value of 'l'

Recall that the magnetic quantum number (\(m_l\)) can have integer values ranging from -\(l\) to \(l\). Let's find the possible values of \(m_l\) for each value of \(l\): - When \(l=0\), the possible value of \(m_l\) is 0, as there is only one component along the z-axis. - When \(l=1\), the possible values of \(m_l\) are -1, 0, and 1. These represent the three magnetic substates for the given value of the angular momentum quantum number. - When \(l=2\), the possible values of \(m_l\) are -2, -1, 0, 1, and 2. These represent the five magnetic substates for the given value of the angular momentum quantum number.

Determine the possible values of \(L_z\)

Finally, we can use the formula \(L_z = m_l \hbar\), to calculate the possible values of \(L_z\) for each value of \(m_l\). We can eliminate the constant \(\hbar\) in this equation, as we are only interested in the relative ratios, and \(\hbar\) is just a scaling factor. The possible values of \(m_l\) are: -2, -1, 0, 1, and 2, which are the possible values of \(L_z\) when scaled by \(\hbar\). Thus, the electron in the second excited state (\(n=3\)) of the hydrogen atom has possible components of angular momentum along the \(z\)-axis as: \(L_z = \{-2 \hbar, -\hbar, 0, \hbar, 2 \hbar\}\)