Discussion
The wave function for an electron in a hydrogen atom is given by
\[ \begin{aligned} \psi(\vec{r}) &= \psi_{200}(\vec{r}) + 2\psi_{211} \\ &\quad + 3\psi_{210} + \sqrt{2}\psi_{21-1} \end{aligned} \]
where \(\psi_{nlm_{l}}(\vec{r})\) is the wave function for an electron in the eigenstate \((nlm_{l})\). The expectation value of z-component of angular momentum, \(\langle L_{z} \rangle\) in the state \(\psi(\vec{r})\) is :
Answer: B
\(\langle L_{z} \rangle\) is \(\hbar/8\).
The state is not normalised: the squared coefficients add to \(1 + 4 + 9 + 2 = 16\). Each component \(\psi_{nlm}\) has \(L_{z} = m\hbar\), so:
\[ \begin{aligned} \langle L_{z} \rangle &= \frac{1(0) + 4(1) + 9(0) + 2(-1)}{16}\hbar \\ &= \frac{2\hbar}{16} = \frac{\hbar}{8} \end{aligned} \]
The states with m = 0 contribute nothing. Forgetting the m = -1 term gives \(4\hbar/16\), and forgetting to divide by the norm gives \(2\hbar\); neither matches another option exactly, while the other options do not follow from the coefficients.
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