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PROBLEM (10 Points) The azimuthal wave function for the hydrogen atom can be obtained from the solution of the azimuthal component of the Schrodinger equation as follows $\Phi(\varphi) = Ae^{im\varphi}$ where $\varphi \in [0, 2\pi)$ is the angle around z-axis in spherical coordinates as shown in the figure and A is the normalization constant. (a) [3 Points] Find the allowed values of m from the periodicity condition $\Phi(\varphi) = \Phi(\varphi + 2\pi)$ and the normalization constant A from $\int_0^{2\pi} d\varphi |\Phi(\varphi)|^2 = 1$. (b) [4 Points] Find the operator $L_z$ in spherical coordinates where full angular momentum vector is given by the operator $\mathbf{L} = \mathbf{r} \times \mathbf{p}$. Note that in spherical coordinates $\mathbf{r} = r\hat{r}$ and $\mathbf{p} = -i\hbar\nabla$ with $\nabla = \hat{r}\frac{\partial}{\partial r} + \frac{\hat{\theta}}{r}\frac{\partial}{\partial \theta} + \frac{\hat{\varphi}}{r\sin\theta}\frac{\partial}{\partial \varphi}$ and unit vectors form an orthogonal system with the cross products $\hat{\theta} \times \hat{\varphi} = \hat{r}$ and cyclic permutation of $\hat{r}, \hat{\theta}, \hat{\varphi}$. (Hint: Consider the dot product $L_z = \hat{z} \cdot \mathbf{L}$ where unit vector along z can be written in spherical coordinates $\hat{z} = \cos\theta\hat{r} - \sin\theta\hat{\theta}$.) (c) [3 Points] Find the expectation value of $L_z$ using your results in (a) and (b).

          PROBLEM (10 Points) The azimuthal wave function for the hydrogen atom
can be obtained from the solution of the azimuthal component of the Schrodinger
equation as follows
$\Phi(\varphi) = Ae^{im\varphi}$
where $\varphi \in [0, 2\pi)$ is the angle around z-axis in spherical coordinates as shown
in the figure and A is the normalization constant.
(a) [3 Points] Find the allowed values of m from the periodicity condition $\Phi(\varphi) = \Phi(\varphi + 2\pi)$ and the normalization
constant A from $\int_0^{2\pi} d\varphi |\Phi(\varphi)|^2 = 1$.
(b) [4 Points] Find the operator $L_z$ in spherical coordinates where full angular momentum vector is given by
the operator
$\mathbf{L} = \mathbf{r} \times \mathbf{p}$.
Note that in spherical coordinates $\mathbf{r} = r\hat{r}$ and $\mathbf{p} = -i\hbar\nabla$ with $\nabla = \hat{r}\frac{\partial}{\partial r} + \frac{\hat{\theta}}{r}\frac{\partial}{\partial \theta} + \frac{\hat{\varphi}}{r\sin\theta}\frac{\partial}{\partial \varphi}$ and unit vectors form
an orthogonal system with the cross products $\hat{\theta} \times \hat{\varphi} = \hat{r}$ and cyclic permutation of $\hat{r}, \hat{\theta}, \hat{\varphi}$. (Hint: Consider the
dot product $L_z = \hat{z} \cdot \mathbf{L}$ where unit vector along z can be written in spherical coordinates $\hat{z} = \cos\theta\hat{r} - \sin\theta\hat{\theta}$.)
(c) [3 Points] Find the expectation value of $L_z$ using your results in (a) and (b).

PROBLEM (10 Points) The azimuthal wave function for the hydrogen atom
can be obtained from the solution of the azimuthal component of the Schrodinger
equation as follows
Φ(φ) = Ae^imφ
where φ∈ [0, 2π) is the angle around z-axis in spherical coordinates as shown
in the figure and A is the normalization constant.
(a) [3 Points] Find the allowed values of m from the periodicity condition Φ(φ) = Φ(φ + 2π) and the normalization
constant A from ∫0^2π dφ |Φ(φ)|^2 = 1.
(b) [4 Points] Find the operator Lz in spherical coordinates where full angular momentum vector is given by
the operator
𝐋 = 𝐫×𝐩.
Note that in spherical coordinates 𝐫 = rr̂ and 𝐩 = -iħ∇ with ∇ = r̂(∂)/(∂ r) + (θ̂)/(r)(∂)/(∂θ) + (φ̂)/(rsinθ)(∂)/(∂φ) and unit vectors form
an orthogonal system with the cross products θ̂×φ̂ = r̂ and cyclic permutation of r̂, θ̂, φ̂. (Hint: Consider the
dot product Lz = ẑ·𝐋 where unit vector along z can be written in spherical coordinates ẑ = cosθr̂ - sinθθ̂.)
(c) [3 Points] Find the expectation value of Lz using your results in (a) and (b).

Added by Sean R.

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