(a) Construct the spatial wave function $\mathbf{\left(}\mathbf{\psi }\mathbf{\right)}$for hydrogen in the state $\mathbf{n}\mathbf{=}\mathbf{3}\mathbf{,}\mathbf{}\mathbf{I}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{m}\mathbf{=}\mathbf{1}\mathbf{.}$Express your answer as a function of $\mathbf{r}\mathbf{,}\mathbf{}\mathbf{\theta }\mathbf{,}\mathbf{\varphi }\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}\mathbf{a}$(the Bohr radius) only—no other variables ($\mathbf{p}\mathbf{,}\mathbf{z}\mathbf{}\mathbf{,}$etc.) or functions ($\mathit{p}\mathbf{,}\mathit{v}\mathbf{,}$etc.), or constants ($\mathbf{A}\mathbf{,}{\mathbf{c}}_{\mathbf{0}}\mathbf{,}$etc.), or derivatives, allowed (π is okay, and e, and 2, etc.).

(b) Check that this wave function is properly normalized, by carrying out the appropriate integrals over, $\mathbf{\theta }\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}\mathbf{\varphi }\mathbf{.}$

(c) Find the expectation value of ${\mathbf{r}}^{\mathbf{s}}$in this state. For what range of s (positive and negative) is the result finite?

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