What is the overage strength of the nucleus of H- - - tom for the given normalized wove function?
$$
\psi_{2,0,0}=\frac{1}{\sqrt{32 \pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2}\left(2-\frac{r}{a_{0}}\right) e^{-r / 2 a_{0}}
$$

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∻Solution∻ ‖ 1 ⋮ **Identify the given wave function**: The normalized wave function for the hydrogen atom is given as: $$ \psi_{2,0,0} = \frac{1}{\sqrt{32 \pi}} \left( \frac{1}{a_{0}} \right)^{3 / 2} \left( 2 - \frac{r}{a_{0}} \right) e^{-r / 2 a_{0}} $$ ‖ ‖ 2 ⋮ **Understand the context**: The question asks for the average strength of the nucleus of a hydrogen atom, which typically involves calculating the expectation value of the potential energy. ‖ ‖ 3 ⋮ **Expectation value of potential energy**: The potential energy $ V(r) $ for a hydrogen atom is given by: $$ V(r) = -\frac{e^2}{4 \pi \epsilon_0 r} $$ The expectation value $ \langle V \rangle $ is calculated using the integral: $$ \langle V \rangle = \int \psi^* V \psi \, d\tau $$ where $ \psi^* $ is the complex conjugate of the wave function $ \psi $. ‖ ‖ 4 ⋮ **Simplify the integral**: Given the spherical symmetry, the integral can be simplified in spherical coordinates: $$ \langle V \rangle = \int_0^\infty \int_0^\pi \int_0^{2\pi} \psi^* \left( -\frac{e^2}{4 \pi \epsilon_0 r} \right) \psi \, r^2 \sin \theta \, d\phi \, d\theta \, dr $$ Since $ \psi_{2,0,0} $ is real, $ \psi^* = \psi $. ‖ ‖ 5 ⋮ **Evaluate the integral**: Substitute the given wave function into the integral and solve. This step involves detailed integration which can be complex, but the result for the expectation value of the potential energy for the $ 2s $ state is: $$ \langle V \rangle = -\frac{e^2}{4 \pi \epsilon_0 a_0} \left( \frac{1}{2} \right) $$ where $ a_0 $ is the Bohr radius. ‖ ∻Answer∻ ⚹ The average strength of the nucleus of the hydrogen atom for the given normalized wave function is $ -\frac{e^2}{8 \pi \epsilon_0 a_0} $. ⚹ ∻Key Concept∻ ⚹ Expectation value of potential energy ⚹ ∻Explanation∻ ⚹ The expectation value of the potential energy for a given wave function provides the average strength of the nucleus in a hydrogen atom. For the $ 2s $ state, this involves integrating the potential energy function with the square of the wave function. ⚹◊What is the significance of the normalized wave function in determining the average strength of the nucleus of a hydrogen atom⍭ Generate me a similar question◊

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