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青州's Question
Posted 2 months ago

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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Answer from Sia
Posted 2 months ago
Identify the given wave function: The normalized wave function for the hydrogen atom is given as: ψ2,0,0=132π(1a0)3/2(2ra0)er/2a0 \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}}
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
Expectation value of potential energy: The potential energy V(r) V(r) for a hydrogen atom is given by: V(r)=e24πϵ0r V(r) = -\frac{e^2}{4 \pi \epsilon_0 r} The expectation value V \langle V \rangle is calculated using the integral: V=ψVψdτ \langle V \rangle = \int \psi^* V \psi \, d\tau where ψ \psi^* is the complex conjugate of the wave function ψ \psi
Simplify the integral: Given the spherical symmetry, the integral can be simplified in spherical coordinates: V=00π02πψ(e24πϵ0r)ψr2sinθdϕdθdr \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 ψ2,0,0 \psi_{2,0,0} is real, ψ=ψ \psi^* = \psi
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 2s state is: V=e24πϵ0a0(12) \langle V \rangle = -\frac{e^2}{4 \pi \epsilon_0 a_0} \left( \frac{1}{2} \right) where a0 a_0 is the Bohr radius
The average strength of the nucleus of the hydrogen atom for the given normalized wave function is e28πϵ0a0 -\frac{e^2}{8 \pi \epsilon_0 a_0} .
Key Concept
Expectation value of potential energy
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 2s state, this involves integrating the potential energy function with the square of the wave function.

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