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MMLU-Pro / 4175 / Assume that the charge of the proton is distributed uniformly throughout the…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Assume that the charge of the proton is distributed uniformly throughout the volume of a sphere of radius $10^{-13} \mathrm{~cm}$. Use perturbation theory to estimate the shift in the ground-state hydrogen-atom energy due to the finite proton size. The potential energy experienced by the electron when it has penetrated the nucleus and is at distance rr from the nuclear center is −eQ/4πε0r-e Q / 4 \pi \varepsilon_0 r, where QQ is the amount of proton charge within the sphere of radius rr. The evaluation of the integral is simplified by noting that the exponential factor in ψ\psi is essentially equal to 1 within the nucleus.
Plain-text mathematical notation (without MathML)
Assume that the charge of the proton is distributed uniformly throughout the volume of a sphere of radius $10^{-13} \mathrm{~cm}$. Use perturbation theory to estimate the shift in the ground-state hydrogen-atom energy due to the finite proton size. The potential energy experienced by the electron when it has penetrated the nucleus and is at distance r from the nuclear center is −eQ/4πε₀r, where Q is the amount of proton charge within the sphere of radius r. The evaluation of the integral is simplified by noting that the exponential factor in ψ is essentially equal to 1 within the nucleus.
Original LaTeX notation
Assume that the charge of the proton is distributed uniformly throughout the volume of a sphere of radius $10^{-13} \mathrm{~cm}$. Use perturbation theory to estimate the shift in the ground-state hydrogen-atom energy due to the finite proton size. The potential energy experienced by the electron when it has penetrated the nucleus and is at distance $r$ from the nuclear center is $-e Q / 4 \pi \varepsilon_0 r$, where $Q$ is the amount of proton charge within the sphere of radius $r$. The evaluation of the integral is simplified by noting that the exponential factor in $\psi$ is essentially equal to 1 within the nucleus.

category

chemistry

options

  1. 1.4 $10^{-8} \mathrm{eV}$
  2. 2.0 $10^{-8} \mathrm{eV}$
  3. 1.2 $10^{-8} \mathrm{eV}$
  4. 3.0 $10^{-8} \mathrm{eV}$
  5. 0.4 $10^{-8} \mathrm{eV}$
  6. 2.4 $10^{-8} \mathrm{eV}$
  7. 0.8 $10^{-8} \mathrm{eV}$
  8. 2.8 $10^{-8} \mathrm{eV}$
  9. 0.6 $10^{-8} \mathrm{eV}$
  10. 1.6 $10^{-8} \mathrm{eV}$

src

scibench-quan

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Source and history

upstream_id / Original problem ID
4175
task_id
eedb40af-1f45-5a64-ab78-4a9d14b36c6f
task_key
default--test--4175
task_revision_id
1
config / split
default / test

Official source

initial import