Hydrogen atom:<K>,<V>,momentum distribution

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SUMMARY

The discussion focuses on evaluating the expectation values of kinetic energy and potential energy , as well as the momentum distribution \varphi(p) for a hydrogen atom in the ground state (n=1). The participants analyze the wave function \varphi(r) = \frac{exp(-r/a)}{\sqrt{\pi a^3}} and derive the expectation value of kinetic energy using integration techniques. They identify errors in the application of the Laplacian operator and discuss the Fourier Transform necessary for calculating the momentum distribution. The correct evaluation of is confirmed to be K = \frac{+me^4}{2 \hbar^2}, and participants suggest using momentum space for easier calculations.

PREREQUISITES
  • Quantum Mechanics fundamentals, specifically the hydrogen atom model.
  • Understanding of wave functions and their properties in quantum mechanics.
  • Familiarity with expectation values and integration techniques in spherical coordinates.
  • Knowledge of Fourier Transforms and their application in quantum mechanics.
NEXT STEPS
  • Learn about the application of the Laplacian operator in spherical coordinates in quantum mechanics.
  • Study the Fourier Transform of wave functions in quantum mechanics.
  • Explore the derivation of expectation values in momentum space.
  • Investigate the properties of the Gamma function and its applications in integrals.
USEFUL FOR

Students and professionals in quantum mechanics, particularly those studying atomic physics, wave functions, and expectation values in quantum systems.

  • #31
ah,ok...yes,i have the solution for <k>...but not for phi(p)...neither in any book!
 
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  • #32
ok, then you can check by evaluating <K> in momentum space, as KDV said, this is much easier than in positions space.
 

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