Integrate Laplacian operator by parts

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SUMMARY

The discussion focuses on the integration of the Laplacian operator by parts as a crucial step in transforming the position space Schrödinger equation to its momentum space equivalent. The transformation process specifically addresses the transition to equation 3.21. Key concepts mentioned include tensor calculus and the Gauss-Ostrogradski theorem in three dimensions, which are essential for understanding the integration of the Laplacian term.

PREREQUISITES
  • Tensor calculus
  • Gauss-Ostrogradski theorem in 3D
  • Schrödinger equation fundamentals
  • Momentum space concepts
NEXT STEPS
  • Study the application of the Gauss-Ostrogradski theorem in physics
  • Explore the derivation of the momentum space Schrödinger equation
  • Learn advanced tensor calculus techniques
  • Investigate the implications of Laplacian integration in quantum mechanics
USEFUL FOR

Physicists, mathematicians, and students studying quantum mechanics or advanced calculus, particularly those interested in the mathematical foundations of the Schrödinger equation.

Eric_J
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upload_2016-1-18_18-33-58.png

This is the key step to transform from position space Schrödinger equation to its counterpart in momentum space.
How is the first equation transformed into 3.21?
To be more specific, how to integral Laplacian term by parts?
 
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Do you know tensor tensor calculus or at least the Gauss-Ostrograski theorem in 3D?
 
dextercioby said:
Do you know tensor tensor calculus or at least the Gauss-Ostrograski theorem in 3D?
upload_2016-1-18_21-6-4.png

Thanks for the hint!
 

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