Angular Momentum and Hamiltonian Commutation

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SUMMARY

The discussion focuses on demonstrating that the angular momentum operator squared commutes with the Hamiltonian operator (H) in spherical coordinates. Participants suggest analyzing the eigenvalues of both the angular momentum operator and its square, as well as examining how these operators interact with the components of H. The commutation relation is emphasized as a key concept to understand the relationship between angular momentum and H, ultimately revealing that the problem is straightforward once the correct approach is identified.

PREREQUISITES
  • Understanding of angular momentum operators in quantum mechanics
  • Familiarity with Hamiltonian mechanics
  • Knowledge of spherical coordinates
  • Basic principles of commutation relations
NEXT STEPS
  • Study the properties of angular momentum operators in quantum mechanics
  • Learn about Hamiltonian operators and their role in quantum systems
  • Explore the mathematical framework of commutation relations
  • Review eigenvalue problems related to quantum operators
USEFUL FOR

Students of quantum mechanics, physicists working with angular momentum, and anyone studying Hamiltonian dynamics in quantum systems.

jtgurkin
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I am working on a problem for homework and am supposed to show that the angular momentum operator squared commutes with H and that angular momentum and H also commute. This must be done in spherical coordinates and everything I see says "it's straightforward" but I don't see it. At least not yet. Can someone help?
 
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I would suggest considering the eigenvalues for both the angular momentum and square of the angular momentum operator. Then think about which parts of either angular momentum (squared or not squared) operator act on the components of the Hamiltonian. Finally, what does the commutation relation tell you about when the operator commutes with the Hamiltonian? I think you should find the problem is actually straight forward, just a little different when you first see it.
 

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