Discussion Overview
The discussion revolves around the conservation of angular momentum in different types of symmetric potentials, specifically comparing spherically symmetric and elliptically symmetric potentials. Participants explore the implications of these symmetries on the conservation laws applicable to particles within these potentials.
Discussion Character
- Debate/contested
- Technical explanation
Main Points Raised
- One participant asserts that angular momentum is conserved in a spherically symmetric potential and questions whether a similar conservation applies in an elliptically symmetric potential.
- Another participant suggests that angular momentum conservation arises from the rotational invariance of the Hamiltonian, implying it should hold in some form for elliptical potentials.
- A different viewpoint argues that angular momentum, defined as \(\vec x \times \vec p\), does not have a comparable conserved quantity in a generic elliptically symmetric potential.
- One participant proposes that if the elliptically symmetric potential is specifically defined, angular momentum along a certain axis (z-axis) could be conserved.
- Another participant mentions that in the context of an elliptic potential well, the total angular momentum is not a good quantum number due to mixing of energy eigenvalues.
- A later reply clarifies that while angular momentum around the azimuthal direction may be conserved, the total angular momentum is not conserved in the case of an elliptical potential.
Areas of Agreement / Disagreement
Participants express differing views on the conservation of angular momentum in elliptical potentials, with no clear consensus reached on the conditions under which conservation may or may not hold.
Contextual Notes
Participants reference specific forms of potentials and quantum mechanical implications, indicating that the discussion may depend on the definitions and assumptions regarding the symmetry of the potentials involved.