Discussion Overview
The discussion revolves around the separation of variables for non-central potentials in various coordinate systems, particularly focusing on the potential V(r,θ) and its separability in spherical and potentially other coordinates. Participants explore theoretical aspects and references related to this topic.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- One participant questions the separability of the potential V(r,θ) in coordinates other than spherical coordinates, suggesting the form V(r,θ)=u(r)+f(θ)/r².
- Another participant proposes the use of parabolic coordinates as a potential alternative for separation.
- References to Landau and Lifgarbagez's work on quantum mechanics are made, indicating a discussion of separability in various coordinates.
- A participant mentions Morse and Feshbach's book, noting it discusses separation of variables in linear PDEs, though the specific volume is not recalled.
- A more specific potential is presented by a participant, which includes terms involving constants b, c, d, and f, and asks about its separability in other coordinates.
- Concerns are raised about the generality of the functions u(r) and f(θ), suggesting that if they are completely general, finding another factorization may not be feasible.
- One participant seeks examples of other non-central potentials that could be separable in different coordinates, expressing a desire for input from experts in the field.
Areas of Agreement / Disagreement
Participants express differing views on the potential for separability in various coordinate systems, with no consensus reached on specific alternatives or the generality of the functions involved.
Contextual Notes
There are limitations regarding the assumptions about the forms of the potentials discussed, and the specific conditions under which separability might hold are not fully resolved.