Discussion Overview
The discussion centers on the conservation of canonical angular momentum in the context of a purely magnetic Hamiltonian involving a vector potential. Participants explore the implications of the Hamiltonian formulation and the definitions of angular momentum in this framework.
Discussion Character
- Technical explanation
- Debate/contested
Main Points Raised
- One participant presents a Hamiltonian of the form $$ H = \dfrac{1}{2m} (\vec{p} - q\vec{A}) $$ and questions the conservation of $$ \vec{L} = \vec{r} \times \vec{p} $$.
- Another participant challenges the correctness of the Hamiltonian, noting that it should be a scalar and suggesting the computation of the Poisson bracket between $$ H $$ and $$ \vec{L} $$ as a method to investigate conservation.
- There is a suggestion that the proper canonical angular momentum might be $$ \partial {\mathcal L}/\partial\dot\phi $$ instead of the ordinary orbital angular momentum.
- Some participants agree that the term $$ (\vec{p} - q \vec{A}) $$ should be squared in the Hamiltonian formulation.
- Clarifications are made regarding the distinction between canonical angular momentum and "naive" angular momentum.
Areas of Agreement / Disagreement
Participants express disagreement regarding the formulation of the Hamiltonian and the definitions of angular momentum. There is no consensus on the correct approach to determine conservation.
Contextual Notes
Participants note potential limitations in the Hamiltonian's formulation and the definitions used, which may affect the conclusions drawn about conservation.