Discussion Overview
The discussion revolves around the justification of a method used in solving partial differential equations (PDEs) in physics, particularly focusing on cases where the equations and boundary conditions are independent of certain variables, such as the azimuthal angle. Participants explore the implications of this method in the context of the heat/diffusion equation and the Schrödinger equation, questioning when it is valid to assume independence from certain variables without justification.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- Some participants propose that in certain PDEs like the heat equation, assuming independence from an angular variable can lead to valid solutions, while in the Schrödinger equation, this assumption may fail due to the presence of angular dependence in the Laplacian.
- Others argue that the boundary and initial conditions can dictate the allowed angular dependence, suggesting that the apparent independence is a result of these conditions rather than the nature of the PDE itself.
- A participant highlights that while the Laplacian in spherical coordinates includes angular derivatives, the radial potential in the Schrödinger equation leads to solutions that are not angle-independent, contrasting with the heat equation.
- Some participants discuss the role of existence and uniqueness theorems in justifying the solutions to the Schrödinger equation, although this does not resolve the question of angular dependence.
- There is a suggestion that the evolution of the system, whether time-dependent or time-independent, does not affect the angular dependence in the Schrödinger equation, unlike in the heat equation.
- One participant questions the validity of assuming that the initial temperature profile in a cylinder does not depend on the azimuthal angle, suggesting that this could influence the solution.
- Another participant raises the possibility that the nature of eigenvalues in the problems might contribute to the differences in angular dependence between the two types of equations.
Areas of Agreement / Disagreement
Participants express differing views on the validity of the method used to assume independence from angular variables in PDEs. There is no consensus on when this trick is justified, particularly in relation to the Schrödinger equation, indicating that multiple competing views remain.
Contextual Notes
The discussion highlights the complexity of boundary conditions and their impact on the solutions of PDEs, as well as the nuances in the mathematical treatment of different types of equations. Participants acknowledge the need for careful consideration of initial conditions and the implications of angular dependence.