Discussion Overview
The discussion revolves around the energy eigenvalue problem for a particle in a box, specifically exploring solutions in momentum space rather than through traditional differential equations. Participants are examining the implications of using momentum eigenvectors and the challenges associated with boundary conditions in this context.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- One participant proposes using the momentum representation to solve the particle in a box problem, suggesting a state expressed as a linear combination of momentum eigenstates.
- Another participant suggests looking for linear combinations of free particle solutions that satisfy the boundary conditions of a perfect box, specifically that the wave function goes to zero at the edges.
- A different participant mentions a previous attempt to apply a similar approach to the infinite square well, concluding that it was not feasible, and references another thread for context.
- Another contribution raises concerns about the Fourier transform of the potential, describing it as complex and noting difficulties with boundary conditions in momentum space, as well as the non-hermiticity of the momentum operator which affects eigenvalue considerations.
Areas of Agreement / Disagreement
Participants express differing views on the feasibility of using momentum space for this problem, with some suggesting it may be possible under certain conditions, while others highlight significant challenges and limitations. No consensus is reached on the approach or its validity.
Contextual Notes
Participants note limitations related to boundary conditions, the nature of the momentum operator, and the implications of using momentum eigenstates in a bound problem. These factors contribute to the complexity of the discussion.