Discussion Overview
The discussion revolves around the concept of bound states in quantum mechanics and their relationship to free-particle momentum eigenstates, particularly through the lens of Fourier transforms. Participants explore the implications of representing bound states as superpositions of free-particle states, including considerations of potential dependencies.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant suggests that bound states can be viewed as superpositions of free-particle momentum eigenstates, using the example of Hermite polynomial eigenfunctions of the harmonic oscillator.
- Another participant confirms that this perspective leads to the momentum space wave function, although it will be time-dependent in the case of bound states.
- A question is raised regarding the validity of this approach in the presence of velocity-dependent potentials, with a focus on whether the physical meaning of the momentum space wave function remains unchanged.
- A later reply discusses the utility of expressing perturbations in momentum space for small perturbations to a free Hamiltonian, referencing the Interaction Picture as a method for analyzing scattering problems.
Areas of Agreement / Disagreement
Participants express varying degrees of understanding and agreement on the mathematical representation of bound states, but there is uncertainty regarding the implications of velocity-dependent potentials and whether the physical meaning of the momentum space wave function holds in those cases.
Contextual Notes
The discussion does not resolve the implications of velocity-dependent potentials on the momentum space wave function, leaving open questions about the physical significance of such representations.