Discussion Overview
The discussion revolves around the treatment of wavefunctions in quantum mechanics, particularly in relation to tempered distributions and Rigged Hilbert spaces. Participants explore the implications of using non-square integrable states, such as exponentially growing states and singular solutions, in the context of physical states and bases.
Discussion Character
- Debate/contested
- Technical explanation
- Conceptual clarification
Main Points Raised
- Some participants note that standard quantum mechanics (QM) texts reject certain states, like exponentially growing ones, for not being in L² space, while others point out that scattered states also fall outside L² spaces.
- There is a proposal that Rigged Hilbert spaces can address the dichotomy between these states and allow for the inclusion of tempered distributions.
- One participant questions how to determine which states are physical, especially when considering singular solutions in central potentials, which are also considered tempered solutions.
- Another participant asserts that states at fixed time must be square integrable, but the wave functions used for analysis do not necessarily need to be.
- There is a reiteration of the idea that the basis in Rigged Hilbert spaces allows for the inclusion of non-physical states, such as Dirac kets, which are used for analysis but do not represent physical states of a system.
- A participant asks for clarification on why singular solutions are not used as a basis for the radial wavefunction of the hydrogen atom, despite being elements of the Rigged Hilbert space.
- It is suggested that while any basis can be used, experience indicates that well-chosen bases lead to more manageable formulas, with continuous bases being preferred for scattering problems and discrete bases for bound state problems.
Areas of Agreement / Disagreement
Participants express differing views on the physicality of certain states and the appropriateness of using singular solutions as bases. There is no consensus on the criteria for determining which states are considered physical or the implications of using Rigged Hilbert spaces.
Contextual Notes
Participants highlight limitations in the understanding of which states can be considered physical and the criteria for selecting bases, indicating a dependence on definitions and unresolved reasoning regarding the use of singular solutions.