Discussion Overview
The discussion revolves around the assumptions and conditions under which time-independent perturbation theory in quantum mechanics can be applied, particularly focusing on the expansion of new states in terms of old states. Participants explore the implications of different Hilbert spaces and the completeness of eigenfunctions in various contexts.
Discussion Character
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant questions the assumption that new states can always be expanded in terms of old states, citing the example of free-particle wavepackets and infinite square well eigenfunctions.
- Another participant suggests that the assumption holds when wavefunctions have support over the region of interest, indicating that infinite square well wavefunctions cannot serve as a complete basis for plane waves but can for other functions with the same support.
- A different viewpoint emphasizes that the Hilbert spaces for the Hamiltonians H_0 and H' are different, noting that the infinite potential outside the well restricts the Hilbert space to L²[a,b].
- One participant mentions that perturbations V should not change the Hilbert space for the expansion to be valid, while also acknowledging that if V restricts the Hilbert space, the expansion may still work.
- Another participant reiterates the importance of H_0 and V being defined on the same dense domain of the Hilbert space and that V is relatively compact with respect to H_0, referencing functional analysis texts for clarification.
- Questions arise regarding the meaning of "with respect to" in the context of compactness, with a participant speculating on its relation to the ranges of V and H_0.
- A later reply provides a definition of a linear operator being compact relatively to H, explaining the conditions under which this holds.
Areas of Agreement / Disagreement
Participants express differing views on the applicability of the assumptions in perturbation theory, with no consensus reached on the conditions under which the expansion of states is valid.
Contextual Notes
Participants highlight limitations related to the definitions of Hilbert spaces and the nature of the perturbations, as well as the mathematical details surrounding compactness and relative compactness, which remain unresolved.