Rigged Hilbert Space Φ ⊂ H ⊂ Φ'

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The discussion revolves around the mathematical framework of Rigged Hilbert Spaces, specifically the relationship Φ ⊂ H ⊂ Φ', where H is a Hilbert space, Φ is a subspace, and Φ' is the dual space of Φ. Participants clarify that H is included in Φ' due to the properties of dual spaces and the Riesz-Fischer theorem, which asserts that H is its own dual. The conversation also touches on the implications of boundary conditions in quantum mechanics, particularly regarding the expectation values of momentum operators in square well potentials. A detailed examination of wave functions and their behavior in different potential scenarios is presented, highlighting the complexities involved in calculating expectation values. The discussion emphasizes the importance of understanding these mathematical concepts for a deeper grasp of quantum mechanics.
  • #61
sweet springs said:
Another idea to remove for the difficulty is modification of energy eigenfunction; [...]
It is usually more profitable to study thoroughly what other people have already achieved before constructing one's own theory.

In this vein, it may be helpful to study distribution theory more deeply. I found Appendix A of Nussenzveig's book on "Causality and Dispersion Relations" quite helpful.
 
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  • #62
Thanks for your comment.
I am excited to know that the square well potential that appears n many QM textbooks seems to have something more to be explored.
I agree with you that I should learn more.
 

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