State Vectors vs. Wavefunctions

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Discussion Overview

The discussion centers around the relationship between state vectors in Hilbert space and wavefunctions in L^2 space within the context of Quantum Mechanics. Participants explore the isomorphism between these two mathematical structures, the nature of linear operators in both spaces, and the implications of notation and representation in quantum theory.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that L^2 space and Hilbert space are isomorphic and seeks to construct an explicit isomorphism between linear operators in both spaces.
  • Others argue that the distinction between kets and wavefunctions is significant, emphasizing that they are not the same, despite being related through an isomorphism.
  • Some participants highlight that an arbitrary operator has different representations in L^2 and Hilbert space, suggesting that notation choice does not affect the underlying mathematical structure.
  • A participant mentions that while some operators can be expressed as differential operators acting on wavefunctions, others, like spin operators, operate in different Hilbert spaces and do not span L^2.
  • There is a discussion about the ambiguity in notation, particularly regarding the distinction between a function and its value at a point, with a participant clarifying that these are different mathematical objects.
  • Another participant points out that the completeness relation can be used to express how operators act on wavefunctions, leading to integral transforms in position space.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and implications of constructing isomorphisms between the two spaces. While some agree on the fundamental relationships, others contest the interpretations and representations, leading to an unresolved discussion with multiple competing perspectives.

Contextual Notes

Participants note that some operators cannot be represented as differential operators acting solely on wavefunctions, and there is a distinction between the Hilbert spaces associated with different physical systems, such as spin states versus position states.

  • #31
bob012345 said:
My takeaway from this discussion is that the physics sometimes gets lost in the math. Very disheartening when intelligent people can't even agree on the mathematics of quantum theory.

I don't think the disagreement is very deep. It's a disagreement about how best to describe what's going on, but there's no disagreement about how to use the mathematics.
 
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  • #32
stevendaryl said:
I don't think the disagreement is very deep. It's a disagreement about how best to describe what's going on, but there's no disagreement about how to use the mathematics.
What is going on that the mathematical treatments are not agreeing. What is the physics? Thanks.
 
  • #33
bob012345 said:
What is going on that the mathematical treatments are not agreeing.
Nothing. Nobody is disagreeing on what the actual mathematics are. The argument is about how to best define one's notation.
 

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