State Vectors vs. Wavefunctions

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SUMMARY

The discussion centers on the relationship between state vectors in Hilbert space and wavefunctions in L^2 space within the context of Quantum Mechanics. The participants explore the isomorphism between these two spaces, specifically defining mappings such as ζ : H → L^2 and Λ : L(H;H) → L(L^2,L^2). Key points include the distinction between kets and wavefunctions, the construction of linear operators, and the implications of Hermitian operators in both spaces. The conversation emphasizes the necessity of understanding these relationships for a deeper grasp of quantum mechanics.

PREREQUISITES
  • Understanding of Hilbert Space and its properties
  • Familiarity with L^2 space and square-integrable functions
  • Knowledge of bra-ket notation in quantum mechanics
  • Concepts of linear operators and Hermitian operators
NEXT STEPS
  • Study the properties of Hilbert Spaces and their completeness
  • Learn about the implications of Hermitian operators in quantum mechanics
  • Explore the mathematical foundations of isomorphisms in functional analysis
  • Investigate the role of basis transformations in quantum state representation
USEFUL FOR

Students and professionals in physics, particularly those studying quantum mechanics, as well as mathematicians interested in functional analysis and the properties of Hilbert spaces.

  • #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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