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I'm studying about Quantum Information and Quantum Mechanics. Right off the bat it's stated that we deal with vectors and operators in Hilbert space. But then I see the definition of a Hilbert space and several questions pop up in my head: like why do we need a complete vector space? Why should it have an inner product structure? etc.

One way to learn could be to not worry about these questions, accept the mathematical structure at face value and methodically follow the consequences of these assumptions.

But I'd prefer to understand the rationale behind each and every assumption about the mathematical structure. Specifically,

  • Why is it not possible to use a weaker set of assumptions?
  • From the physical point of view what's the point of each assumption, etc.

My question is, how do I go about answering the bulleted questions as I learn QM?

Qmechanic
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user9343456
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    Possible duplicates: https://physics.stackexchange.com/q/20822/2451 , https://physics.stackexchange.com/q/41719/2451 and links therein. – Qmechanic Mar 15 '21 at 20:41
  • Why do we use Eigenvalues to represent Observed Values in Quantum Mechanics? – mmesser314 Mar 15 '21 at 21:23
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    Even if your space is not complete, it is a mathematical theorem one can construct a larger space called the completion which is complete. Even if there was no physical need for completeness, there is no harm in having it and not using it, a bit like having a hard drive with more memory space than you will actually need. Also, for an explanation of math hypotheses from physical principles, the book by Dirac on the principles of QM is hard to beat. Likewise in QFT with vol 1 of the books by Weinberg. – Abdelmalek Abdesselam Mar 15 '21 at 21:33
  • @Qmechanic : Thanks for the links! I did go through them but my question pertains to the general way of finding the reason for a mathematical assumption. The linked questions deal with specific examples of such assumptions – user9343456 Mar 15 '21 at 21:35
  • @AbdelmalekAbdesselam: Thanks! Will look at that reference. But isn't the Dirac book very old and potentially outdated? Some of the mathematical underpinnings may be based on more modern work? – user9343456 Mar 15 '21 at 21:37
  • I said "is" hard to beat, present tense. – Abdelmalek Abdesselam Mar 15 '21 at 21:38

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