Does this equality have a name?

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SUMMARY

The equality discussed in the forum, represented as llψ+øll^2 + llψ-øll^2 = 2llψll^2 + 2lløll^2, is identified as the parallelogram law in the context of linear algebra. This law holds geometric significance and indicates that if a norm satisfies it, the polarization identity can be utilized to define an inner product. While the user initially associated it with quantum mechanics (QM), the discussion clarifies that this concept is fundamentally rooted in basic linear algebra principles.

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  • Understanding of Hilbert spaces
  • Familiarity with wave functions in quantum mechanics
  • Knowledge of norms and inner products in linear algebra
  • Comprehension of the polarization identity
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  • Study the properties of Hilbert spaces in quantum mechanics
  • Explore the implications of the parallelogram law in linear algebra
  • Learn about the polarization identity and its applications
  • Investigate the role of norms in defining inner products
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M. next
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Hey there,

I was wondering if this equality in QM has a name or not.

It goes as follows:

llψ+øll^2 + llψ-øll^2 = 2llψll^2 + 2lløll^2

where: ψ and ø are wave functions in Hilbert's space.

If it doesn't specifically have a name, what is its significance in QM?

Thankis
 
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It's just the parallelogram law. It has the usual geometric meaning and moreover if a norm satisfies it then the polarization identity can be used to define an inner product using the norm. The converse of course is trivial. This has nothing to do with QM it's just basic linear algebra.
 
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Oh, okay. I see! The reason, I said QM, is because I saw it in QM notebook. Thanks!
 

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