Can someone with access to Shankar's QM book help me (vector spaces)?

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The discussion centers on the interpretation of scalar multiplication in quantum mechanics as presented in Shankar's "Principles of Quantum Mechanics." Specifically, the assertion that 1|V⟩ = |V⟩ for all vectors |V⟩ is examined. Participants clarify that this relationship is typically treated as an axiom rather than derived from other axioms. The consensus is that while Shankar implies this relationship, it is indeed an established axiom in the context of vector spaces.

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On pg. 2, Shankar seems to just assume (I'm guessing) that [itex]1|V\rangle = |V\rangle[/itex] for all vectors [itex]|V\rangle[/itex] when he does the exercise at the bottom of the page. Is this true, or is it possible to prove [itex]1|V\rangle = |V\rangle[/itex] from the axioms he lists?
 
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I think that this is what the author means by "do what is natural". Usually, however, 1|v> = |v> is just listed as an axiom ("there is a scalar multiplication r|v> with 1|v> = |v> and 0|v> = |0>").
 

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