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How should Dirac notation be understood?

If vectors $|\vec{r}⟩$ and $|\vec{p}⟩$ are defined as $$ \hat{\vec{r}} |\vec{r}⟩ = \vec{r} |\vec{r}⟩ \\ \hat{\vec{p}} |\vec{p}⟩ = \vec{p} |\vec{p}⟩ $$ then one can see that products like $$ ⟨\vec{ …
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