Alternate formulation of Dirac Notation

[IttyBittyBit]

I was reading some more quantum mathematics, and a question occurred to me. In the current treatment of the topic, the bra-ket notation is defined as a shorthand notation for more complex mathematical operations. But couldn't bra-ket notation be defined separately from quantum physics? In other words, couldn't we start with some basic properties (things like <.|.> being a function on the cross product of function space, and <f| = |g>*, etc.) and the definition of the dirac delta, then derive all the rest of the properties (such as the integral the notation represents) from there? Forgive me if the answer to my question is obvious and I've failed to see it.

I think that would be good for two reasons:

1. It would be less abstract, allowing easier access for first-time readers,

2. Some problems in quantum mechanics could be removed from their details and studied separately.

 

[Fredrik]

But couldn't bra-ket notation be defined separately from quantum physics?

Yes. See e.g. this post. Note that this doesn't define the "eigenstates" of unbounded operators like momentum. The momentum component operators don't actually have eigenvectors, and explaining why we can get away with pretending that they do goes far beyond simply defining bra-ket notation, and is much too difficult for an introductory QM class. The discussion in this thread touches on that subject. One of Strangerep's posts contains a link to an article that explains the "rigged Hilbert space" concept pretty well.

In other words, couldn't we start with some basic properties (things like <.|.> being a function on the cross product of function space, and <f| = |g>*, etc.) and the definition of the dirac delta, then derive all the rest of the properties (such as the integral the notation represents) from there?

Cross product? I hope you mean inner product, or scalar product.