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.

 

[Entropia]

I appreciate your curiosity and critical thinking in questioning the traditional formulation of Dirac notation. It is always important to explore alternative approaches and see if they can offer any advantages or insights.

In response to your question, I believe that while it is possible to define the bra-ket notation separately from quantum physics, it may not necessarily be as useful or intuitive for understanding the concepts and operations in quantum mechanics. The bra-ket notation was specifically developed by Dirac to represent the fundamental principles and mathematical operations in quantum mechanics, and it has been widely adopted and used by physicists to simplify and communicate complex ideas.

Furthermore, the bra-ket notation is not just a shorthand notation, but it also represents the inner product between two vectors in a Hilbert space. This is a crucial concept in quantum mechanics, as it allows for the calculation of probabilities and expectation values of observables. Deriving the properties of the notation from basic functions and the Dirac delta may not fully capture the essence of the notation and its significance in quantum mechanics.

However, I do agree that exploring alternative formulations of Dirac notation could potentially offer new insights and perspectives. This could be a fruitful area of research for mathematicians and physicists alike. But for now, I believe that the traditional formulation of Dirac notation remains the most effective and widely accepted approach for understanding and applying quantum mechanics.