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I would love to read this paper, however I'm stumped in the very first equation.

[itex]|\psi \rangle =\sum|a_k\rangle\langle a_k|\psi\rangle[/itex]

I actually have some basis in the ideas behind QM, and a general love of math, but I just have no idea how to approach it mathematically. I have tried to look up the Dirac notation before and got some answer, but it was as equally hard to understand as my original question. This seems to be my main problem with even beginning to try and teach myself some semblance of QM, all I find as explanations to my questions are more confusing than my original question!

I wouldn't be against buying a book, but I suspect that any book I buy will have equations similar to this and I will be in the exact same predicament, but a 100$ poorer. I don't think this math is hard, I just don't know what it is. Like if you tried to teach yourself Einstein Notation but had no idea what a summation over i was; it just wouldn't work. If I could get some basic understanding of the notation used throughout the linked paper, I think I would be well on my way to being able to more fully understand the real quantities of QM and not just some 'kind of' answers that I am forced to search for because of my lack of mathematical understanding.

A little about me background: I just graduated with my masters in civil - structural engineering, so I have some basic understanding of tensors and higher level analysis etc, I just have never needed to be exposed to the higher mathematics which are so crucial for things like QM and GR (And finally, string theory or LQG). I hope to be able to pursue a degree in theoretical physics next, as my currently chosen career progresses, but in the mean time I would love to be able to pursue these topics without being completely overwhelmed in simple (or maybe not so simple?) notation.