I'm Patrick, and I'm new to this forum.
However, I'm studying Applied Natural Science in germany and at the moment I'm trying to recapture my knowledge about quantum physics and quantum theory. Well, since my QT lecture was in 3rd semester I can barely remember anything and I did not do a great job in understanding the mathematics behind the theory.
Right now I'm trying to work through the lessons from lectures of the University of Oxford (I discovered they are offered in iTunes U for free) about QT. The lecturer does a great job in teaching the mathematic basics, but sometimes its not enough (for me...).
At the moment I'm stuck at the following problem:
We just discovered, that any operator, which is defined like the Hamiltonian operator: Sum over k qk |qk><qk| (is there any way to insert a latex code or something, because if I always insert formula like this I will spend a lot of time explaining what I mean ^^). We just covered that, if squeezed between a bra and a ket the result is the expectation value. And further we said that, if that Operator Qk works on the bra <qi| it gives the Eigenvalues times the Eigenket of the Operator Q. Thats what I understood, at least...
However, later we worked with this Operator Qk and "bra'd through" from the left with a bra (of course). And now my problem: How does one calculate or simplify the following:
<j|Q|i>, where Q is the Operator described above and j and i are normal bras and kets. If I substitute Q I know what happens to <qk|i> (Kroneka delta and so on), but what happens if a ket meets a bra? so |qk><qi|?
I hope someone can help me and give me more information about operators or can recommend a book where the mathematical basics are described (best for dummies ^^) well...
I'm looking foreword to many, hopefully helpful answers.