What does the hat notation imply?
It's a very usual notation in Quantum mechanics: Did you notice the observable also weared one? Well, the hat just means "this is an observable". So yes, (position) is also an observable. Whitout further knowledge the only explanation that can be given is "you can certainly observe (measure) the position of a particle".
If you know about Fourier transform, I can expand it - and when you have all the basis, it's fairly trivial that (or in 1D) and (linear momentum) are observables.
Please post any questions you have. It's the best way to motivate me to contribute. --Jorge 15:21, 3 February 2007 (UTC)
Why "I"? 
Why is this called Quantum Mechanics "I"? Which criteria do we follow to separate on courses? --Jorge 17:08, 3 February 2007 (UTC)
Wavefunction as 'the basis of quantum mechanics' 
It is arguable that the wavefunction is the basis of QM. Aside from this, it is more logical in practice that students should deal with Dirac's notation for quantum systems, which can be thought of as an amalgam of Schrödinger's and Heisenberg's approach to QM. In the real world, hardly anyone talks about wavefunctions anymore. They're simply not as useful!
States should be represented as vectors in a vector space:
Then the expectation value of some measurement (say of position using the position operator ) is given by
Would you agree that this is the way to go?
Dmonkey 00:06, 6 February 2007 (UTC)
- I can talk you about my experience: A few years ago I knew very well about Fourier transforms, and (from it) about identifying functions with vectors of infinite and continous dimension (and transforms with matrix multiplication), and also about wavefunctions, but haven't been thaught yet about what those <s, |s and >s meant. Every time I saw that notation in a WP article I couldn't understand a single equation!! I even didn't link the different <s, |s and >s with what I know is "their content", and thougth they were some kind of mysterious operators, rather than mere special parenthesis. When they tougth it to me, I had no more problems with them, and of course I also "fell in love" with the burden the notation removed from me, and the smartness of identifying a vector (and matrices) without relying on its coordinates (which depended on the basis, e.g. x, or p).
- I agree with you that the state space (although not necesarily the quantum one) should be presented and used earlier, and that the Dirac notation should also be presented before QM, in a Mathematics course, so the student doesn't have to discover by himself that the notation isn't something pertaining to QM, but rather mathematical language. (And a language that contributes to alienate even more QM to the new student). The notation can be presented as earlier as in first year's Algebra, after Quadratic Forms (associating bras with row vectors, kets with column vectors, operators with matrices, and yuxtaposition with matrix multiplication - at this point the student knows yet about diagonalizing and bases of eigenvectors). A year or two later, after Hermitic
operatorsmatrices are taugth, it should be showed why converting kets to bras (transposing complex vectors) implies conjugating the complex scalars within them.
- We must be careful that the notation is well explained and understood, because no material on QM is comprehensible to who doesn't understand it. I, however, disagree with the removal of the wavefunction, since it is a lot easier for the student to understand and believe interference if he has acces to the image of a wave in space. It is also more suited to explain the current of probability than kets, for the explicit graphical interpretation it provides.
- Let's see what everybody else believes. --Jorge 02:10, 6 February 2007 (UTC)
Jorge, I have never seen a treatment of the nomenclature at the level you describe above. I think a math refresher course aimed at reviewing/showing the math concepts and tying them to the nomenclature you describe above would be a very useful course/module for people like me twenty years away from the advanced math concepts we handled in College. Perhaps such an outline of lesson plans could rely heavily on Wikibooks. I think their math shelves are fairly advanced. Mirwin 05:50, 6 February 2007 (UTC)
- http://en.wikibooks.org/wiki/Waves/Fourier_Transforms This has twelve pages of pretty good review of math related to waves. Probably should link to it anywhere someone might need a review of waves treated mathematically. Mirwin 05:56, 6 February 2007 (UTC)
state machines in QM 
Is there value in discussing state machines in the QM study guide? This would include not only finite state machines but also abstract state machines. ASMs provide an analysis of the concept of state, which can be beneficial in some discussions. FSMs have the advantage of simplicity. I suspect that both could be used in the foundations of QM. Can you provide references to previous publications or wiki discussions of state machines in QM? Thanks. Telecomtom 07:04, 27 September 2011 (UTC)