5. What are “off-diagonal elements”? I tried to get to grips with off-diagonal long range order, when I was looking at the work of Mazur and Chapline, but got no further than realizing it had something to do with the organization of particles in a quantum state.
Ok above we defined classic physics as a state of QM so now we have to deal between pure and mixed states. Just like in real world we have mixture of ice, water and water vapour the states don't exist in isolation.
I am going to do a separate post in a bit on this because it's interesting lets just see how many bits I need to tidy up first.
For now what I would like you to look at is the example in the following link example Beam A prepared in superposition and Beam B organized 50% spin up and 50% spin down. See if you can follow why the result.
http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/DensityMatrix.htm 6. My understanding of the Hamiltonian is that it quantifies the total energy (kinetic and potential) of the particles in a system, but beyond that it is a mass of mathematical esoterica.
That is correct and the mathematics basically reflects observation. It wouldn't be valid to pose an answer gravity is only valid when you get more or less than X amount of matter or the universe is X. The mathematics of gravity observation and subsequent calculation does not indicate a limit and so it is not valid, the same is true of QM.
So it's is saying as QM is involved in energy it clearly goes from the smallest size to the largest size and over all time scales.
in many cases, the quantum-classical transition can actually be observed (at the predicted place)
Could you give an example of such an observation, please.
Just google "macro quantum entanglement" there are now thousands of them. All these experiment are large macro objects you would clearly call classical yet you can make them behave in a manner that only QM predicts.
7. I’ve no idea what the preferred basis vectors of the Hilbert space are, but this bit seems to be saying that in the classical world what we observe is our reality, and it’s not going to just vanish.
I think I get the bit about consistent histories, but the link between Schrödinger’s cat and the Hamiltonian eludes me.
You almost got it spot on except add in that the world exists with or without you it's just in a quantum superposition. The Schrödinger’s cat reference is needed because of Many World Interpretation and for that I will refer you to this reading
http://plato.stanford.edu/entries/qm-manyworlds/8. I thought I was doing quite well with this one until I got to the “diagonalization of the density matrix in the preferred basis”.
Lost!
Yeah that's back to point 5 and the density matrix and mixed states and we will cross that off in detail next if you like.
If you are happy with that we will try and get you understanding mixed and pure states and the density matrix it's not as hard as you may think.
