Look, if a asystem is in a state,which is superposition of eigenstates, and expressed as a linear combination of basis kets multiplied with some coef. the probability for you to measure an eigenvalue, is equal to the square of coef. of the corresponding eigenstate, this is the simple case of non degenerative and discrete basis
But in my example, I am having particles with pozitive z-spin direction and I want to measure the x-spin. ANd I am asked about the result of this experiment, which are the states present
I don't know what "usually" means - you just told me that if you want to measure an operator, you need to figure out how to express your state as a superposition of eigenstates of that operator
And the best part is when the professor hits you with that : You don't know about groups? SO3 symmetry and Clifford algerbra? ooohhh you should have done that in high school
@ACuriousMind later on in the same sub section of the example, I am asked :Is the state of the system after the measurement a pure state or a mixed state and why?
sure, but if you just know "a x-spin measurement has occured, but I don't know what the result was", then it's perfectly reasonable to say that the state is currently a mixed state with 50% x-spin up and 50% x-spin down. if you know the result, then it's just the pure state corresponding to that result
@ACuriousMind Usually when you express the density matrix for a system, that is an indicator of whether the system is in a statistical mixture of states, or in a pure state. But then if you have to give the density matrix of N-particles, N system, how do you do it?
Usually in my class we mostly confine ourselves to one system
I think the exercise at hand, is TOTALLY unrelated to what we do in the lectures, since the person who does the lectures and the one who does homework exercises are two different people, that do not interact at all
that's why
right now, I feel like I am blindfolded and asked to walk over a narrow bridge
but out of curiosity in the same example that we were discussing, I am asked to give the density matrix of a system of N particles, from which n are in the |z,-> state and the rest in the |x,+> state
Also how am I supposed to take this seriously when one object is called the tractor bundle
"The term tractor is a portmanteau of "Tracy Thomas" and "twistor", the bundle having been introduced first by T. Y. Thomas as an alternative formulation of the Cartan conformal connection"
@ACuriousMind In the micro canonical ensemble, the microstates are eigenstates of the Hamiltonian. But what about in the case of a canonical ensemble? They shouldn't,right? Since energy changes