Measure destroys the state?

[sanctus]

COnsider the Stern gerlach experiment:
1. Measure the spin along the y axis-->50%up 50%down
2.measureright away after the spin along the x axis of let's say the particles with spin up on the y axis--> 50% spin left 50%spin right
3. Take the particles with spin left and measure again spin on the y axis-->50%up AND 50% down (while classicaly 100% should be up).

In this case the measure of of spin-x completely changed the spin-state of the system. My question is: is there a quantity/mathematical construct/... that tells us how much the measure of a state changes the other state in which the system was in.
A drastic answer would be if the two observables commute they don't influence each other, otherwise do Do influence each other, my question is how mcuh do they influence? Is it always total if they don't commute?

[Qfwfq]

The mathematical construct for spin 1/2 is an operator having the Pauli matrices as components (it represents the 3-vector angular momentum). For a relativistic treatment the Dirac gamma matrices are appropriate.

Originally Posted by sanctus

A drastic answer would be if the two observables commute they don't influence each other, otherwise do Do influence each other, my question is how mcuh do they influence? Is it always total if they don't commute?

It depends on how much they don't commute! The commutator of two operators is an operator,it might be very different, or not much different, from zero

How much a measurement influences the state depends on how the measurement is performed. QM formalism only places a minimal influence which would be the case for a so-called ideal measurement of 1st species. This means that the new state is given by applying the projector assoicated with the eigenvalue that was the result of the measurement, to the previous state, and then normalizing.

Originally Posted by sanctus

(while classicaly 100% should be up)

Hmmm, classically, it wouldn't even be "either up or down"

[Little Bang]

Qfwfq, if you could explain spin for a 12 year old kid I would forever be in your debt.

[Qfwfq]

Perhaps I could, for a 12 year old kid that can understand SU(2, C) and its homomorphism onto SO(3, R).

[sanctus]

Thanks Qwfwq, but you were very mean to little bang

Little bang, of course Qfwfq is right to understand spin you must know group theory, but speaking with hands (which as usual is wrong) electron has spin 1/2 means just that it turns around itself with angular momentum (that measures how much it turns) 1/2*h were h is the planck constant... this is obviously wrong because how can a point-like particle turn around itself? But that's the classical idea behind.

[sanctus]

And qfwfq, what does big and small mean in eigenvalue of a commutator? I mean what is the scale?
I mean one should be able to say that if we take an ideal measure and the commutator is x, then we should be able to say x is bigger (always when applied to state, logically) than some threshold value and therefore the system won't have any memory. Or x is smaller than the threshold and then we know that instead of having 1/2 &1/2 we get let's say 1/3 &2/3.

[C1ay]

Originally Posted by Little Bang

Qfwfq, if you could explain spin for a 12 year old kid I would forever be in your debt.

Try here and here.

[Little Bang]

Thank you C1ay that was exremely helpful, and thank the sanctus.

[Little Bang]

C1ay, the Stern-Gerlach experiment raises about a hundred questions to replace the one that I asked, again thank you.

[Qfwfq]

Originally Posted by sanctus

And qfwfq, what does big and small mean in eigenvalue of a commutator?

Oh, yeah, right! It depends on the metric you choose in the Hilbert space, or in the space of operators. Of course the metric is usually set by the norm which a Hilbert space has anyway by def, and this is quite a hint for your ops etc.

Alright, sorry for being mean to LB, maybe it's 'cause the name seemed familiar, I must have recently seen it around here.

Originally Posted by Little Bang

the Stern-Gerlach experiment raises about a hundred questions to replace the one that I asked.

Questions are a great way of learning. Keep asking them Little Bang.