zonde said:
Ok, you are identifying definite state with definite (pure)
quantum state. I don't.
Hmm, may this is exactly the same problem in my discussion with
@stevendaryl.
Well, the difference between a macroscopic object, such as a measuring device, and a microscopic object, such as an electron, is that for any given macroscopic state (what we would intuitively, pre-quantum mechanics, think of a state, such as "the readout shows the number 32" or "the pointer points to the left" or "the left light is on") there are many, many microscopic states that correspond to it.
I don't have the mathematical sophistication to accurately describe the situation using quantum mechanics, but perhaps it's something like the following:
The complete system perhapse can be described by three variables: ##|s, S, j\rangle##, where ##s## is the observable corresponding to the system being measured (an electron's spin, maybe), ##S## is the corresponding value of the "pointer variable", and ##j## represents all the other degrees of freedom.
To make it both simple and definite, we will assume that there are two possible values for ##s##:##u## and ##d##, and three possible values for ##S##: ##0, U, D##. There are many (possibly infinitely many) values for the other degrees of freedom, ##j##.
To say that the pointer accurately measures the z-component of spin is to say something like the following:
- If you start in the state ##|u, 0, j\rangle##, and you allow the system to evolve, then you will end up most likely in a superposition of the form
- ##\sum_k c_{ujk} |u, U, k\rangle##
- If you start in the state ##|d, 0, j\rangle##, and you allow the system to evolve, then you will end up most likely in a superposition of the form
- ##\sum_k c_{djk} |d, D, k\rangle##
- It follows from the linearity of the evolution operator that if you start in a superposition of the form ##\alpha |u, 0, j\rangle + \beta |d, 0, j\rangle##, then you will end up in a superposition of the form ##\alpha \sum_k c_{ujk} |u, U, k\rangle + \beta \sum_k c_{djk} |d, D, k\rangle##
By "end up", I mean applying the evolution operator ##e^{-iHt}##.
Now, what I'm a little hazy about is how to deal with irreversibility in quantum mechanics. A measurement is irreversible. I don't know whether the irreversibility is completely explained by the fact that the final state is massively degenerate, compared to the initial state. That's the classical explanation. I don't know whether anything we have to say hinges on the interpretation of irreversibility.
Anyway, given the above assumptions about the evolution, we can always add classical uncertainty, by letting the initial state be an incoherent mixture of
##\alpha |u, 0, j\rangle + \beta |d, 0, j\rangle##
for different values of ##j##.
