Re: Spin measurement problem


Yung wrote:
>
> Your arguement is of course correct in explaining why the result of
> measuring many times a single spin paritcle is the same as a single
> measurement of many identical spin particles. However, this is
> accurate only for a single shot type expectation values.
>

This is a common lack of knowledge of bounded measures properties. The
well-known strong law of large numbers may be generalized to an
infinite sequence of independent random variables (An) with different
expectation values assuming just some weak additionnal constraint on
the second order momentum:
* sum_n Var(An)/n^2 <+oO in a physical application ~ finite energy:
i.e. it is a physical reasonable constraint.
* sum_n <An>/n <+oO
=> sum_n An/n = sum_n <An>/n (almost surely)

> Here the main point of my question is that why the wave function
> collapse concept does not apply to a macroscopic system (say a
> collection of spin-half particles). For example, we can perform
> measurement on the magnetization of a bulk materials continuously,
but
> we cannot make continuous measurement on a single spin particle. What
> is the fundmental difference between them?

The collapse postulate always applies in a QM measurement to any
system:
We can always do a measurement, at least formally (this is the only way
to say a system has a property in QM). However, you must take care on
the meaning of "collapse" in QM. Most of the times, people confuse the
logical meaning of collapse (the QM logical formalism) with the
interpretation problem (~ philosophy).
A collapse applies to a macroscopic system each time you say a given
property of that system is true (once you have defined a measurement).
Therefore, in QM formalism, you just build a global projector for this
system property and you say that the measurement value of this
projector gives "1". Once you say value "one" of this measurement
is true, you say that the system state is in the projected state
(collapse) of this projector (only logical statements, that's all).
Now, if you say that a "local" property (e.g. a single spin) of this
system is true, in the same manner, you have implicitly defined a
"local" projector that applies to the whole system.

Now, the main problem with QM, is the non commutative projectors
properties(logic "ordering"). In other words, you cannot say that two
properties of two different projectors that do not commute are true "at
the same time" as this has no meaning in QM logic formalism. Therefore,
for your spin systems, if the global projector property commutes with
the local projector (depends on what property you measure exactly: the
total spin, the mean value of local spins etc ...), you just recover
the known results of probability. You get the same defined value for
each trial on the global projector while you get different values on
the local projector. In other words (QM words), the measured global
state is degenerated (different state vectors - different "local"
properties- lead to the same "global" measured value).
However, if the projectors do not commute, the question "what is the
local value if the global value is A" has no meaning (like saying for
an electrical signal s(t): it has a unique time value s(to) and
frequency value s(w)).

This is for the formal part. Now you have to implement these ideal
measurements with the tools you have: interactions between systems
(e.g. coils, etc...). However, at the end, you will apply a
"measurement statement" to the measurement apparatus to know the
state of the system through the interaction between the measurement
apparatus and the system.

Seratend.

.

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