The mathematics of measurement?

[homology]

So when we try to "observe" an observable through an experiment we, in the process, "collapse" the state vector into one of the eigenstates of the observable. So my question is, how do we represent such an experiment mathematically? Are there "measurement" operators that do this collapsing?

Thanks in advance,

Kevin

 

[speeding electron]

Yes. In quantum mechanics the act of observing any property of a system is represented by an operator. The values of these observables are always eigenvalues of the corresponding operator, even though the system may not be in an eigenstate of the the operator. Thus by measuring the property we have "forced" it to enter some eigenstate, and we detect the corresponding eigenvalue.

 

[homology]

Yes. In quantum mechanics the act of observing any property of a system is represented by an operator. The values of these observables are always eigenvalues of the corresponding operator, even though the system may not be in an eigenstate of the the operator. Thus by measuring the property we have "forced" it to enter some eigenstate, and we detect the corresponding eigenvalue.

okay, but how does this work? Say you have |psi> and you want to observe Q, an observable, how does |psi> end up as |q>? (let's say that |q> is the eigenstate of Q that the system ends up in after measurement and that it corresponds to the eigenvalue q)

Kevin

 

[Haelfix]

It depends what interpretation of QM you use. In the Copenhagen picture it jumps states, more or less instanteously.

But you can make it time dependent if you want, which is what they do for the decoherence picture. It becomes a many body problem unfortunately, so the details become nontrivial.

 

[Eye_in_the_Sky]

In the "standard presentation" of QM, the degrees of freedom corresponding to the measuring device do not appear in the formalism at all. We have a Hilbert space corresponding to the quantum object, but the measuring device is something which "selects", according to the probabilistic rules, "projections" (normalized back to unity) into orthogonal subspaces. In brief: the measuring device and quantum object are not on the same "footing" in the theory ... at least with regard to the "standard presentation".

From another angle, this "schism" presents itself in the form of two types of "evolutionary" processes: (i) unitary evolution of the object state-vector in the absence of any "measurements"; (ii) reduction of the object state-vector in the presence of a "measurement".

Here are two distinct approaches towards a resolution of the problem, found at "opposite" ends of the "spectrum":

(1) Say that the above "schism" is merely an "accident" of the presentation. By a suitable "reformatting" of the presentation, one can arrive at a consistent account of the measuring process in which the measuring device and quantum object are represented on the same "footing".

(2) Say that the above "schism" is already "built" into the formalism. In order arrive at a consistent account of the measuring process, one must discover a new theory T for which the "standard presentation" of QM is a "special" or "limiting" case.


According to the point of view afforded by (1), one - at least initially - has the expectation that "reduction" of the object state-vector has an equivalent representation as "unitary evolution" on a "larger" Hilbert space (which incorporates the measuring device) followed by a "tracing-out" of the non-object degrees of freedom. This expectation, in its most simple formulation, however, turns out to be a mathematical impossibility.

One might then consider settling for the lesser expectation of specifying a unitary evolution (again) on a larger Hilbert space followed (again) by a "partial tracing-out", now, for which the overall effect is to map the "restricted" state |psi><psi| of the quantum object alone to a mixed state

Sigma_i { p_i |a_i><a_i| }

in the object Hilbert space, where the |a_i> are the eigenvectors of the measured observable (assumed, for simplicity, to have a nondegenerate (discrete) spectrum) and p_i = |<a_i|psi>|². The problem here, however, is that the quantum object is in a state described by an "improper" mixture.

Continuing along this branch, with a more careful analysis, one includes "pointer" states of the apparatus into picture, and through unitary evolution gets the required correlations between object and pointer, but is still at loss for "reduction" of the joint object-pointer system ... and one is right back a "square one" again.

I have heard about the inclusion of "environment" degrees of freedom into the picture, and so called "quantum decoherence", … but I don't see how this can give anything but an "improper" mixture.

There are many, many other approaches within the context of (1) above, all of which (at least, to my knowledge) fail to give an acceptable account.


Regarding the point of view afforded by (2), all versions of a "new" theory T that I have come across are no more that "extensions" of QM, introducing nonlinear and/or stochastic elements into the Schrödinger equation. If you want to feast your eyes on one such prospect (just for the fun of it), then here:

Stochastic extensions of the Schrödinger equation have attracted attention recently as plausible models for state reduction in quantum mechanics. Here we formulate a general approach to stochastic Schrödinger dynamics in the case of a nonlinear state space of the type proposed by Kibble. We derive a number of new identities for observables in the nonlinear theory, and establish general criteria on the curvature of the state space sufficient to ensure collapse of the wave function.

http://theory.ic.ac.uk/~brody/DCB/dcb24.pdf

... I haven't seen or heard of any sort of theory T which completely abandons the QM formalism and opts for something which would only give QM back after the non-object degrees of freedom are properly "averaged-over". (Just how a Hilbert space, in this way, can "pop" out of something that bears no resemblance to one at all remains to be understood.)

 

[reilly]

Then there's the simple minded 'shut up and compute' like approach, which treats measurements as eliminating uncertainty -- like the difference before and after a coin toss. Sir Rudolf Peirels, a Nobelist, hence not too simple minded, is an advocate of this approach. Treat QM and classical probabilities as expressions of human uncertainty; the collapse then becomes physical, the process by which one's brain goes from "Could be A or B or..." to "It is A..."

Regards,
Reilly Atkinson

 

[selfAdjoint]

English and other languages have a "subjunctive mood" to handle this. The giant in Jack and the Beanstalk sang "Be he alive or be he dead, I smell the blood of an Englishman." Could apply to Schroedinger's cat.