How do we associate real measurements with theoretical quantities?

[pellman]

Let us take for granted for this discussion that we are clear by what we mean by macroscopic measurements: reading a number from a ruler, a scale, a gauge of any sort. Now suppose we perform a (macro) measurement of some aspect of a real laboratory system whose result, we presume, depends on some quantum-level phenomena. Now although this generally amounts to measuring the position of something on our equipment, we then, supposedly, infer the value of a quantum observable Q. How do we do this?

Measurements of the position observable itself may be straightforward (or not, as far as I know). But what about other observables? Given, say, the observable momentum operator P, how do we conclude that our real physical measurement has anything to do with this specific theoretical quantity P? Textbook quantum mechanics says nothing about this. QM tells us the probability of "observing" the system to be in the state for a particular value of P. But it tells us nothing about how to recognize such a momentum eigenstate. Does it? What does a momentum eigenstate "look like?" It seems we need a second theory about how QM relates to the results of the lab.

Let's not try to rigorously answer this in general. Let's just understand one simple case first. Please provide one. (I myself know nothing of experimental physics.) It does not have to be momentum. Use energy, spin, whatever you like. Describe the procedure and then tell us, if possible, how we conclude that the measurement should be associated with the given quantum operator and not some other quantum operator.

Granted that if we have observed something there exists some quantum observable operator for that something, how do we know that this observable operator is the specific observable we want? That is, if we are interested in the energy E, and we perform a (macro) measurement with our real physical lab equipment, we assume there is some quantum observable Q whose eigenstate we have reduced the quantum state to by our observation... ok, but how do we conclude that Q = E?

 

[zonde]

Let's not try to rigorously answer this in general. Let's just understand one simple case first. Please provide one. (I myself know nothing of experimental physics.) It does not have to be momentum. Use energy, spin, whatever you like. Describe the procedure and then tell us, if possible, how we conclude that the measurement should be associated with the given quantum operator and not some other quantum operator.

I am trying to imagine how momentum measurement can be performed.
Say we have box with small hole and we direct electrons trough that hole. Inside the box be have electric field that deflects electrons to one side of the box and there we have arranged number of detectors. Depending how far electron gets before hitting detector on the side wall we can make some conclusions about momentum of particle. Of course this measurement will be a bit spoiled by electron diffraction at the hole.
So I think that you use reasoning from classical physics to associate measurement with some physical quantity and appropriate observable.

Polarization measurement of photons can be performed by putting polarizer in path of photon beam and measuring intensity or detection rate at different orientations of polarizer. Here polarization measurement can be spoiled if there happens to be interference for some reason.
Again it's classical physics (Malus law).

 

[Demystifier]

Textbook quantum mechanics says nothing about this.

Some textbooks do, like D. Bohm (1951), or L. E. Ballentine (1998).
If you don't have them, see also
http://lanl.arxiv.org/abs/quant-ph/0605180

Perhaps the best explanation is presented in P. Holland (1993). This is a textbook on dBB interpretation of QM, but the theory of quantum measurements explained there is mostly independent on the dBB interpretation.
See also the appendix of my
http://xxx.lanl.gov/abs/quant-ph/0208185 [Found.Phys.Lett. 17 (2004) 363]

Texts on many-world interpretation and decoherence theory also often have fine explanations of the theory of measurements.

 

[pellman]

Say we have box with small hole and we direct electrons trough that hole. Inside the box be have electric field that deflects electrons to one side of the box and there we have arranged number of detectors. Depending how far electron gets before hitting detector on the side wall we can make some conclusions about momentum of particle.

How? How do we conclude that the measurement of deflection has anything to do with eigenstates of the operator $$i \frac{\partial}{\partial x}$$ ?

Suppose that we assume there is a point particle traveling through our system with velocity v and we apply the Lorentz force, and so the deflection measurement (assuming the classical trajectory) can tell us v .. how do we get back to an eigenstate of $$i \frac{\partial}{\partial x}$$ from that?

It is true that $$m\frac{d}{dt}\langle \hat{x}\rangle = \langle \hat{p}\rangle$$, but this is only for expectation values. Whatever the expectation value may be, a single measurement can still take on any value.

Certainly we can perform many measurements and determine the momentum-representation of the wave-function from the statistical distribution. But this amounts to performing many position measurements, inferring the position representation of the wave-function, and taking the Fourier transform. In the case of the 2-slit experiment with an incoming beam of definite momentum we could use this method to determine that momentum from the interference pattern.

But still this does not tell us how to perform a single momentum measurement--that is, how to relate an observation to eigenstates of the momentum operator.

 

[pellman]

Some textbooks do, like D. Bohm (1951), or L. E. Ballentine (1998).
If you don't have them, see also
http://lanl.arxiv.org/abs/quant-ph/0605180

Thanks, D. This looks good and I plan to read through it carefully. I especially like the analogy with the Monty Hall problem. But after skimming through it, I don't think it covers my question.

I will look for Holland's book.

 

[peteratcam]

The momentum measurement example you use is a bad one because it involves slamming the particle into a wall of detectors. You don't need any quantum reasoning to conclude that the measurement disturbs the state of the system!

To think about the ideal quantum measurement, you need to do the minimum possible to extract the value of the observable being measured. The minimal model is to couple the system to a measuring device with an interaction hamiltonian which is a direct product of the observable of interest and the operator which moves the pointer on the measuring device.

This procedure necessarily leaves the system in an eigenstate of the observable associated with the eigenvalue which the measuring device points to.

 

[Demystifier]

But after skimming through it, I don't think it covers my question.

I will look for Holland's book.

Good, because I think the Holland's book better covers your question. It constructs (a toy model for) an interaction Hamiltonian appropriate for measuring any given observable. Moreover, it explains how essentially the same interaction Hamiltonian should be used for both classical and quantum measurements. Thus, in order to build a good measurement device, you don't need to know in advance that the system obeys quantum laws. In particular, you don't need to know that momentum has anything to do with the derivative operator (times i). The crucial point is that quantum operators in the Heisenberg picture obey the same equations of motion as the corresponding classical quantities.

 

[pellman]

peteratcam and Demystifier, your posts' mention of the interaction Hamiltonian got me thinking and I sort of start to see how it comes together. Sounds like the Holland book will do the trick.

Thanks, all.

 

[zenith8]

peteratcam and Demystifier, your posts' mention of the interaction Hamiltonian got me thinking and I sort of start to see how it comes together. Sounds like the Holland book will do the trick.

Thanks, all.

Holland's measurement stuff is summarized in Lecture 4 - "The theory of measurement and the origin of randomness" - of this http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" .

 

[peteratcam]

For some previous ramblings of mine on this, see https://www.physicsforums.com/showthread.php?p=2618661