A Quantum Mechanical model of measurement

peteratcam

Following a short discussion in another thread, I promised to start a new thread for this.
I asked myself the question "how would I describe the process of measurement in a fully quantum mechanical way?" and came up with this.

1) A model for the general measuring device
The aim is to have a QM system which describes a measuring device, with a finite dimensional space of states so that the whole thing can be written down with finite matrices.

I imagine a dial, with a pointer (see picture coming soon).
Any position of the pointer corresponds to a different quantum mechanical state for the pointer (much like a 1-d particle on a ring).
To make things easier, discretise the possible positions of the pointer, so that we have N possible pointer states, all equally spaced.
Assume that the Hamiltonian for the dial is H = 0 so there is no dynamics at all.
So the measuring apparatus is an N-state quantum system.

2) Operators for the measuring device.
There is only one operator we need for the measuring device: the operator that moves the pointer. By analogy with the momentum operator, we want the generator of pointer rotations, which I will call R.
Define it so that the unitary matrix exp(-iR) advances the pointer clockwise by pi radians.

So far so good, we have a device which we can start in the state pointing to 0, and we have an operator which rotates the pointer round by whatever angle we want.
We can calibrate the device by letting the maximum reading be 'm', corresponding to pi radians.

3) Performing a measurement of the number '4'.
Suppose we want to measure a number, called y, using our device, the way to do it is:
In units where hbar = 1, change the hamiltonian to H = R y/m for a duration of 1 unit of time, then change back to H = 0.
The time evolution operator will be exp(-i(y/m)R), which will swing the pointer round by (y/m) of a half circle.
Eg, we might choose m = 20, with N = 40, so the states on the dial run from -19,-18,....0,...19,20.
Acting on the initial state with exp(-i(4/20)R) will move the pointer to point at '4'. ie, we have measured the number 4 using our measuring apparatus.

[EDIT at 9am GMT: Despite numerous replies already, I will continue from here, but address concerns in a reply below]
4) Measuring the properties of another quantum system
So far we have shown how to measure a number on our dial. Now we want to measure the properties of another system.
Generalising what we did to measure a number, the following procedure will 'measure' the value of observable [tex]\hat O[/tex]. (I've put the hat on it so it doesn't look like zero).
The full hamiltonian, which acts on the product space [tex]{\mathcal H}_M\otimes{\mathcal H}_S[/tex] will
initially be [tex]H = \mathbf 1 \otimes H_S[/tex] since our measuring device has H=0.
Now turn on the measurement hamiltonian. Again, 'm' sets the maximum reading on the dial.
[tex]H_{meas} = \frac{1}{m\tau}R\otimes \hat O[/tex]
After time tau, set the measurement hamiltonian back to H = 0. Ideally, our clever experimentalist is able to make 'tau' very short, so we have an instantaneous measurement.

4a) Explicit example, measuring the energy of an harmonic oscillator.
Suppose the maximum reading on my dial is 3, and I have N = 12. Label the states of the apparatus by [tex]|-2.5\rangle,|-2.0\rangle,|-1.5\rangle,\ldots|0.0\rangle,\ldots,|2.5\rangle,|3.0\rangle [/tex].
Also, suppose the system we are interested in is a Harmonic oscillator, currently in the state
[tex]\frac{1}{2}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle+\frac{1}{2}|2\rangle[/tex].

Therefore the inital state of the composite system is the product state:
[tex]|0.0\rangle \otimes [\frac{1}{2}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle+\frac{1}{2}|2\rangle][/tex]
Now the experimentalist comes along, and attaches his measuring device to the harmonic oscillator in such a way as to measure the energy. The effect of this is that the measurement hamiltonian becomes
[tex]H_{meas} = \frac{1}{m\tau}R\otimes H_s[/tex].
After time tau (which is much shorter than the period of the oscillator), he disconnects the measuring device.
The final state of the composite system is
[tex]\frac{1}{2}|0.5\rangle \otimes |0\rangle+\frac{1}{\sqrt{2}}|1.5\rangle \otimes |1\rangle+\frac{1}{2}|2.5\rangle \otimes |2\rangle[/tex]
The apparatus state is now entangled with the measured state.
*So the readings on the apparatus are the eigenvalues of the observable.
*Moreover, the state of the system is the eigenvector for the measured value.
Crucially, the apparatus states are orthogonal, so there is no interference possible between the two options for the system.
Applying the probability interpretation to the entangled system is consistent with applying it to just the system we measured originally, except the measuring process and apparatus are described in a QM way too now.

I believe the points with (*) were the points in the other thread under point 2, which I wanted to address.

5) Generalisations.
This is completely general. My measuring apparatus will measure any observable you like and in performing the measurement, it is clear why only eigenvalues are allowed, and why eigenvectors is what you get afterwards.

[edit: more may need to be edited if it is unclear, but I thought it better to have all of it in one place if there is going to be a discussion]

humanino

There is no big breakthrough in this attempt, it is just decoherence

SpectraCat

I don't necessarily think so ... this was spawned from a discussion in this thread where someone brought up the point (often made) that decoherence only solves part of the measurement problem, i.e. how to provide a description of the apparently irreversible resolution of a quantum state into the eigenstate of an operator for which a measurement was being performed, that is consistent with the inherently time-reversible laws that govern the evolution of quantum systems. Decoherence does indeed provide such an explanation, however, as far as I know it does not explain how that resultant eigenstate is registered by the measurement device.

This thread was started by peteratcam to give his perspective on that point, and I for one am eager to see it.

conway

I am pleased to agree with Spectracat (for a change). Could we not refrain from the scoffing and give the man a chance to make his point?

humanino

I do not agree that Bell's FAPP purpose argument holds. There is no established general mathematical theorem to prove either side of the argument. I will let Spectracat answer, as indeed it is not a short and straightforward argument, but it is discussed at length in the decoherence literature, for instance Omnes "The interpretation of quantum mechanics" (Princeton 1994) 7.7 where one can find the core of the discussion, preceded and followed by many explicit calculations and numerical estimates. Decoherence has also been measured in the lab, and agree with those estimates.

Bell's arguments holds at a quite general level, if one is willing to accept as a postulate that every observable can be measured in principle. Since it is quite an interesting discussion, but too long for me to go into right now, I will just copy below some parts of Omnes' conclusion in chapter 12 section 7 to the question "can one circumvent decoherence ?"

Fredrik

I thought you were going to end up with something very similar to von Neumann measurement theory, but the quote is a major difference that ensures that your model isn't compatible with QM. The Hamiltonian isn't changed at random into one of several possibilities by the interaction between the measuring device and the system.

meopemuk

I wrote some (possibly relevant) thoughts about the QM measurement problem in http://physicsforums.com/showpost.ph...41&postcount=4

Eugene.

peteratcam

(I have now finished the original post)

I'm not claiming a big breakthrough (perhaps a pedagogical one at most). But I've read some of the introductory reviews on decoherence and I didn't feel that it addressed what I've tryed to address by the toy model above. I feel that the decoherence program is worried more about irreversibility, but *all* I wanted to demonstrate to myself with the model above is that the mysterious postulates that "we always measure an eigenvalue of an operator, and in doing so, collapse the wavefunction to the associated eigenvector" can be understood through quantum dynamics of composite systems.

peteratcam

Fredrik's reply was written before I had finished the exposition in the original post, but I will reply to it.
My model is compatible with QM. Unitary evolution of states according to a hamiltonian - what more do you want!
I don't claim the Hamiltonian is changed *at random*. Rather, a clever experimentalist arranges his levers, electrical wires, and filters, such that the hamiltonian which describes the interaction between the system and the apparatus is one which will measure the desired outcome. It is what happens all the time in labs.

Fredrik

It sure sounded like you were describing a random change of the Hamiltonian. But now it sounds even worse, like an experimentalist can choose what outcome to get.

peteratcam

My word 'outcome' was a terrible choice. I mean to say that in an ideal world, an experimentalist can choose what to measure (be it energy, position, etc), but obviously he can't choose the result of the measurement. Choosing what property to measure equates to choosing a hamiltonian for the interaction between system and apparatus.

Dmitry67

So why outcome is different in identical setups, when people measure the same observables?

peteratcam

Because that is what QM predicts, I don't really know how else to answer that. My model was to make an explicit example of the quantum dynamics of a measurement process, which I believe it does - unless there are any objections...

Dmitry67

No, QM itself does not predict it, because QM math and Decoherence are deterministic.

Now, when collapse int are ruled out, we have non-collapse interpetations (+macroscopic realism aka shut up and calculate).

Decoherence removes non-diagonal elements, but it does not tell us what outcome is "real". All non-collapse interpretations must provide a mechanism to resolve that issue. Using hidden variables (BM) or by claiming that all outcomes are real (MWI).

You can not, in principle, derive non-deterministic view of the world from deterministic QM without some sorts of additional assumtions (well, in MWI these assumptions are null, but anyway...)

Dmitry67

I tried to pinpoint the problem:

Saying "aha, I have calculated the result, now let’s apply the probability interpretation" is an attempt to resurrect the Mr. Wavefunction Collapse from it’s grave. You can't do it.
Born rule is either predicted (BM) or denied (but accepted FAPP) in MWI. Both theories are deterministic.

Fredrik

Peteratcam, your approach is just von Neumann measurement theory, as I initially suspected. The time evolution that you describe is sometimes called a "premeasurement". It's just an interaction that creates a correlation between states of the measuring device and states of the system. von Neumann didn't know this of course, but the actual measurement is a second interaction between the measuring device and its environment that makes the the system FAPP indistinguishable from a mixed state (i.e. a state that has "collapsed" into one of the alternatives).

I think you would find this article interesting, at least the first few pages. I'm too tired right now to try to explain the consequences of what he's saying there, so you'll have to read it for yourself (if you're interested).

meopemuk

This is the biggest mystery of nature. Nobody knows the reason for this randomness.

QM formalism does not predict/explain the random behavior of nature. This behavior is simply accepted as a given physical fact, and the entire formalism is built upon this assumption. QM math is fully deterministic, but this math is directed at calculation of probabilities. QM cannot say what exactly will be measured. It can only predict probabilities of different possible outcomes. The random behavior of nature and statistical QM formalism are in perfect agreement with each other.

Eugene.