# I Ontology of spin measurement

#### Elias1960

This is a very questionable statement.

What is the probability distribution of something of which the information is given that 5 times someone observed 1, twice 3 was observed, and once 6 was observed? One cannot maximize the entropy given this information. But all information one can gather about aspects of the universe is information of this kind.
One can. One starts with the state of zero additional information. This maximizes entropy for 1/6 for each number of the dice. Then one adds the information given and uses the formula for "Bayesian updating" to compute the probability distribution which takes into account that new information.

#### PeterDonis

Mentor
One starts with the state of zero additional information. This maximizes entropy for 1/6 for each number of the dice. Then one adds the information given and uses the formula for "Bayesian updating" to compute the probability distribution which takes into account that new information.
What are you updating? The probabilities for the 6 possible rolls of the die?

Mentor

#### romsofia

A passage about measurement from Bryce DeWitt that I enjoy:
"Much of the earlier work on the measurement problem, influenced no doubt by Bohr's shadow, emphasized the need for permanent information storage and hence, for complexity, metastability, ergodicity, etc. This emphasis was misplaced. One does not gain understanding by making a problem more complicated. A measurement is simply the establishment of a correlation between a "system" observable an an "apparatus" observable. It is the function of the apparatus to "observe" the system, not vice versa, and hence there is a fundamental asymmetry between them. It turns out that there are two prominent features that characterize a good apparatus: Its "pointer" must be in a localized quantum state, and it must be massive compared to the system. That is all". (Taken from his essay "Decoherence without Complexity and without an Arrow of Time" in the book"Physical origins of time asymmetry").

So to OP, yes, the measurement is real. That's the whole point of the experiment! I think an important point, especially when considering a simple experiment like the SG is: "One does not gain understanding by making a problem more complicated." If you turn the experiment upside down, to the left, to the right isn't the point of the experiment. It's that orientation of the electrons of the silver atoms is probabilistic, and is CORRECTLY predicted by the math of QM!

#### A. Neumaier

What is the probability distribution of something of which the information is given that 5 times someone observed 1, twice 3 was observed, and once 6 was observed? One cannot maximize the entroy given this information. But all information one can gather about aspects of the universe is information of this knd.
One can. One starts with the state of zero additional information. This maximizes entropy for 1/6 for each number of the dice. Then one adds the information given and uses the formula for "Bayesian updating" to compute the probability distribution which takes into account that new information.
Did you ever apply what you recommend to others?
Please tell us the updated probability distribution after having recorded the information described above.

#### A. Neumaier

A measurement is simply the establishment of a correlation between a "system" observable an an "apparatus" observable.
???

Which correlation is established in the Stern-Gerlach experiment between a "system" observable an "apparatus" observable when you take a single measurement of a spin?

#### romsofia

???

Which correlation is established between a "system" observable an an "apparatus" observable when you take a single measurement of a photon?
OP is talking about the SG experiment, not measuring photons.

#### A. Neumaier

OP is talking about the SG experiment, not measuring photons.
Yes, corrected.

#### romsofia

Yes, corrected.
I will outline the steps taken by Bryce DeWitt in his book (as it is an old book, and I don't think many members will have it off hand!): "Dynamical theory of groups and fields" (starting on page 16). So note, this is not my argument, but I believe his argument should be presented.

He argues that the mathematical form for analyzing a single observable "D" for a coupling between a system and apparatus is the total action functional: $S+S_A+gxD$ Where S is the action for the system, $S_A$ is the action for the apparatus, and gxD where: g is the (adjustable) coupling constant, x is some convenient apparatus variable, and D is the observable.

The observable in this case is the spin which we will refer to as the "system", and the "apparatus" will be refered to as: the atom (ignoring spin here), the magnetic field, and a coordinate framework.

The atom is massive compared to the spin, so the dynamical motion of the system S can be considered constant. The apparatus functional will take the form: $S_A = \int \frac{1}{2} m(\dot{x_2}^2 +\dot{x_3}^2)dt$
Here $(x_2, x_3)$ are the apparatus coordinates in the plane, and we save the $x_3$ axis for the direction of the magnetic field. m is the mass of the atom. He makes the assumption that the atom will move in this plane, so he will ignore $x_1$.

He then argues that the coupling term that correlates spin and atomic motion has the form ($\hbar = 1)$:
$\int \mu D H dt$

If left undistrubed, the atom (essentially the apparatus) will follow the trajectory $x_2 = vt, x_3=0$ which is a stationary trajectory for the action $S_A$ He then argues, once again, if the atom is massive, it won't change much from this trajectory.

He then makes some assumptions about the strength of the magnetic field, and approximates it to:
$H = \theta(x_2) \theta(L-x_2) x_3(\frac{\partial H}{\partial x_3})|_{x_3=0}$ Where L is the length of pole pieces of the magnet, and the minimum time experiment is to be: $0 < t < \frac{L}{v}$

and $\theta$ is some step function defined by: $\theta(e) = \frac{1}{2}(1+\frac{e}{|e|})$ which are defined by: 1 for e>0, $\frac{1}{2}$ for e = 0, and 0 for e<0.

With these approximations in mind, he argues that the coupling term will reduce to the form gxD where:
$x = \int \theta(x_2) \theta(L-x_2) x_3 dt$, $g = \mu (\frac{\partial H}{\partial x_3})|_{x_3=0}$

Later on in the book, he talks about elementary vs complete measurements, and adds more rigor to this arguments in the SG experiment, but I would rather just refer to the book at that point as it is several pages to build it up properly.

#### A. Neumaier

A measurement is simply the establishment of a correlation between a "system" observable an an "apparatus" observable.
Which correlation is established in the Stern-Gerlach experiment between a "system" observable an "apparatus" observable when you take a single measurement of a spin?
I will outline the steps taken by Bryce DeWitt in his book (as it is an old book, and I don't think many members will have it off hand!):
What you outline only says that by the formal part of quantum mechanics, the system state and the detector state become entangled through the interaction.

Measurement, i.e., recording a particular value of the spin, is not yet involved. A single measurement establishes no correlations at all. A sufficiently long sequence of measurements therefore does not establish anything either, but just reveals the preexisting correlations created by the entanglement.

#### Elias1960

Those aren't formulas for updating the probabilities of the 6 rolls of the die. They're formulas for updating the probabilities of hypotheses. What hypotheses are you updating the probabilities of?
There are six relevant hypotheses, namely that throwing the dice the next time will give 1 resp. 2,3,4,5,6.

#### PeterDonis

Mentor
There are six relevant hypotheses, namely that throwing the dice the next time will give 1 resp. 2,3,4,5,6.
Well, then it's easy to falsify all but one of them just by rolling the die once more. Which obviously isn't right, so you must be misidentifying the hypotheses. Try again.

#### vanhees71

Gold Member
Another important difference is that a POVM measurment makes no claim about which values are measured.
So there's no meaning whatsoever in the POVM formalism? I thought it's a binary decision "click or no-click"?

#### vanhees71

Gold Member
???

Which correlation is established in the Stern-Gerlach experiment between a "system" observable an "apparatus" observable when you take a single measurement of a spin?
The observable is the spin component in a direction determined by the magnetic field. The "apparatus observable" (usually called the pointer observable) is the location of the particle. The SG magnet establishes within some accuracy (which can at least in principle be made as good as you wish) an entanglement between the spin component and the position of the particle. Rightly calibrated selecting particles in the appropriate spatial region leads to a FAPP 100% determination of the spin component, i.e., is a preparation procedure for the corresponding pure spin state.

#### A. Neumaier

Another important difference is that a POVM measurment makes no claim about which values are measured.

It just says that one of the detectors making up the detection device responds with a probability given by the trace formula.
So there's no meaning whatsoever in the POVM formalism? I thought it's a binary decision "click or no-click"?
This is not what I was saying.

The POVM gives probabilities for a particular detector element responding "click or no-click", without assigning a numerical value to it. The latter must be assigned independently. POVM and value assignment together define a numerical observable. Given only the POVM, the value assignment can be done in principle arbitrarily. To make the detector produce a measurement of the intended observable, the value assignment must be properly calibrated.

Just as a classical pointer just points somewhere, without assigning a numerical value to it. The latter must be assigned independently. This is done by adding a scale with numbers on it a reasonable interpolation scheme implied by additional ticks. Pointer and scale together define a numerical observable. Given only the pointer, the scale can be chosen in principle arbitrarily. To make the detector produce a measurement of the intended observable, the scale must be properly calibrated.

The situation is therefore completely analogous to the classical case.

#### vanhees71

Gold Member
A probability is a numerical value (between [0,1]). To measure it you prepare a lot of systems and count in how many cases the detector clicks. Then, if the POVM is an accurate description, the relative frequency of the clicks should converge to the predicted value of the probability.

One should not complicate the issue even further by discussing the trivial fact that you have to calibrate your measurement device to give a value for an observable in a given unit.

#### A. Neumaier

A probability is a numerical value (between [0,1]). To measure it you prepare a lot of systems and count in how many cases the detector clicks. Then, if the POVM is an accurate description, the relative frequency of the clicks should converge to the predicted value of the probability.
Yes, of course.
One should not complicate the issue even further by discussing the trivial fact that you have to calibrate your measurement device to give a value for an observable in a given unit.
But POVMs also cover measurements of arbitrary observables, not only of binary ones. A numerical assignment is necessary in all cases where one wants to measure nonbinary observables (such as relative position, particle spin, interference patterns, or optical angular momentum states). In this case one needs to assign different numbers to different detector elements.

#### Elias1960

Did you ever apply what you recommend to others?
Only if I find it useful for me to do this. Fortunately, I do not have to make money by computing something for other people.
Please tell us the updated probability distribution after having recorded the information described above.
Feel free to make a financial offer for computing something for you, but I doubt I will accept it, given that I have no necessity to work for money.
Well, then it's easy to falsify all but one of them just by rolling the die once more. Which obviously isn't right, so you must be misidentifying the hypotheses. Try again.
The possibility of falsification shows that they are not hypotheses? I think hypotheses even have to be falsifiable, else they are not empirical hypotheses (even if string theorists think otherwise).

Similar to the remark above, I see no reason to try something upon your request. The time when I was a pupil who had to answer questions in examinations was in the last Millenium, and there is also no contract for me which obliges to teach you something. If you want an introduction into the objective Bayesian probability interpretation, there are textbooks for this, I would recommend

Jaynes, E.T. (2003). Probability Theory: The Logic of Science

for this.

#### A. Neumaier

@Elias1960: Asking these questions was a polite way of pointing out how poorly you understand what you are talking about. By doing the exercise you would have found this out without having to be told explicitly, saving your face.
I doubt I will accept it, given that I have no necessity to work for money.
But probably you also feel that you have no necessity to save your face. Well, as you wish....

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#### Elias1960

@Elias1960: Asking these questions was a polite way of pointing out how poorly you understand what you are talking about. By doing the exercise you would have found tis out without having to be told explicitly, saving your face.
If I make errors, I prefer open explanations of the errors. And, BTW, everybody understands very well that your method of asking questions, like a teacher to pupils, is a way of saying that I don't understand anything. Given that this is combined with not openly telling what is wrong, this is much worse than an explicit explanation of what was wrong with what I wrote.

#### A. Neumaier

Well, I am a university teacher, and know how to impart knowledge efficiently.

#### Elias1960

Well, I am a university teacher, and know how to impart knowledge efficiently.
To put oneself into a superior position, namely by asking questions in a way expected from a teacher to pupils, in a discussion with somebody you don't know at all, and who has not asked you for help, is not very polite behavior. I doubt that such impolite behavior is a good way to impart knowledge efficiently.

You also seem to have ignored that my last post was also a sort of request for an explicit description of what was wrong with what I wrote. So, I repeat this request, if there was IYO something wrong with what I wrote, explain what was wrong, and what would be IYO the correct way to describe this.

#### PeterDonis

Mentor
everybody understands very well that your method of asking questions, like a teacher to pupils, is a way of saying that I don't understand anything.
No, it was a way of trying to get you to show your work explicitly. Which was a perfectly valid request (I made it too) and you refused to do it. Which means that you are now banned from further posting in this thread.

#### vanhees71

Gold Member
But POVMs also cover measurements of arbitrary observables, not only of binary ones. A numerical assignment is necessary in all cases where one wants to measure nonbinary observables (such as relative position, particle spin, interference patterns, or optical angular momentum states). In this case one needs to assign different numbers to different detector elements.
Fine, but how can one make this concrete for the most fundamental observable in non-relativistic QT, i.e., position measurements. I've been googling a long time, but couldn't find a concrete description of this fundamental example. I tried to make it up myself, but I'm unsure, whether it's correctly defining a POVM. At least it would make some sense to me as a simple model to describe a position measurement of a pixel detector with finite resolution. The details are in the other thread:

"Ontology of spin measurement"

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