What is the POVM generalization of Born's rule for quantum measurements?

[vanhees71]

I admit that I haven't found a satisfactory introduction to QM 1 either, but in your paper you simply define q-expectation values using the most general notion of a state as a statistical operator. I don't believe that this is simpler for the beginners of QT than the more standard approach starting with pure states.

The approach with polarized light, using just plane waves, is the most simple I can think of. It's also already known from the elctrodynamics lecture. To use the complete Stokes-vector formalism seems also a bit overdone for an introductory lecture.

That's not a criticism against your paper per se, but I doubt that this approach starting from the most general and sophisticated case is useful for the introductory course.

 

[A. Neumaier]

in your paper you simply define q-expectation values using the most general notion of a state as a statistical operator. I don't believe that this is simpler for the beginners of QT than the more standard approach starting with pure states.

It is not postulated but derived by a simple linearity argument, and has a very intuitive form. Whereas Born's original formula is quadratic and strange at first sight - Born even got it wrong initially!

In addition, Born's approach needs the whole spectral machinery including multiplicities to produce the expectation formula, quite a messy process.

 

[A. Neumaier]

The approach with polarized light, using just plane waves, is the most simple I can think of. It's also already known from the elctrodynamics lecture. To use the complete Stokes-vector formalism seems also a bit overdone for an introductory lecture.

In an introductory lecture in quantum mechanics, I'd regard it to be a bit overdone using as motivation a very special case of the Maxwell equations, which govern a completely different field. The latter are appropiate as motivation when one is going to describe QED, which merges the two fields!

Just as one commonly states the results of the Stern-Gerlach experiment without prior fundamental theory, the simple laws for partially polarized light can also be introduced in a purely phenomenological way. When asking about the physical description of ordinary light, students should not be left in the dark!

Unpolarized light is much more common than completely polarized light. The laws that describe both polarized and unpolarized light are easly stated; they were known since 1852, even before Maxwell's equations were formulated. Just as Stern-Gerlach was known before its explanation through spin and the Schrödinger equation.

 

[A. Neumaier]

I admit that I haven't found a satisfactory introduction to QM 1 either,

You should not criticize my work on the basis of standards that you are not able to follow yourself.

What I proposed in my paper is a detailed, fully working and well motivated alternative to the traditional approach to quantum mechanics. My proposal may not be perfect but it is as good as the latter, and has the advantages of greater simplicity anf greater generality.

 

[RUTA]

The discussion about how one should best introduce QM is interesting. With applications to nuclear physics, solid state physics, quantum optics, quantum information/computing, particle physics, and foundations to name a few, one could start almost anywhere. There are three of us teaching intro courses in these areas for engineers at my small undergrad institution and there is very little overlap in content between the courses, even though all the courses analyze actual experiments and/or applications.

 

[vanhees71]

You should not criticize my work on the basis of standards that you are not able to follow yourself.

What I proposed in my paper is a detailed, fully working and well motivated alternative to the traditional approach to quantum mechanics. My proposal may not be perfect but it is as good as the latter, and has the advantages of greater simplicity anf greater generality.

You asked for comments. I'm sorry if you take it as undue criticism.

 

[A. Neumaier]

You asked for comments. I'm sorry if you take it as undue criticism.

Well, it is the way you phrase your comments. For example you wrote:

In your paper you just give an axiomatic mathematical scheme. This is something you can understand in a very specialized lecture after the students have heard at least the two standard course lectures (QM I and II).

But in two different introductory courses you also just gave an axiomatic mathematical scheme! This double standard is a bit unfair, even when only regarded as a comment.

 

[vanhees71]

The discussion about how one should best introduce QM is interesting. With applications to nuclear physics, solid state physics, quantum optics, quantum information/computing, particle physics, and foundations to name a few, one could start almost anywhere. There are three of us teaching intro courses in these areas for engineers at my small undergrad institution and there is very little overlap in content between the courses, even though all the courses analyze actual experiments and/or applications.

The problem is to get the radically new physical thinking as compared to classical physics which (apparently) is straight-forward from the very first moment. I don't think that just to put the "trace formula" for expectation values on top of a axiomatic is good for the introductory course lecture. It's very good for a treatment in an advanced lecture on quantum foundations, quantum information, or something similar.

The criticism about my "photon-polarization first" approach is also well justified. I made this choice, because the lecture about electrodynamics, which is part of my course on theoretical physics for high-school teachers, ends with a quite complete treatment of electromagnetic waves, including a discussion of plane-wave solutions, polarization and also of course diffraction on slits and gratings.

Then my idea was that it is best to start with the most simple case of a 2D quantum-mechanical Hilbert space, as is done in many textbooks I like using spin 1/2 and the Stern-Gerlach experiment. This is however in my opinion also not so good for an introductory QM lecture, i.e., one where the students never have heard about spin before, i.e., and that's why I start rather with the polarization of electromagnetic waves and then argue heuristically (and I don't think that you can introduce QM at this level in any sensible way purely deductively without a big portion of heuristics) what happens if you use single-photon sources (without of course being able to explain the subtle and in fact important difference between "dimmed laser light" and real one-photon Fock states). I see the advantage in the fact that you have all the material of electromagnetism lecture at hand, particularly the fact that the most intuitive observable of light, which is intensity (i.e., "brightness" as a qualitative entity), is an adequate (temporal) average over the energy density of the electromagnetic field, i.e., for a plane wave something $\proportional |\vec{E}^2|$.

Then you have with the most simple example of using polaroid foil(s) doing experiments with polarized light (which is nowadays really no problem to do even in the introductory expermental-physics lecture, since one has lasers at hand to produce nicely coherent light; even with "natural light" it's no big deal to get polarized light using a polaroid filter) all the arguments at hand to motivate the QT formalism:

-polarization states as preparation procedure: Just take a polaroid to provide a source of linearly polarized light in a given direction

-self-adjoint operators as representants of polarization states: Just use another polaroid and describe it as projection operarator

-the superposition principle and complete orthonormal systems: You work within a framework of linear optics, where the em. field equations are linear equations, and you can find new solutions by superposition and even all solutions in terms of complete sets of orthonormal function systems (a concept also known from the E&M lecture, where this is treated in the context of Fourier series and Fourier integrals).

Of course there are also some drawbacks of this method, because it introduces photons at the introductory stage, and there's some danger that the students get the wrong idea as if one could treat photons in a kind of 1st-quantization approach, which of course then follows directly after the introduction with photon polarization, where the Hilbert space for one particle is introduced in the usual heuristics a la Schrödinger (in a very simplified form, not taking recourse to the Hamilton-Jacobi theory of classical mechanics).

 

[vanhees71]

Well, it is the way you phrase your comments. For example you wrote:

But in two different introductory courses you also just gave an axiomatic mathematical scheme! This double standard is a bit unfair, even when only regarded as a comment.

The only manuscript for an introductory lecture is the manuscript for teachers students, and with Theorie III I'm not really satisfied, but this is far from an axiomatic treatment. It's very heuristic and hand-waving. It's not even complete, as indeed in an axiomatic treatment you have to start with statistical operators, and I even think that for that purpose your axiomatics is pretty elegant (though I still don't believe that one should identify the q-expectation values with observables, for the already mentioned reason).

 

[RUTA]

The problem is to get the radically new physical thinking as compared to classical physics which (apparently) is straight-forward from the very first moment. I don't think that just to put the "trace formula" for expectation values on top of a axiomatic is good for the introductory course lecture. It's very good for a treatment in an advanced lecture on quantum foundations, quantum information, or something similar.

The criticism about my "photon-polarization first" approach is also well justified. I made this choice, because the lecture about electrodynamics, which is part of my course on theoretical physics for high-school teachers, ends with a quite complete treatment of electromagnetic waves, including a discussion of plane-wave solutions, polarization and also of course diffraction on slits and gratings.

Then my idea was that it is best to start with the most simple case of a 2D quantum-mechanical Hilbert space, as is done in many textbooks I like using spin 1/2 and the Stern-Gerlach experiment. This is however in my opinion also not so good for an introductory QM lecture, i.e., one where the students never have heard about spin before, i.e., and that's why I start rather with the polarization of electromagnetic waves and then argue heuristically (and I don't think that you can introduce QM at this level in any sensible way purely deductively without a big portion of heuristics) what happens if you use single-photon sources (without of course being able to explain the subtle and in fact important difference between "dimmed laser light" and real one-photon Fock states). I see the advantage in the fact that you have all the material of electromagnetism lecture at hand, particularly the fact that the most intuitive observable of light, which is intensity (i.e., "brightness" as a qualitative entity), is an adequate (temporal) average over the energy density of the electromagnetic field, i.e., for a plane wave something $\proportional |\vec{E}^2|$.

Then you have with the most simple example of using polaroid foil(s) doing experiments with polarized light (which is nowadays really no problem to do even in the introductory expermental-physics lecture, since one has lasers at hand to produce nicely coherent light; even with "natural light" it's no big deal to get polarized light using a polaroid filter) all the arguments at hand to motivate the QT formalism:

-polarization states as preparation procedure: Just take a polaroid to provide a source of linearly polarized light in a given direction

-self-adjoint operators as representants of polarization states: Just use another polaroid and describe it as projection operarator

-the superposition principle and complete orthonormal systems: You work within a framework of linear optics, where the em. field equations are linear equations, and you can find new solutions by superposition and even all solutions in terms of complete sets of orthonormal function systems (a concept also known from the E&M lecture, where this is treated in the context of Fourier series and Fourier integrals).

Of course there are also some drawbacks of this method, because it introduces photons at the introductory stage, and there's some danger that the students get the wrong idea as if one could treat photons in a kind of 1st-quantization approach, which of course then follows directly after the introduction with photon polarization, where the Hilbert space for one particle is introduced in the usual heuristics a la Schrödinger (in a very simplified form, not taking recourse to the Hamilton-Jacobi theory of classical mechanics).

There is nothing wrong with "heuristics," indeed we must use intuitively understood, but strictly undefined, concepts from the outset in physics, e.g., normal force, string tension, friction. Arguably, since we still do not have a unified axiomatic basis for all of physics, heuristics are unavoidable. Your approach is pedagogically sound because it introduces the new topic starting with what the student already knows. Serway and Jewett's intro physics text introduces the wave function using $u \propto \vec{E}^2$ from E&M and $E = hf$ assuming monochromatic radiation, then asking how many photons exist per unit volume. It's a reasonable (and heuristic) way to introduce the probability amplitude.

 

[Morbert]

As an aside: I think a straightforward link can be made between POVMs associated with detectors referenced in OPs paper, and the typical von Neumann notion of properties/measurement results as projectors. Given the POVM $\{P_k\}$ associated with measurement outcomes $\{k\}$, we can construct a spectral decomposition for each $P_k$ $$P_k = \sum_j\lambda_{jk}\xi_{jk}$$ (where $\lambda_{jk}$ are scalars and $\xi_{jk}$ are projectors) such that if we observe measurement outcome $k$, we can infer property $\xi_{ik}$ with probability $$\lambda_{ik}\frac{\mathbf{Tr}\left[\rho\xi_{ik}\right]}{\mathbf{Tr}\left[\rho P_k\right]}$$
where $\rho$ is the preparation as usual.

 

[A. Neumaier]

As an aside: I think a straightforward link can be made between POVMs associated with detectors referenced in OPs paper, and the typical von Neumann notion of properties/measurement results as projectors. Given the POVM $\{P_k\}$ associated with measurement outcomes $\{k\}$, we can construct a spectral decomposition for each $P_k$ $$P_k = \sum_j\lambda_{jk}\xi_{jk}$$ (where $\lambda_{jk}$ are scalars and $\xi_{jk}$ are projectors) such that if we observe measurement outcome $k$, we can infer property $\xi_{ik}$ with probability $$\lambda_{ik}\frac{\mathbf{Tr}\left[\rho\xi_{ik}\right]}{\mathbf{Tr}\left[\rho P_k\right]}$$
where $\rho$ is the preparation as usual.

But what should be the meaning of this property?

 

[A. Neumaier]

Yes, and Heisenberg was wrong and was corrected immediately by Bohr at the time.

You got the history completely wrong - please read (or reread) some of the old papers! Bohr criticised details of Heisenberg's argument, not the general principle; see this post. In Bohr's paper

• N. Bohr, The Quantum Postulate and the Recent Development of Atomic Theory, Nature Suppl., April 14, 1928, 580-590. (German version: Das Quantenpostulat und die neuere Entwicklung der Atomistik, Naturwissenschaften 16(1928), 245-257.)

he says explicitly on p.583:

The product of the least inaccuracies with which the positional co-ordinate and the component of momentum in a definite direction can be ascertained is therefore just given by formula (2).

Thus Bohr reaffirms Heisenberg's statement that you had wrongly claimed Bohr had corrected. Such joint measurements of position and momentum with limited accuracy cannot be described in terms of Born's statistical interpretation.

That Heisenberg's three views of the uncertainty relation (limits on preparation, unavoidable disturbance through measurement, and uncertainty of joint measurements) are all valid according to the modern understanding is discussed in detail in the references in this post from another thread.

 

[Morbert]

But what should be the meaning of this property?

I don't think it would necessarily have a classical interpretation, other than perhaps the implication that there is in principle a perfect apparatus that can resolve the alternatives $\{\xi_{jk},I-\xi_{jk}\}$. Questions about the nature of quantum properties in general likely lead us into the usual candy shop of QM interpretations.

 

[*now*]

You got the history completely wrong - please read (or reread) some of the old papers! Bohr criticised details of Heisenberg's argument, not the general principle; see this post. In Bohr's paper

• N. Bohr, The Quantum Postulate and the Recent Development of Atomic Theory, Nature Suppl., April 14, 1928, 580-590. (German version: Das Quantenpostulat und die neuere Entwicklung der Atomistik, Naturwissenschaften 16(1928), 245-257.)

he says explicitly on p.583:

Thus Bohr reaffirms Heisenberg's statement that you had wrongly claimed Bohr had corrected. Such joint measurements of position and momentum with limited accuracy cannot be described in terms of Born's statistical interpretation.

That Heisenberg's three views of the uncertainty relation (limits on preparation, unavoidable disturbance through measurement, and uncertainty of joint measurements) are all valid according to the modern understanding is discussed in detail in the references in this post from another thread.

What is your view of the Bohr Rosenfeld arguments and so forth?

 

[A. Neumaier]

What is your view of the Bohr Rosenfeld arguments and so forth?

Which of these arguments relate to the present thread?

 

[*now*]

To start, Zur Frage der Messbarkeit der elektromagnetischen Feldgrössen, 1933.

 

[A. Neumaier]

To start, Zur Frage der Messbarkeit der elektromagnetischen Feldgrössen, 1933.

The thermal interpretation is consistent with the observation that only smeared (q-expectations of) fields without significant spatial or temporal high frequency oscillations can be measured reliably.

 

[*now*]

I see, thank you.