Many measurements are not covered by Born's rule

[Prathyush]

I decribed it clearly and even had boldfaced the contradictions with Born's rule, in case of the Helium atom. No coarse-graining is involved!

It does not contradict QT (so everybody is right to take it very seriously) but it very clearly contradicts Born's rule in the usual formulation.

Born's rule is not QT but only a very fallible part of it, with a very limited domain of applicability!

Why doesn't Born's rule with work with Helium atom? You predict the energy levels and you see them? You can precisely calculate transition matrices, and the results can be interpreted in an experiment using a probe particle. How we measure the probe particle can be postponed into a separate investigation. To interpret the results of such an investigation we use the Born's rule.

 

[Prathyush]

But in contrast to what Born's rule claims, the measured energy levels are not the exact eigenvalues of H.

Now why do you say this?

 

[ftr]

Arnold, Let's say we measure the position of electron in many hydrogen atoms in ground state, are you saying the expectation value will be only 3/2a0(a0 Bohr radius) and we cannot actually prove Born rule even in principle because it cannot be done in practice. So that we can only use the density matrix to calculate expectation value only, is that correct. If not, what are you saying for such case, i.e. what is the electron doing, is it playing hide and seek, dancing or what.

 

[Paul Colby]

Born's rule is silent about the value of the measured mass of a single brick of iron. Here N=1N=1N=1 in the above formula. The values can take any of an astronomically large number of values, and the Born probability of measuring any of these is extremely tiny. Since only a single measurement is made, the above derivation based on the law of large numbers does not apply.

It's unclear to me one may even assure the total number of iron atoms is fixed. The act of just looking at the brick may knock atoms on or off. I think the hair you're trying to split here is too fine for me to see so let me concede defeat and go back to my calculations.

 

[vanhees71]

I decribed it clearly and even had boldfaced the contradictions with Born's rule, in case of the Helium atom. No coarse-graining is involved!

It does not contradict QT (so everybody is right to take it very seriously) but it very clearly contradicts Born's rule in the usual formulation.

Born's rule is not QT but only a very fallible part of it, with a very limited domain of applicability!

Again, there is no contradiction between QT and the Helium spectrum nor is there any contradiction between observations and Born's rule, which is an integral part of QT as a physical theory. If such a contradiction were discovered, this would mean the most sensational result since the discovery of QT itself, and we'd be well aware of this. We really discuss in circles.

 

[A. Neumaier]

Why doesn't Born's rule with work with Helium atom? You predict the energy levels and you see them? You can precisely calculate transition matrices, and the results can be interpreted in an experiment using a probe particle. How we measure the probe particle can be postponed into a separate investigation. To interpret the results of such an investigation we use the Born's rule.

I only claimed that Born's rule doesn't work for the measurement of the total energy of a helium atom. For this it doesn't matter whether other things can be interpreted with the Born rule, but only that Born's rule, applied to the observable $H$, does not apply. A measurment of $H$ (if it is at all possible) never gives a single energy level (neither exactly as claimed by Born's rule nor even approximately as a relaxed version would perhaps claim), picked at random from the full list according to the probabilities stated in the rule. I'd like to see the experimental arrangement that would do this!

[''the measured energy levels are not the exact eigenvalues of H.''] Now why do you say this?

I had explained it already in the post cited by you. The exact energy levels are still (and will probably always be) unknown, in spite of the fact that the energy levels have been measured many times.

Lets say we measure the position of electron in many hydrogen atoms in ground state

This is an essentially imposiible thing to do. independent of that, this is not the problem addressed by me. I just point to examples of measurements where Born's rule clearly doesn't make sense. This is not meant to address all possible or impossible other applications of Born's rule.

It's unclear to me one may even assure the total number of iron atoms is fixed.

In the grand canonical ensemble the number of atoms need not be fixed. nevertheless, the total energy is a well-defined entity.

Born's rule, which is an integral part of QT as a physical theory.

No. it is part of the interpretation, and as such like any other interpretation not an integral part of QT.

 

[vanhees71]

Again, not knowing the exact Hamiltonian, has nothing to do with the foundations of QT we are discussing here. You made a very bold claim, namely that the very foundation of QT (and thus QT itself) is disproven by claiming that Born's rule is wrong. You should give a clear evidence for that bold statement, but all you do is to give examples for incomplete knowledge about sufficiently complicated systems.

Indeed we don't know exactly how to describe even a proton, which is a complicated bound state of "partonic constituents", where even this phrase is not completely understood. All this, however, has nothing to do with the fundamental structure of QT and thus doesn't disprove QT as a fundamental theory, including Born's rule.

 

[A. Neumaier]

You made a very bold claim, namely that the very foundation of QT (and thus QT itself) is disproven

You deliberately misread what I write. I never claimed this. Born's rule is needed only to interpret some measurements, approximately, and is therefore not an intrinsic part of quantum theory, which is a theory supposed to hold exactly (i.e., to arbitrary precision), not only approximately.

 

[ftr]

I just point to examples of measurements where Born's rule clearly doesn't make sense.

Are you saying in effect that the expectation value of the density matrix( wavefunction derivative) has a physical significance, is that correct.

 

[A. Neumaier]

Are you saying in effect that the expectation value of the density matrix( wavefunction derivative) has a physical significance, is that correct.

The expectation value and the associated uncertainty have indeed a physical significance, independent of measurement. They are the only values that can be consistently be assigned to an observable in a given state before a measurement is made. The expectation value gives a prediction for the observed value within this uncertainty, the best possible prediction without having actually performed the measurement.

 

[ftr]

The expectation value and the associated uncertainty have indeed a physical significance, independent of measurement. They are the only values that can be consistently be assigned to an observable in a given state before a measurement is made. The expectation value gives a prediction for the observed value within this uncertainty, the best possible prediction without having actually performed the measurement.

But we still have to assign probabilities to the spectrum to calculate expectation values, don't we.

Edit: in that case what does those assignments mean/imply.

 

[vanhees71]

I'm also unable to understand, what A. Neumaier means. On the one hand, he doesn't want to use probabilities, on the other hand, all he says is using probabilities, because he has not defined, what the words "prediction of the observed value" and "within this uncertainty" means.

In normal language, what's behind these words is probability theory, and if I know about a random variable only its expectation value and its standard deviation, I'd use the maximum-entropy method to associate a probability distribution with "least prejudice" in the sense of the Shannon entropy and end up with a Gaussian distribution.

To repeat it one zillionth time again: The state (defined as an equivalence class of preparation procedures) is described in the formalism by a positive semidefinite self-adjoint operatator $\hat{\rho}$ (the statistical operator) implies all probabilitis (or probability distributions) to measure a value of any observable definable on the system in question. In the formalism an observable (defined as an equivalence class of measurement procedures) is described by a self-adjoint operator $\hat{A}$. If then $|a,\beta \rangle$ is a complete set of orthonormalized (generalized) eigenvectors of $\hat{A}$, the probability (distribution) for measuring $A$ given the state described by $\hat{\rho}$ is
$$P(a)=\sum_{\beta} \langle a,\beta|\hat{\rho}|a,\beta \rangle,$$
where the sum has to be understood in the usual sense as a sum over the discrete part of $\beta$ and an integral over the continuous part of $\beta$.

 

[Paul Colby]

In the grand canonical ensemble the number of atoms need not be fixed. nevertheless, the total energy is a well-defined entity.

Well defined so what? The number of atoms present in the brick isn't constant from measurement to measurement in any real world measurement. To argue so is to apply an accuracy limit or averaging argument. Born rule looks safe to me.

 

[vanhees71]

The (non-relativistic) grand-canonical ensemble by definition applies to the situation that you consider a smaller subsystem of a large (also macroscopic) system where both energy and (a conserved!) particle number can be exchanged within the subsystem and the "rest". Only the average energy and particle number are fixed via their "conjugate" thermodynamic quantities $\beta=1/T$ and $\alpha=-\mu/T$. The corresponding statistical operator is
$$\hat{\rho}(\beta,\alpha) =\frac{1}{Z} \exp(-\beta \hat{H} -\alpha \hat{N}),$$
where
$$Z=\mathrm{Tr} \exp(-\beta \hat{H} -\alpha \hat{N}).$$

 

[A. Neumaier]

he has not defined, what the words "prediction of the observed value" and "within this uncertainty" means.

In general, this is an approximate notion independent of probability. The position of a car is always uncertain to within about 1 meter, a statement that can be verified without recourse to probability

The meaning of observed values and uncertainty are discussed in the NIST Reference on Constants, Units, and Uncertainty, which may be regarded as the de facto standard for representing uncertainty. This source explicitly distinguish between uncertainties ''which are evaluated by statistical methods'' and those ''which are evaluated by other means''. For the second class, it is recognized that the uncertainties are not statistical but should be treated ''like standard deviations''.

The number of atoms present in the brick isn't constant from measurement to measurement in any real world measurement.

So what? Measured positions of a moving car, or measured times or measured currents or measured temperatures or whatever else people measure are not constant either, and still people trust their single measurements. Except when the noise is so large that repetition is necessary. Only then statistics enters.

But we still have to assign probabilities to the spectrum to calculate expectation values, don't we.

Expectations are calculated from the defining formula $\langle A\rangle =Tr \rho A$, and probabilities nowhere enter.

all he says is using probabilities

I nowhere use them.

if I know about a random variable only its expectation value and its standard deviation

But quantum observables are not random variables, as you know very well! Random variables always commute!

(a conserved!) particle number

Only locally conserved. The total particle number in the ''rest'' may well be infinite, and hence globally meaningless. Only the particle density matters, and figures in the formulas.

 

[Paul Colby]

Measured positions of a moving car, or measured times or measured currents or measured temperatures or whatever else people measure are not constant either, and still people trust their single measurements. Except when the noise is so large that repetition is necessary. Only then statistics enters.

Like all things, it depends on what one is doing. I think you're conflating engineering measurement with measurement as it might be defined in an "ideal" case. In current theory measuring the position of a car (how is that defined exactly?) is a quantum mechanical problem, though, as you point out one may choose to forgo QM for expedience without too much error. QM is never far away from an actual measurement. One would use some form of measuring device like a camera or such which has pixels which have counting statistics which are QM in origin. So there is a Hamiltonian for the car and it's interaction with the electromagnetic field. All of this matters at some level, even for cars.

 

[A. Neumaier]

measuring the position of a car (how is that defined exactly?)

That's the point: It cannot be defined exactly (even classically), just like the phase-space-position of a quantum particle. Different definitions of how to measure the car position will agree only within an uncertainty of about one meter, just as different ways of performing a phase-space-position measurement (i.e., a joint approximate measurement of position and momentum) of a particle will only agree to within the limits of Heisenberg's the uncertainty relation.

conflating engineering measurement with measurement as it might be defined in an "ideal" case.

If Born's rule is not to be meant to be about real measurements but about imaginary ones, it doesn't give the claimed connection between theory and experimental practice.

 

[dextercioby]

The assumption behind Born rule is: one can in principle perform real-life experiments on systems composed of 1,2,3,... , 10^30, ... subsystems (let's call them particles) to measure certain physical (individual) properties of these subsystems (e.g. energy - observable = Hamiltonian). One can measure only a restricted set of values for energy: the spectral values of the Hamiltonian. This is a fact which is interpretation-independent, it's a mathematical assumption on what one can measure in a lab. A very strong one. I believe the debate is here: vanHees71 says that this is true, experiments are limited only to mathematically known values, while I perceive that Arnold Neumaier is saying there is no mathematical limitation to the actual (with a certain degree of technological inaccuracy) values measured in experiments.

The probabilistic view I have read is summed up below and is part of the interpretation:

For 100^100 experiments done at the same time (this is called a virtual statistical ensemble), there are 100^100 results which follow a statistical spread around the arithmetic mean. Born's rule simply gives the probability to obtain a value "a" out of all possibly measurable "a,b,c, etc." (see the assumption above) for an arbitrary system out of all the 100^100, in case all of them have been prepared (by absurd) to a known state.

 

[A. Neumaier]

one can in principle perform real-life experiments on systems composed of 1,2,3,... , 10^30, ... subsystems (let's call them particles) to measure certain physical (individual) properties of these subsystems (e.g. energy - observable = Hamiltonian).

1. ''In principle'' and ''real-life'' are opposites.

2. Please explain the principle according to which one can measure the energy of these subsystems.

3. Please explain how this should be related to the measured valued of the total energy of a brick of iron. Note that the latter is neither an average nor a sum of the energies of the subsystems but also contains the effects of numerous interactions.

 

[dextercioby]

1. ''In principle'' and ''real-life'' are opposites.

2. Please explain the principle according to which one can measure the energy of these subsystems.

3. Please explain how this should be related to the measured valued of the total energy of a brick of iron. Note that the latter is neither an average nor a sum of the energies of the subsystems but also contains the effects of numerous interactions.

"In principle" that I used are two missing words I believe to be necessarily written when one states the Born rule and its assumption. I believe in principle one can measure the energy of a single H atom, but the mere fact that one hasn't done it yet makes me think it is nothing that wishful thinking.

There is no way to write down the quantum Hamiltonian of an iron brick. This is a limitation of human knowlege. I strongly believe its energy cannot be measured anymore than classical relativity tells us.

 

[A. Neumaier]

There is no way to write down the quantum Hamiltonian of an iron brick. This is a limitation of human knowlege. I strongly believe its energy cannot be measured

Thermodynamics tells how to measure the energy of a brick of iron with several digits of accuracy. You measure its volume, pressure, and temperature and convert it to total energy by means of the experimentally known equation of state of iron. Without using any probability or statistics.

To do this, there is no need to know the quantum Hamiltonian. However, the latter can be written down to some reasonable accuracy, too.

There are even computer packages that do quantum calculations for iron crystals and related things and match them with the thermodynamic results.

 

[Paul Colby]

If Born's rule is not to be meant to be about real measurements but about imaginary ones, it doesn't give the claimed connection between theory and experimental practice.

As I said before, I don't understand the point you're trying to make. This is no big deal. Real measurements either are or can be analyzed using the Born rule as you seem to concede. For me QM predicts the frequency of these measurement results and not the individual measurement values. I see no problem with a theory of nature having this property. I see no problem with this being a fundamental aspect of such a theory.

 

[A. Neumaier]

QM predicts the frequency of these measurement results

Born's rule neither predicts the possible values nor the frequencies of the results of measuring the total energy of any atom or molecule, though this is one of the most basic observables of quantum mechanics.

 

[dextercioby]

Thermodynamics tells how to measure the energy of a brick of iron with several digits of accuracy. You measure its volume, pressure, and temperature and convert it to total energy by means of the experimentally known equation of state of iron. Without using any probability or statistics.

To do this, there is no need to know the quantum Hamiltonian. However, the latter can be written down to some reasonable accuracy, too.

There are even computer packages that do quantum calculations for iron crystals and related things and match them with the thermodynamic results.

No, no, I meant E =mc^2, where the m is the mass of the iron brick measured with the most sensitive balance at 0 m sea level and at equator.

 

[A. Neumaier]

No, no, I meant E =mc^2, where the m is the mass of the iron brick measured with the most sensitive balance at 0 m sea level and at equator.

You confuse mass and weight...

 

[dextercioby]

I don't. The balance shows me miligrams or even micrograms by properly transforming a gravitational effect (force in case of negligable space-time curvature effects) into mass via the gravitational acceleration.

Edit: this is an over simplified version. The working principles behind an electronic pharmaceutical balance showing the mass of a brick to be 1200,456 grams involve a very intricate way to mass calculation in terms of piezoelectric effects, microcurrents and quartz microcrystal physics and geometry.

 

[Paul Colby]

Born's rule neither predicts the possible values nor the frequencies of the results of measuring the total energy of any atom or molecule, though this is one of the most basic observables of quantum mechanics.

Is the total energy of an atom measurable? If so how is it done? I'm familiar with detecting photons either emitted or interacting with various atomic states but I know of no direct measurement of an atoms total energy. Perhaps this is more a theoretical construct than a direct observable? Usually the statement, "I have an hydrogen atom in state x" is not the result of direct measurement of the atoms energy but rather deduced from some other measurement. I worked with negative polarized ion sources in my formative years. Neutral hydrogen atoms were excited in an RF discharge. Then the proper spin ones were selected by focusing with a 6-pole magnet. Even though we "know" the atomic state (and by theory it's energy) it's only through indirect means that this is known. I know for a fact the Born rule was used 6 ways to Tuesday in the development of that polarized ion source.

 

[Paul Colby]

Actually, I would go one step further and point out that the energy of an atomic state has a natural line width due to vacuum polarization and such. So, even atoms don't have perfectly well defined energy levels.

[Edit] I see this is what you are pointing out. The Born rule does predict the average state energy. Yep, still don't know what you're on about.

 

[A. Neumaier]

the statement, "I have an hydrogen atom in state x" is not the result of direct measurement of the atoms energy

Of course. It is always the result of a preparation of the atom in the given state (though a judicious arrangement of sources and filters). That''s how the inputs to quantum experiments are created. Born's rule doesn't enter. To find out the state if it is not known (e.g., to find out whether the given state is an eigenstate of H or a thermal state) is impossible unless one has a large number of identically prepared atoms. Here Born's rule enters, but applied to very simple observables different from the total energy - typically binary observables of yes-no type, in case of photons also of quadrature measurements.

The Born rule does predict the average state energy.

The Born rule, taken literally, predicts that each measurement of the total energy produces exactly one energy level, at the exact value given by an eigenvalue of the Hamiltonian (normalized so that the ground state has zero energy). For a large ensemble of identically prepared atoms one can deduce from Born's rule a prediction of the average value of the total energy obtained from all these measurement values. This average value equals the expectation value computed from the theory. But because there is no way to perform the first part one cannot perform the second part which needs the first part as input. Thus Born's rule does not even predict the average total energy!

Yep, still don't know what you're on about.

An average interpretation makes not even sense if the individual measurements are fictitious of which it should be an average.

Thus if one starts with Born's rule one needs substantial handwaving to arrive for any observable $A$ at an interpretation of $Tr\rho A$ as an expectation value!

 

[A. Neumaier]

I know for a fact the Born rule was used 6 ways to Tuesday in the development of that polarized ion source.

Yes, Born's rule is often used, but quite often its use is not justified by its formulation, because most often it is used without direct reference to measurement, although the latter figures explicitly in its definition. And its conventional formulation is faulty because (unlike with any other physical rule) the assumptions are never stated under which it is valid, namely the conditions I gave in my post #28.

Thus everything surrounding Born's rule and its application contains a large dose of hand-waving!

This doesn't matter for experimental practice. But it is the cause of the century-long and wide-spread dissatisfaction with the foundations of quantum mechanics.

 

[A. Neumaier]

the energy of an atomic state has a natural line width due to vacuum polarization and such. So, even atoms don't have perfectly well defined energy levels.

My criticism still applies, though now to the more general version of Born's rule for measuring observables with a continuous spectrum.

On the other hand, in statistical mechanics it is commonly assumed that the (for a macroscopic object very densely spaced) spectrum of H is discrete, since otherwise the partition sum makes no longer sense. Thus I based my arguments on this widely used idealization.

 

[Prathyush]

I only claimed that Born's rule doesn't work for the measurement of the total energy of a helium atom. For this it doesn't matter whether other things can be interpreted with the Born rule, but only that Born's rule, applied to the observable HH, does not apply. A measurment of HH (if itisi at all possible) never gives a single energy level (neither exactly as claimed by Born's rule nor even approximately as a relaxed version would perhaps claim), picked at random from the full list according to the probabilities stated in the rule. I'd like to see the experimental arrangement that would do this!

Point is the following, if you create the atom(say the ground state). If you probe it with particle x, we can precisely calculate the probability amplitudes of the given process. Consider situations where the helium atom remains in the ground state for instance. Situations where helium atom remains in an excited state is somewhat subtle to analyze(atleast for me.)

an amplitude such as |x,He> -> |y,He>, where x is some initial state of the photon. Y could be a multi particle state treated suitable using an appropriate second quantized framework.

This amplitude given by $<y,He|e^{-iHt}|x,He>$ has a sharp and precise meaning. And that is what is measured in a lab experimentally. The probability is $|<y,He|e^{-iHt}|x,He>|^2$ this quantity has a precise meaning and can only be interpreted using the Born's rule. How we measure the test particles is irrelevant for these considerations.

 

[A. Neumaier]

This amplitude ... has a sharp and precise meaning. And that is what is measured in a lab experimentally. ... and can only be interpreted using the Born's rule.

Yes, but it measures a state transition and not the total energy. I have no problem accepting Born's rule as the appropriate tool for interpreting scattering processes.

But it is already nontrivial to say which observable (in the sense of the conventional formulation of Born's rule) corresponds to the transition measured. In any case, a different observable figures for different transitions. And there are many more possible transitions than there are energy levels, hence one cannot even figuratively take one for the other.

 

[Prathyush]

Yes, but it measures a state transition and not the total energy.

That is not particularly important. Any experiment you construct can be analyzed using the framework of quantum mechanics using the Born's rule, which in general assigns probabilities to processes. I am trying to understand what it means to amplify microscopic information into macroscopic observables. The point is the following, the stage for final amplification can be separated from interaction of the probe particle with the system.

 

[ftr]

Arnold, what do you think of this experiment

https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.110.213001