Clarifying Neumaier's Interpretation of Quantum Mechanics: Wave or Particle?

[A. Neumaier]

I don't know, how Schrödinger interpreted multi-particle wave functions either.

He didn't. I think he just remarked that his 1-particle recipe no longer worked.

According to Schrödinger each single (!) electron should give a wide-spread continuous distribution, while we always find a point-like track

Why should it? In the situation you describe, the charge distribution of the single electron before the first ionization is not measurable at all!

The attempt to measure it heavily affects the state of the electron. As discussed in another thread in the context of Mott's analysis, the very first ionization effected by the single electron changes its state to one with a tiny cross section (reasonably well-defined transversal position) and a reasonably well-determined longitudinal momentum only, with a cigar-shaped charge distribution, whose track is measured!

The only situations where the charge distribution of a single electron can be measured are those where the measurement doesn't influence this distribution significantly, so that repeated measurements can be made. This is the case

• (i) for an electron in a well-collimated beam, and
• (ii) for an electron in a temporally stable stationary state.

In the first case, the charge distribution is cigar shaped and the experimental tracks confirm this. In the second case, the electron must be in the ground state, the charge distribution is shaped according to an equipotential surface of the static electric potential it experiences, and atomic resolution microscopy confirms this, too.

 

[vanhees71]

Sure, and that's why Schrödinger's original interpretation doesn't make physical sense!

Case (i) measures not a single-electron position but the position of many electrons. Very often you can consider the electrons in such a "macroscopic beam" as independent particles (which, however, is no longer the case with modern accelerators of high-intensity beams; the space-charge effects are a complicated problem there!) and thus take it as (approximate) ensemble. So you measure (approximately) a single-particle distribution function through ensemble averaging.

Case (ii) describes what's measured with a tunnel microscope!

 

[A. Neumaier]

Sure, and that's why Schrödinger's original interpretation doesn't make physical sense!

Of course it does. In all cases where one can measure the charge distribution of a single electron it conforms to his interpretation. Moreover, his interpretation is just a special case of the interpretation of the expectation of the charge density operator as the observable charge density, which is something that is verifiable both macroscopically through the classical limit of electrodynamics and microscopically through a tunneling microscope.

Thus it is safe to assume it also in cases where one cannot measure it. After all, we also assume that the standard properties of macroscopic bodies exist and behave according to the laws of physics as found where we can measure it - even in cases such as the deep interior of the Earth or the sun where we cannot measure anything (except indirectly by assuming the laws)!

 

[A. Neumaier]

Case (i) measures not a single-electron position but the position of many electrons.

?

A single track measures a single electron, as you had asserted before:

each single (!) electron [...] we always find a point-like track

 

[vanhees71]

If we have a single electron and measure its position we see a pointlike trace on a screen (or nowadays a pixel in a digital device) but not a spread charge distribution. Only an ensemble of electrons, each leaving such a pointlike trace, measures the spread position probability distribution. If you were right, and Schrödinger's interpretation were correct, we'd not have all these discussions on interpretation at all, because then quantum theory would be simply a classical field theory as classical electrodynamics, but that's not what's observed in nature! I don't know, if you've ever seen this in a lab. As physicists we have had to do such experiments in the mandatory labs, and that's why it's clear to us!

BTW: I tried to find a paper by Schrödinger on many-electron atoms. So far to no success. He only discusses the single-particle wave function and clearly rejects the probabilistic interpretation (in the last section of the book containing his "Four lectures on wave mechanics").

 

[A. Neumaier]

clearly rejects the probabilistic interpretation (in the last section of the book containing his "Four lectures on wave mechanics").

Can you please quote his rejection statement and his reasons for it?

 

[vanhees71]

Schrödinger, "Four Lectures on Wave Mechanics" (1928)

An obvious statistical interpretation of the $\psi$-function
has been put forward, viz. that it does not relate to a
single system at all but to an assemblage of systems,
$\psi \bar{\psi}$ determining the fraction of the systems which happen
to be in a definite configuration. This view is a little
unsatisfactory, since it offers no explanation whatever
why the Quantities $a_{ki}$ yield all the information which they
do yield*. In connexion with the statistical interpretation
it has been said that to any physical quantity which would
have a definite physical meaning and be in principle (prin-
cipiell) measurable according to the classical picture of the
atom, there belong definite proper values (just as e.g.
the proper values $E_k$ belong to the energy); and it has
been said that the result of measuring such a quantity
will always be one or the other of these proper values,
but never anything intermediate. It seems to me that
this 1st statement contains a rather vague conception, namely
that of measuring a quantity (e.g. energy or moment of
momentum), which relates to the classical picture of the
atom, i.e. to an obviously wrong one. Is it not rather bold
to interpret measurements according to a picture which
we know to be wrong? May they not have quite another
meaning according to the picture which will finally be
forced upon our mind? For example: let a beam of
electronic rays pass through a layer of mercury vapour, and
measure the deflection of the beam in an electric and in a
magnetic field before and after the beam has traversed
the vapour. According to the older conceptions this is
interpreted as a measurement of differences of energy-
levels in the mercury atom. The wave-picture furnishes
another interpretation, namely, that the frequency of
part of the electronic waves has been diminished by an
amount equal to the difference of two proper frequencies
of the mercury. Is it quite certain that these two
interpretations do not interfere with one another, and that the
old one can be maintained together with the new one?
Is it quite certain that the conception of energy,
indispensable as it is in macroscopic phenomena, has any
other meaning in micro-mechanical phenomena than the
number of vibrations in h seconds?

 

[A. Neumaier]

If we have a single electron and measure its position we see a pointlike trace on a screen (or nowadays a pixel in a digital device) but not a spread charge distribution.

As mentioned before in post #31, this is because what is measured is the track after the first ionization, where the single electron has a cigar-shaped charge distribution and moves along the cigar's long direction. Nobody expects to measure a spread charge distribution in this case.

 

[vanhees71]

Exactly! So why do you claim the correctness of Schrödinger's original interpretation of the single-particle wavefunction, which clearly is a misconception.

 

[A. Neumaier]

Exactly! So why do you claim the correctness of Schrödinger's original interpretation of the single-particle wavefunction, which clearly is a misconception.

Because in all cases where one can measure the charge distribution of a single electron it conforms to his interpretation. Before the first ionization, we cannot measure the charge distribution of the electron. So we have either deny that it has a charge distribution (which is the take of the Copenhagen interpretation), or to find a definition of the charge distribution that makes sense both in the observable and in the unobservable case (which, for a single electron, is Schroedinger's interpretation). Both points of view completely agree on the measurable part, hence are equally valid.

But the Copenhagen view has a very strange side effect: The mean of something nonexistent exists!?? How can this be? The ensemble interpretation inherits this strange property, though in a less outspoken way.

 

[vanhees71]

No they do not agree on the measurable part, because the one states that we measure a point-like location and the other states that we measure a smeared distribution. Everybody who has ever made an experiment with a weak radioactive source finds points on the scintillation screen. No single event gives a smeared distribution. QT is about what's measurable and observable not about theoretical constructs.

Scanning a probe with a tunnel microscope measures ensembles and not locations of a single particle! That's why it can resolve the distributions.

The ensemble interpretation doesn't claim that something nonexistent exists. Also the no-nonsense flavor of Copenhagen (Bohr himself included; Heisenberg has to be taken with a grain of salt ;-)) hasn't ever claimed such a thing. The only, in my opinion correct, claim is that an observable has a predetermined value if and only if the state of the system says so, i.e., if the probability to find this value is 1. That's a tautology, if you ask me!

Take the example of an electron in a hydrogen atom in its ground state. It has no definite position but a probability distribution of positions. Whenever measured you find a point-like particle (for which you usually have to destroy the atom, i.e., you have to kick out the electron, if you want a sufficient spatial resolution). If you want to measure the probability distribution you need an ensemble of hydrogen atoms. You can also use the same atom many times, you only have to make sure that you always prepare it in its ground state again before you measure the electron's position again.

 

[A. Neumaier]

Take the example of an electron in a hydrogen atom in its ground state. It has no definite position but a probability distribution of positions.

What does it mean for a single electron to have a probability distribution of positions? The latter is a property of an ensemble, not of a single system!

QT is about what's measurable and observable not about theoretical constructs.

Then what is measurable and observable about the probability distribution of positions of a single electron?

 

[A. Neumaier]

Everybody who has ever made an experiment with a weak radioactive source finds points on the scintillation screen. No single event gives a smeared distribution.

Again this is since, by Mott's argument, the first ionization changes the state of the electron to one with a point-like geometric cross section, which is then measured, and of course leads to a point-like flash. The measurement (irreversible amplification) is not one of the unperturbed electron in its spherical state emanating from the source but one of the perturbed state caused by the interaction leading to the first ionization.

 

[A. Neumaier]

Take the example of an electron in a hydrogen atom in its ground state. It has no definite position but a probability distribution of positions. Whenever measured you find a point-like particle (for which you usually have to destroy the atom, i.e., you have to kick out the electron, if you want a sufficient spatial resolution). [...] you need an ensemble of hydrogen atoms.

Don't you agree that destroying the atom definitely changes the state of the electron?

Consider the following analogous situation: To measure the chemical structure of a complex molecule you have to destroy sufficiently many copies of it, split them into pieces that can be analyzed by a mass spectrometer, and afterwards find out how overlapping fragments can be combined to a whole. At the end, if you are able to reconstruct a unique chemical formula, your measurement was successful.

What you try to argue is like saying that in the analogous situation that whenever you measure the complex molecule you always find fragments, and conclude that the complex molecule must be an ensemble of fragments with the remarkable property that they seem to come from a single molecule that is never observed in the measurement process. Whereas the natural interpretation is that there is a single molecule, which changes its state during the measurement, so that only its fragments are measured.

In the same vein, rather than arguing that when analysing an atom one always measures rays impinging on a screen that seem to come from a spherical charge distribution which is only visible in the ensemble, it is far more natural to assume that the spherical charge distribution exists but is destroyed by the measurement process, so that the latter has to be repeated often enough to recover the original, spherical structure.

The example discussed in posts #47 and #49 of the thread Are tracks in collision experiments proof of particles? suggest the same.

 

[vanhees71]

What does it mean for a single electron to have a probability distribution of positions? The latter is a property of an ensemble, not of a single system!

Then what is measurable and observable about the probability distribution of positions of a single electron?

As you say: You prepare an ensemble of independently prepared single-electrons and measure their positions, leading to the probability distribution by the usual statistical evaluation procedures. The position of the single electron is not determined when prepared in the hydrogen-atom ground state, but you know the probability distribution by solving the time-independent Schrödinger equation. You can verify the predicted probability distribution only on an ensemble. This is the standard point of view of the ensemble interpretation.

 

[vanhees71]

Again this is since, by Mott's argument, the first ionization changes the state of the electron to one with a point-like geometric cross section, which is then measured, and of course leads to a point-like flash. The measurement (irreversible amplification) is not one of the unperturbed electron in its spherical state emanating from the source but one of the perturbed state caused by the interaction leading to the first ionization.

That's what makes such a scientillator a good position-measurement device. It's one way to define the position observable operationally, and without such a definition the idea of observables is empty.