How Does the Statistical Interpretation Address Quantum Interference Effects?

[Quantum River]

I thought Statistical Interpretation of quantum mechanics is a very good alternative besides the orthodox interpretation. But I just can't explain the particle-wave duality from the statistical point.

How could the Statistical Interpretation explain the interference effects in the Quantum mechanics?

 

[marlon]

But I just can't explain the particle-wave duality from the statistical point.

Forgive me my ignorance but the p/w duality is described by statistics per definition. I mean, if you look at the dubble slit experiment, the probability of a particle "passing through both openings" is NOT equal to the sum of the probabilities of "passing through one opening" (while the other one is closed).

Conclusion : the classical additivity for probabilities does no longer work here ! That is the entire point of the p/w duality. Statistics are an inherent part of this concept from the beginning on.


marlon

 

[zbyszek]

I thought Statistical Interpretation of quantum mechanics is a very good alternative besides the orthodox interpretation. But I just can't explain the particle-wave duality from the statistical point.

How could the Statistical Interpretation explain the interference effects in the Quantum mechanics?

Perhaps I could be of service here. Let's illustrate the idea on the double-slit experiment.

Single electrons (i.e. one at the time) pass through a screen with two slits and hit a
detector that tells their positions. One electron leaves one spot on the detector. There
are no waves.

After the detector collects many spots a pattern in their positions appears: the interference
fringes.

Does this emergent pattern mean that electrons are waves?
Not at all. It means that there is a correlation between all the electrons.

How can they be correlated if they have never been in contact with each other, you may ask?
The preparation procedure of the electrons is common to all of them. They all are
prepared in the same way. And this preparation procedure is responsible for the correlations.

If one prepares the electrons in different ways, no interference fringes will show up, in general.

Cheers!

 

[marlon]

Does this emergent pattern mean that electrons are waves?
Not at all. It means that there is a correlation between all the electrons.

I don't think this is a very good conclusion. First of all, one should not be asking the question whether electrons ARE waves. This is NOT what the double slit exp tell us. It tells us that particles can BEHAVE like waves. the interference that is shown is a typical wavelike behaviour in classical physics. Particles do NOT exhibit this type of behaviour in that particular physical regime. However, the double slit exp proves us that on the atomic scale, particles (as defined in classical physics) DO exhibit this wavelike behaviour that we defined in classical physics.

Also, what do you mean by saying "the electrons are correlated" ? How are they correlated ?

If one prepares the electrons in different ways, no interference fringes will show up, in general.

Cheers!

Are you saying that the outcome of the double slit experiment depends on how the electrons are prepared ? If you look at the formal description of this experiment in any intro QM book, can you tell me what they say about "electron preparation" ? Finally, what exactly do you mean by "electron preparation" ?

regards

marlon

 

[Careful]

The preparation procedure of the electrons is common to all of them. They all are
prepared in the same way. And this preparation procedure is responsible for the correlations.

If one prepares the electrons in different ways, no interference fringes will show up, in general.

Cheers!

Ohw, do you have experimental evidence to back that up ?! I mean, you have some hidden messengers in mind here I guess which communicate wheter both slits are open or not depending upon outcomes of previous measurements ?

 

[zbyszek]

I don't think this is a very good conclusion.

What is the very good conclusion, my dear physics guru?

First of all, one should not be asking the question whether electrons ARE waves. This is NOT what the double slit exp tell us. It tells us that particles can BEHAVE like waves.

And which particle behaved in such a way? In my eyes no one. The interference shows up
on the level of an ensemble of electrons. No single electron has been captured behaving
like this.

the interference that is shown is a typical wavelike behaviour in classical physics. Particles do NOT exhibit this type of behaviour in that particular physical regime. However, the double slit exp proves us that on the atomic scale, particles (as defined in classical physics) DO exhibit this wavelike behaviour that we defined in classical physics.

I don't know what you are writing about. The original question had nothing to do with
classical physics and as far as interference is concerned one can do without.

Also, what do you mean by saying "the electrons are correlated" ? How are they correlated ?

The probability of finding a spot in some regions of the detector is greater than in others.
By the "correlation" I mean that all the electrons where drawn from the same probability
density (i.e. a single particle correlation function) which in turn is determined by the
incident electron preparation procedure.

Are you saying that the outcome of the double slit experiment depends on how the electrons are prepared ?

Yes. I'm saying precisely that.

If you look at the formal description of this experiment in any intro QM book, can you tell me what they say about "electron preparation" ? Finally, what exactly do you mean by "electron preparation" ?

One more example that reading without understanding has no use. Check out
the electron preparation procedure in the Tonomura experiment (A.Tononmura et al. ,
Am. J. Phys. 57, 117 (1989)).
Since you seem to like classics check out what has to be done to observe fringes in
the Young experiment with a white light source. Or better yet try to do this pretty
old experiment on your own. Is it possible to see fringes there with the use of only two
screens (one with the slits and one as a detector), my guru?
Or, perhaps, I should not be asking this question either?

Cheers!

 

[zbyszek]

Ohw, do you have experimental evidence to back that up ?! I mean, you have some hidden messengers in mind here I guess which communicate wheter both slits are open or not depending upon outcomes of previous measurements ?

Hi Careful,

Yes, I do. The electrons have to be selected very carefully. For example, if their momenta
vary too much the interference fringes go away. This is because incident electrons with different momenta give interference fringes in different places.
More the colimation of the beam of electrons has to be good. Take a look at the
figure 3 in A. Tonomura ..., Am. J. Phys. 57, 117 (1989). It shows what it takes
to prepare the electrons.

Another example are experiments with light. It is very easy to see fringes if you have
a laser at your disposal. With a white light source it is not so simple.

No messengers in mind :).

Cheers!

 

[Gokul43201]

If you look at the formal description of this experiment in any intro QM book, can you tell me what they say about "electron preparation" ?

marlon, see Sakurai Ch.1 Section 1.4 and Ch.3 Section 3.4 for preparation of a pure ensemble (unless this is not what zbyszek is referring to).

 

[marlon]

What is the very good conclusion, my dear physics guru?

The one i gave you.

And which particle behaved in such a way? In my eyes no one.

I am not talking about one single particle. Remember that in my very first post in this thread, i clearly stated that the "statistics" are inherent to the double slit experiment. So for me to be talking about one single particle, would contradict my own words.

What i said was that on the atomic scale, EVERY particle exhibits wavelike properties. The socalled duality does not apply to certain particles and not to others. One can replace the electrons by photons etc etc...The outcome of the double slit exp would still reveal the same kind of information.

The interference shows up
on the level of an ensemble of electrons. No single electron has been captured behaving
like this.

But I WAS talking about ensembles...STATISTICS remember...

I don't know what you are writing about. The original question had nothing to do with
classical physics and as far as interference is concerned one can do without.

Actually, what i wrote there is the ONLY and correct way in which you should interprete the p/w duality. We don't call this atomic scale behaviour a DUALITY between waves and particles for no reason, you know.

The probability of finding a spot in some regions of the detector is greater than in others.
By the "correlation" I mean that all the electrons where drawn from the same probability
density (i.e. a single particle correlation function) which in turn is determined by the
incident electron preparation procedure.

One does NOT need all that to properly describe the particle wave duality and how it can be shown by the double slit experiment.

Yes. I'm saying precisely that.

Well, my point is that within the context of this thread, that is not the key aspect because you don't need it to describe the duality.

One more example that reading without understanding has no use. Check out
the electron preparation procedure in the Tonomura experiment (A.Tononmura et al. ,
Am. J. Phys. 57, 117 (1989)).

For that matter, check out how the double slit exp is introduced in Bransden and Joachain's INTRO TO QM.

Again, i know what the experimental requirements are, but that is NOT what i was talking about. I already told you that i was objecting against the way you look at the actual duality (ie the point i made, to which you so eloquently replied "i don't know what you are talking about").

Since you seem to like classics check out what has to be done to observe fringes in
the Young experiment with a white light source. Or better yet try to do this pretty
old experiment on your own. Is it possible to see fringes there with the use of only two
screens (one with the slits and one as a detector), my guru?

LOL...This is irrelevant. I am not talking about "white light" versus laser light etc etc. i know what happens with each light source. You are missing the point : in the case of the double slit exp, the formalism is important and is very clear. I objected against your interpretation of what happens with the "particles" in the double slit exp.

Or, perhaps, I should not be asking this question either?

Cheers!

I dunno...

Cheers

 

[Careful]

No messengers in mind :).

Cheers!

So, what do you have in mind then if I may ask ?

Careful

 

[zbyszek]

I am not talking about one single particle. Remember that in my very first post in this thread, i clearly stated that the "statistics" are inherent to the double slit experiment. So for me to be talking about one single particle, would contradict my own words.

What i said was that on the atomic scale, EVERY particle exhibits wavelike properties.

Thank you for your patience, my guru! Let me ask for some more, please.

Your are not talking about any single particle, right?
Still, "EVERY' one exhibits wavelike properties. Is "EVERY" somewhat different from 'every'? If it is not, then can one infere from the above that each particle behaves like that? If this is so, could I humbly ask about the source of such wisdom?

I have heard a heresy that members of the ensemble do not necessarily inherit its properties.

Actually, what i wrote there is the ONLY and correct way in which you should interprete the p/w duality. We don't call this atomic scale behaviour a DUALITY between waves and particles for no reason, you know.

Next time my weak mind deviates from THE WAY, I promise to punish (i.e. enrich, of course) myself with contemplating the depth of your original post.

Cheers!

 

[marlon]

Thank you for your patience, my guru! Let me ask for some more, please.

Your are not talking about any single particle, right?
Still, "EVERY' one exhibits wavelike properties. Is "EVERY" somewhat different from 'every'?

Well, let me explain. In every intro to the double slit experiment, people talk about a bunch of particles that behave like waves. This is obviously correct. If you take out one particle, and both slits are open, we cannot deduce which slit the particle went through. Nor can we explain what happens with the actual trajectory from source to destination. All we can do is "sum up over all possible trajectories". If a particle would NOT respect the wavelike duality (which you seem to imply) this behaviour would NOT be possible for very obvious reasons. One particle with certain initial conditions will appear somewhere on the detector. Another particle with the exact same initial conditions has a large probability of appearing in another position of the detector. Moreover, the spread in momentum will be INFINTE. Hey, guess what, this story sounds familiar, no ? THAT IS MY POINT.


Also, you are clearly not famliar with the QM concept of self interaction. This applies TO ONE SINGLE PARTICLE, and guess what this stuff means in the context of the double slit exp. YES : Shooting a single photon (or electron) at a wall that contains two slits will result in an interference pattern on the wall (detector plate) beyond the holes.

If it is not, then can one infere from the above that each particle behaves like that? If this is so, could I humbly ask about the source of such wisdom?

Like you said yourself, you should not only read without understanding. Check out any intro QM book and read between the lines as well !

I have heard a heresy that members of the ensemble do not necessarily inherit its properties.

True yet irrelevant in this case. One single particle does NOT behave "classically" on the atomic distance scale. What do YOU think " NOT behaving classicaly" might mean. Try to find out this apparent mystery, my dear.

Next time my weak mind deviates from THE WAY, I promise to punish (i.e. enrich, of course) myself with contemplating the depth of your original post.

Cheers!

Not necessary, any self respecting intro QM text will do just fine.
Besides, another think for you : destructive interference does NOT mean NO interference...my dear

Good Luck

marlon

 

[zbyszek]

So, what do you have in mind then if I may ask ?

I am not sure if we connect. I say that in statistical interpretation the fringes in the double
slit experiment come from the preparation procedure and not from some assumptions
about wavelike nature of the electrons.
In other words: from the fact that the ensemble of electrons is prepered in some
wave function that exhibit fringes under position measurement.
Let me emphasize that the wave function describes an ensemble and not a single member
in that interpretation.

No need for any communication between any two electrons or a to-be-send electron and the detector.

If this doesn't help, could you formulate a specific question?

Cheers!

 

[Careful]

I am not sure if we connect. I say that in statistical interpretation the fringes in the double
slit experiment come from the preparation procedure and not from some assumptions
about wavelike nature of the electrons.
In other words: from the fact that the ensemble of electrons is prepered in some
wave function that exhibit fringes under position measurement.
Let me emphasize that the wave function describes an ensemble and not a single member
in that interpretation.

No need for any communication between any two electrons or a to-be-send electron and the detector.

If this doesn't help, could you formulate a specific question?

Cheers!

I presume that in double slit experiments, the electrons can be controlled in such a way that they hit the screen one by one with a considerable time delay. Even then, we are told, the wave pattern builds up. I do not see how you could get such result by preparing the source in some special way unless you assume that there is some physical wave guiding the particle or so. To phrase it correctly, what is the dynamics at the single particle level you have in mind to get the outcome of the double slit experiment.

 

[marlon]

I am not sure if we connect. I say that in statistical interpretation the fringes in the double
slit experiment come from the preparation procedure and not from some assumptions
about wavelike nature of the electrons.
In other words: from the fact that the ensemble of electrons is prepered in some
wave function that exhibit fringes under position measurement.

WRONG. You keep telling me that in some cases the wavelike behaviour of particles cannot be seen from the double slit experiment. Even in the case on destructive interference, there is wavelike behaviour because interference IS BY DEFINITION a wavelike property. You are using the terminology of waves to say, hey in some cases the duality is not clear. Can't you see the contradicitio in terminis, my dear ?

Let me emphasize that the wave function describes an ensemble and not a single member
in that interpretation.

WRONG AGAIN !

Look at the intensity profile of the double slit diffraction. One peak occurs in between the openings etc etc. Such behaviour CANNOT be explained when the particles are "only particle like". This violates the laws of physics. Again, you are forgetting the concept of self interaction here.

marlon

 

[zbyszek]

If a particle would NOT respect the wavelike duality (which you seem to imply) this behaviour would NOT be possible for very obvious reasons.

What would you say about Bohmian trajectories, my guru. Particles there do not behave like waves at all and still give correct predictions as to the fringes. Does this Bohm's speculation
have bad karma? What invalidates the interpretation? Why is it impossible?

Moreover, the spread in momentum will be INFINTE. THAT IS MY POINT.

Spread of what in momentum?

True yet irrelevant in this case. One single particle does NOT behave "classically" on the atomic distance scale. What do YOU think " NOT behaving classicaly" might mean. Try to find out this apparent mystery, my dear.

You do not surprise me with your knowledge of other branches of physics. But still
your revelations would be better understood if you had given a reason behind them.
How do you know the behavior of a single particle? Does it behave classically on 1 meter scale? One kilometer? Is there any theory in your textbooks of single quantum particles?

Not necessary, any self respecting intro QM text will do just fine.

Why go after second-hand truths, if the guru is in the mood for explanations?

Cheers!

 

[marlon]

I presume that in double slit experiments, the electrons can be controlled in such a way that they hit the screen one by one with a considerable time delay. Even then, we are told, the wave pattern builds up. I do not see how you could get such result by preparing the source in some special way unless you assume that there is some physical wave guiding the particle or so. To phrase it correctly, what is the dynamics at the single particle level you have in mind to get the outcome of the double slit experiment.

The guy is talking about the initial wavelike properties that need to be respected for destructive interference to occur (no interference pattern on the detector). A necessary condition for acquiring a interference is determined by the difference in pathlength between two paths that light can take to reach a zone of constructive interference on the viewing screen. This difference must be the wavelength of the light that is used, or a multiple of this wavelength. Newton rings learn us that light does not have to be coherent to produce interference patterns but such patterns are very clear when monochromatic or near-monochromatic light is used. Laser light gives the best (most visible) interference pattern.


He concludes from that that particles do not behave like waves in that case but that is WRONG since all kinds of interference are wave like behaviour, obviously.


marlon

 

[Careful]

The guy is talking about the initial wavelike properties that need to be respected for destructive interference to occur (no interference pattern on the detector). A necessary condition for acquiring a interference is determined by the difference in pathlength between two paths that light can take to reach a zone of constructive interference on the viewing screen. This difference must be the wavelength of the light that is used, or a multiple of this wavelength. Newton rings learn us that light does not have to be coherent to produce interference patterns but such patterns are very clear when monochromatic or near-monochromatic light is used. Laser light gives the best (most visible) interference pattern.


He concludes from that that particles do not behave like waves in that case but that is WRONG since all kinds of interference are wave like behaviour, obviously.


marlon

Yes, that is why I asked him about the single particle dynamics. You know that I would not say that particles behave like (or are sometimes) waves but that the wavelike property is due to a self interaction effect (actually, you could make a (classical) particle theory of waves, but that's rather difficult to control mathematically). But clearly, something wavelike needs to be present at the single particle level; or you could think about a random walk with complex amplitudes with ``ket's'' and ``bra's'' and define a particle event as the merging of a ket and bra, then you give up the notion of a fundamental particle which exists all the time but this leads to other kinds of undesirata.

 

[zbyszek]

I presume that in double slit experiments, the electrons can be controlled in such a way that they hit the screen one by one with a considerable time delay. Even then, we are told, the wave pattern builds up. I do not see how you could get such result by preparing the source in some special way unless you assume that there is some physical wave guiding the particle or so. To phrase it correctly, what is the dynamics at the single particle level you have in mind to get the outcome of the double slit experiment.

O.K. Now I get it.

I have no idea, what is the dynamics of a single particle. It could be bohmian trajectories
or another non-local hidden variables speculation. The theory of single particles
is not available so far.

However, in the statistical interpretation as formulated by Ballentine in Rev.Mod. Phys. 42, 358 (1970), one can read where the followers place the origin of the interference fringes.
Instead of blaming single objects of behaving like waves it is pointed out that the preparation
procedure will suffice.

Cheers!

 

[marlon]

What would you say about Bohmian trajectories, my guru. Particles there do not behave like waves at all and still give correct predictions as to the fringes.

Don't bring in other topics. Let's stick to what we are talking about : the double slit experiment and how it proves the wavelike nature of particles. Particles, in QM, ALWAYS exhibit wavelike properties : THAT IS THE ENTIRE POINT OF THE DUALITY. This duality does NOT give us an "either wave or either particle like" picture.

Spread of what in momentum?

LOL. Don't you know what momentum is ?

HINT : momentum of a particle + uncertainty principle, my dear believer

You do not surprise me with your knowledge of other branches of physics. But still
your revelations would be better understood if you had given a reason behind them.

Self interaction IS an essential part of QM.

How do you know the behavior of a single particle? Does it behave classically on 1 meter scale? One kilometer? Is there any theory in your textbooks of single quantum particles?

Are you being serious here ? When did i ever claim we knew the exact boundary between classical physics and QM ? There is NO such boundary. We are ONLY talking about what actually happens at the atomic scale where basic QM principles like the p/w duality, superposition or the HUP manifest themselves.

But to be clear, are you saying that a particle's self interaction does not exist ?

Why go after second-hand truths, if the guru is in the mood for explanations?

Cheers!

That's not good policy to be doing since. Take THAT as lesson nr 1 from your guru.

marlon

 

[marlon]

Yes, that is why I asked him about the single particle dynamics. You know that I would not say that particles behave like (or are sometimes) waves but that the wavelilke property is due to a self interaction effect (actually, you could make a (classical) particle theory of waves, but that's rather difficult to control mathematically).

Ok got it.

But i just want to be clear here : the particle/wave duality does NOT say that particles are waves or something like that. this duality just states that at the atomic scale, particles exhibit wavelike behaviour like in the double slit experiment. This ALWAYS happens for each kind of particle like electrons photons, etc etc. Wavelike behaviour is a general term that denotes all the types of interference, diffraction, etc etc...

We need this duality because we look at QM through "classical eyes". I mean, we need this duality because we were first familiar with concepts like particles and waves. Should we look at QM through QM glasses, we would never make a distinction between a particle and a wave. The duality would just be a physical behaviour inherent to our QM world. You see my point ?


Now, onto the next discussion

greets
marlon

 

[zbyszek]

WRONG. You keep telling me that in some cases the wavelike behaviour of particles cannot be seen from the double slit experiment. Even in the case on destructive interference, there is wavelike behaviour because interference IS BY DEFINITION a wavelike property. You are using the terminology of waves to say, hey in some cases the duality is not clear. Can't you see the contradicitio in terminis, my dear ?

Actually, it was for Carful. Anyway, where have you seen a destructive interference of one particle?

WRONG AGAIN !

Look at the intensity profile of the double slit diffraction. One peak occurs in between the openings etc etc. Such behaviour CANNOT be explained when the particles are "only particle like". This violates the laws of physics. Again, you are forgetting the concept of self interaction here.

You keep writing ONLY, EVERY, CANNOT without any argument.
Is there any noexistence theorem for the interference explanation if one skips
the wavelike behavior of particles? Again, what is wrong with the counter example involving Bohmian trajectories?
I still hope for something beyond your statement WRONG. Do you have less guru and more
scientific justifications as well?

Cheers!

 

[Careful]

Ok got it.

But i just want to be clear here : the particle/wave duality does NOT say that particles are waves or something like that. this duality just states that at the atomic scale, particles exhibit wavelike behaviour like in the double slit experiment. This ALWAYS happens for each kind of particle like electrons photons, etc etc. Wavelike behaviour is a general term that denotes all the types of interference, diffraction, etc etc...

Yes, yes we know this to be true (dont worry).

We need this duality because we look at QM through "classical eyes". I mean, we need this duality because we were first familiar with concepts like particles and waves.

Well, that is why the logical step would be to try to formulate either a wave theory of particles or a particle theory of waves or something entirely new (like this ket and bra random walk) which captures both. The first was tried by Barut and others with partial succes, I for instance am digging in getting a decent formulation for the second (but it is hard), the last alternative is always cool.

Should we look at QM through QM glasses, we would never make a distinction between a particle and a wave. The duality would just be a physical behaviour inherent to our QM world. You see my point ?

Your point is merely the standard Copenhagen view, which brings you into trouble with relativity and realism. I think a solution to the wave particle duality problem is still of considerable importance although not that flashy to work on.

 

[zbyszek]

Don't bring in other topics. Let's stick to what we are talking about : the double slit experiment and how it proves the wavelike nature of particles. Particles, in QM, ALWAYS exhibit wavelike properties : THAT IS THE ENTIRE POINT OF THE DUALITY. This duality does NOT give us an "either wave or either particle like" picture.

If you read again our discussion you might notice that I have just given you an explanation
of the fringes without assuming the wavelike nature of particles.

LOL. Don't you know what momentum is ?

Yeah, I heard something. But more interested in the momentum of what ("of what" is the key here) has INFINITE spread.

Self interaction IS an essential part of QM.

So you keep telling me. But again without any justification.

Are you being serious here ? When did i ever claim we knew the exact boundary between classical physics and QM ? There is NO such boundary. We are ONLY talking about what actually happens at the atomic scale where basic QM principles like the p/w duality, superposition or the HUP manifest themselves.

You claimed that "One single particle does NOT behave classically on the atomic distance
scale". I ask you if there is any scale where the particle behaves classically.
And more, what gives you the basis to make claims regarding the behavior of single
quantum objects?

But to be clear, are you saying that a particle's self interaction does not exist ?

Does it? What exactly do I need it for? Is this one of the postulates of QM?
Or is it derivable from QM?

 

[zbyszek]

Let me emphasize that the wave function describes an ensemble and not a single member
in that interpretation.

WRONG AGAIN !

BTW, do you know what the statistical interpretation of QM is? From the quote above one
can unjustly deduce that some more reading for guru is in order.

Ballentine's paper "The statistical interpretation of QM" (1970) would be a good start.
Preferably, before any further discussion on the subject.

You can do it!
Cheers!

 

[Quantum River]

zbyszek, could you recommend more readings of statistical interpretation of QM besides Bellentine's review?
Thanks!

 

[zbyszek]

zbyszek, could you recommend more readings of statistical interpretation of QM besides Bellentine's review?
Thanks!

There is Einstein's "Reply to criticism" available on-line.

The book "Quantum Theory: Concepts and Methods" by Asher Peres assumes statistical
interpretation and has large parts devoted to hidden variables.

I heard good things about Ballentine's textbook "Quantum Mechanics: Modern developement"
but don't have access to it yet.

Cheers!

 

[Quantum River]

Bohm's pilot wave theory should also be a good candidate to consider.
Could one connect both Bohm's theory and statistical interpretation together?

 

[zbyszek]

Bohm's pilot wave theory should also be a good candidate to consider.
Could one connect both Bohm's theory and statistical interpretation together?

Bohm already did it. The initial conditions for Bohmian trajectories are drawn
from a probability density i.e. a wave function modulus squared. Thus, the wave function
is associated with an ensemble rather than with a single particle.


Cheers!

 

[Demystifier]

Bohm already did it. The initial conditions for Bohmian trajectories are drawn
from a probability density i.e. a wave function modulus squared. Thus, the wave function
is associated with an ensemble rather than with a single particle.

Not quite!
In the Bohmian approach, the wave function has a double role. It is associated with an ensemble just as you said, but it also determines the trajectory of a single particle. You may, of course, argue that these trajectories are useless, meaningless, or whatever you want, but you cannot say that, in this particular interpretation, the wave function is not associated with properties of single particles.