Rigorous: Where is the Quantum System Prior to Measurement?

In summary, the conversation discusses the use of rigorous terminology and definitions in the interpretation of arguments in the field of quantum mechanics. It also explores the distinction between the physical world and the mathematical/graphical representation of experimental set-ups. The use of a finite region of space and the limitations of causality and the speed of light in quantum experiments is also discussed. Finally, the concept of possible paths in these experiments is introduced.
  • #1
Lynch101
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Based on feedback, I want to attempt a more rigorous argument about the 'location' of the QM system prior to measurement.
In the other thread of a similar name it was stated, and probably rightly so, that I wasn't using rigorous terminology or that I wasn't using them in a rigorous way. While I was making certain assumptions about the ability to interpret the 'jargon' I was using, it seemed to be a serious impediment to interpreting the argument that was being put forward. For that reason, I am going to attempt to define or describe the terms I am using with a bit more rigour, in the hope that that argument I am putting forward will be more intelligible. This will either highlight the error in my reasoning or, with an infinitesimal probability, prove to be a convincing argument.

This is strictly meant as an argument against the completeness of any interpretation in which the mathematics only predicts the measurement outcomes of an ensemble of similarly prepared systems.

I would ask that people apply the principles of 'steelman' (the opposite of 'strawman').The Physical Set-up
We can apply the reasoning to several different set-ups but I will have both the Stern-Gerlach and double-slit experiments in mind. I will attempt to make it clear where I am talking about one or the other.

So, in the laboratory, we have the experimental set-up like such:
1630833736734.png
1630834213209.png


These are graphical representations of real-world experimental set-ups. The equipment in the lab, the lab, and the universe are what we refer to as 'the real world' or 'the physical world'. This is contrasted with the the mathematical/graphical representation of the experimental set-up.

In the reference frame of any given user, the physical world is observed to be 3 Dimensional.

The Mathematical/Graphical Representation
We can represent the 3D world mathematically and graphically by using a co-ordinate reference system, where the three dimensions of space are represented using X, Y, and Z axes. These axes can be extended indefinitely in each direction to allow us to represent the entire 3D universe or, at least, enough of the 3D universe as we need for this argument. Using this co-ordinate system, the lab equipment and everything else in the 3D universe can be represented.Location Within Finite Regions of Space
For simplicity sake, we can imagine that there is a wooden box in the laboratory. Inside this box there is space enough for us to place an array of different items. The space inside this box, as enclosed by the boundaries/sides of the box, is a finite 'region of space'.

We can also represent the box mathematically/graphically along with the finite region of space it encloses. We can do this by drawing a cuboid with the corresponding measurements and shading the area inside the cuboid i.e. inside the lines which represent the sides. The shaded area represent the finite region of space, inside the box.

As we mentioned, we can place an array of objects (or systems) inside the box. Something is said to be 'located within the finite region of space [of the box]' if it is anywhere within the shaded area i.e. anywhere inside the box. To illustrate this point, we could place a pregnant cat inside the box and close the lid. Now, we might not be able to represent then exact position of the cat inside the box, but we can say that the cat is, most certainly, 'located within the finite region of space of the box'.

Quantum experiments are somewhat different from putting cats in boxes although thought experiments with cats in boxes can provide useful analogies.

Quantum Experiments
With our experimental set-ups we have a piece of equipment which prepares the quantum system and at some later time, t, we have a measurement outcome on a screen.

One of our assumptions is that the preparation equipment can only exert a causal influence which propagates through 3D space at a maximum speed c i.e. the speed of light and that any system created by the preparation device can travel no faster than c. If this is the case, then that allows us to define a finite region/volume of 3D space* within which the system prepared by the device must be located. If it were possible to measure the system outside this finite region of space i.e. if there were a non-zero probability of measuring it outside this region, that would necessitate that the system had traveled FTL or that it does not operate in 3D space.

*a sphere with radius ct.

Possible Paths
There are an infinite number of possible paths from the preparation device to the measured outcome and we could, theoretically, represent those infinite number of possible paths, graphically. Some of those would all fall within our sphere of radius ct. Those that fall within the sphere represent the possible paths a system limited to the speed of light could take.

Intuitively, we might think of the paths taken being those of a point particle [with a definite position and momentum], but we can't necessarily do that for quantum systems. Instead, what we can say is that whatever path the system took, from preparation device to measured position, it must have been within the finite region of space enclosed by the sphere with radius ct.

To say that something 'passes through' the finite region of space we simply mean:
1) it took one of the possible paths within the finite region of space enclosed by the sphere
2) it took every possible path within the finite region of space enclosed by the sphere
3) it too more than one possible path within the finite region of space enclosed by the sphere.

If the possible paths of the system from preparation device to measured position must fall within the sphere of radius ct, then we can say with certainty, that the system was located within the finite region of space i.e. that its position was somewhere within that finite region of space. Where position doesn't have to take a single, pre-defined value.

In the case of the Stern-Gerlach experiement, we can set-up two SG magnets (A and B) and two detector screens. But we need use only one preparation device. We can define (and graphically represent) the finite regions of space occupied by the different magentic fields. If we can say the system 'passes through' the region of space occupied by magetic field A but doesn't 'pass through' the adjacent magnetic field B, then we can narrow down the position/location of the system to the finite region of space occupied by magnet A.

Probability of Position
There is a difference between the following statements:
1) There is a non-zero probability of measuring the system in position X.
2) The system is in position X with a probability of 1 or 0.

As mentioned, we can [in principle] represent all possible paths the system could have taken from preparation device to measured outcome. These don't have to be the paths of a point particle with definite position and momentum. There are possibly an infinite number of such paths, but the probability that the system took each path is either 1 or 0 i.e. it either took Path 1, or it didn't; it either took path 2 or it didn't. This is true if the system operates in 3 dimensional space.

This means that prior to measurement, for any possible position within the finite region of space, the system is in that position with a probability of 1 or 0. It is not necessary that the system be in a single, pre-defined positon and that single, pre-defined position only, since it is possible for it to be in multiple positons with a probability of 1.

This is a separate proposition to the probability that when we measure the system it will always be in a single, well-defined position. However, if a system is always in a given positon with a probability of 1 or 0, and cannot be only one single position with a probability of 1, it must be in multiple positions simultaneously. If we always measure it in only one well-defined position then there must be some sort of physical, FTL collapse of the system.

Alternatively, the system does not operate in 3 dimensions.Schroedingers Kittens
To use an analogy. Imagine that we have never witnessed kittens being born before. For some reason, the birthing process only ever seems to occur when we we put a kitten preparation device (i.e. a cat) into a closed, finite region of space e.g. a closed box. We have an intuitive idea about how the birthing process works but our calculations tell us that cats giving birth to kittens don't behave the same way that other animals do.

So, we put the pregnant cat in the box and we know that at some time t when we open the box, we will find a single kitten in a well defined position inside the box. But, prior to opening the box we don't know what the position of the kitten is. However, we know for absolute certain that it must have some position inside the box. The probability of the position of the kitten in the box cannot be 0 everywhere in the box, it must be 1 in at least one position.

Our calculations tell us that it cannot be 1 in just one position, so therefore, it must be 1 in multiple positons. Otherwise, the birthing process does not occur in only 3 dimensions.
 
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  • #2
It might be helpful to think of this question in another way.

We can define the finite region of space representing all possible paths of the system, as the sphere with radius ct. We can then sub-divide that region into individual values for position, that would usually represent the well-defined values for point particles. We can then say, each position in that finite region of space is either occupied by something or it is not, with a probability of 1 or 0.

If we return values of 0 for every position, then there is nothing in that finite region of 3D space and our quantum system does not operate in 3 dimensions.
 
  • #3
Lynch101 said:
It might be helpful to think of this question in another way.

We can define the finite region of space representing all possible paths of the system, as the sphere with radius ct. We can then sub-divide that region into individual values for position, that would usually represent the well-defined values for point particles. We can then say, each position in that finite region of space is either occupied by something or it is not, with a probability of 1 or 0.

If we return values of 0 for every position, then there is nothing in that finite region of 3D space and our quantum system does not operate in 3 dimensions.
Why 1 or 0? What if for some place the probability is 0.1?
 
  • #4
The probability of the position of the kitten in the box cannot be 0 everywhere in the box, it must be 1 in at least one position.
I don't have much knowledge of quantum mechanics, I hope you'll forgive my meddling.
Probability 1 for one position means that all position is false, except for this position. Assigning only 0 and 1 isn't appropriate when we're dealing with a superposition where multiple positions are possibles and that's why values between 0 and 1 are used.

The certainty that the particle is in some space beetween a and b is well described by the integral:
##[ tex ]
\int_a^b | psi(x) |^2 \ dx = 1
[ /tex ]

it isn't necessary to assumy that the probability is 1 somewhere in the box and 0 in the others, but that the sum of them for all possible position measurements is equal to1.
 
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  • #5
martinbn said:
Why 1 or 0? What if for some place the probability is 0.1?
Just to make the distinction again, there is a difference between the probability of measuring the system with a single well-defined position and the system occupying a position prior to measurement. I think you might be referring to the former.

Any given region of space is either occupied or it is not. That is, there is a probability of 1 or 0 that it is occupied. We can extend this to the quantum system. Either the given region of space is occupied by the quantum system or it is isn't, this is with a probability of 0 or 1. If we ascribe 0 to every position in the finite region of space or indeed the universe, then the system either doesn't exist or it doesn't operate in 3D space.

This is separate to the probability of measuring it in a given region of space.
 
  • #6
Ghost Quartz said:
Probability 1 for one position means that all position is false, except for this position. Assigning only 0 and 1 isn't appropriate when we're dealing with a superposition where multiple positions are possibles and that's why values between 0 and 1 are used.
Not necessarily. There could be a probability of 1 for multiple positions. There cannot, however, be a 0 for all positions.

Ghost Quartz said:
The certainty that the particle is in some space beetween a and b is well described by the integral:
##[ tex ]
\int_a^b | psi(x) |^2 \ dx = 1
[ /tex ]

it isn't necessary to assumy that the probability is 1 somewhere in the box and 0 in the others, but that the sum of them for all possible position measurements is equal to1.
We don't need to assume it. It is either true or it is false. We can explore the consequences of it being true and the consequences of it being false.
 
  • #7
Lynch101 said:
Any given region of space is either occupied or it is not. That is, there is a probability of 1 or 0 that it is occupied.
It is obvious that you haven't understood the meaning of probability.
 
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  • #8
WernerQH said:
It is obvious that you haven't understood the meaning of probability.
If something has a probability of 1 then it is 100% certain.
 
  • #9
Lynch101 said:
Any given region of space is either occupied or it is not. That is, there is a probability of 1 or 0 that it is occupied. We can extend this to the quantum system. Either the given region of space is occupied by the quantum system or it is isn't, this is with a probability of 0 or 1. If we ascribe 0 to every position in the finite region of space or indeed the universe, then the system either doesn't exist or it doesn't operate in 3D space.

This is separate to the probability of measuring it in a given region of space.
None of this has any basis in the actual math of QM. So either you are proposing a personal theory, or you are simply failing to ground your interpretation in the actual math of QM.

Neither of those things are a valid basis for PF discussion.

You are claiming to make a rigorous argument, but without a grounding the actual math of QM, that claim is simply false. If you cannot give a grounding for your argument in the actual math of QM, then there is no valid basis for discussion and this thread will be closed.
 
  • #10
PeterDonis said:
None of this has any basis in the actual math of QM. So either you are proposing a personal theory, or you are simply failing to ground your interpretation in the actual math of QM.

Neither of those things are a valid basis for PF discussion.

You are claiming to make a rigorous argument, but without a grounding the actual math of QM, that claim is simply false. If you cannot give a grounding for your argument in the actual math of QM, then there is no valid basis for discussion and this thread will be closed.
I don't think I can claim the argument that the statistical interpretation of QM does not give a complete description of physical reality as my own. I think EPR among others beat me to the punch. I also don't think what I've written would even come remotely close to being considered a theory. I think you are giving me way too much credit in that regard.

As for making a rigorous argument: there appeared to be some difficulty interpreting the idea of 'location within a finite region of space' in the previous thread, so I did my best to outline what was meant using some very basic mathematical principles. Hopefully that idea is now easier to understand.

With regard to being grounded in the mathematics of QM: the point being made is that the statistical interpretation is not a complete description of the system. There are interpretations which are potentially complete descriptions such as Bohmian Mechanics, Everettian Many Worlds, Spontaneous Collapse theories, etc. The reasoning outlined above can be applied to those interpretations so, if it isn't grounded in the mathematics of QM it would imply that those interpretations are also not grounded in the mathematics of QM.
 
  • #11
What still seems unclear to me is how you are connecting this idea of local things with the desire for completeness. Your supposition that there is a spatial boundary, or or a requirement for effective FTL, is relatively straightforward.

But that does not in any way connect with the desire to make QM behave like classical physics.

Why does having the QM experiment in a lab in Paris, and declaring the QM system was within that local part of the universe cause things to become complete?
 
  • #12
Lynch101 said:
I don't think I can claim the argument that the statistical interpretation of QM does not give a complete description of physical reality as my own.
That's not the claim you are trying to make a rigorous argument for in this thread. The claim you are trying to make a rigorous argument for has to do with the "location" of a quantum system prior to measurement, which has nothing whatever to do with the statistical interpretation. It is a separate claim and needs to be judged on its own merits.

Lynch101 said:
As for making a rigorous argument: there appeared to be some difficulty interpreting the idea of 'location within a finite region of space' in the previous thread, so I did my best to outline what was meant using some very basic mathematical principles.
"Using some very basic mathematical principles" is not enough. You need to ground your interpretation in the basic math of QM. This forum is for discussing interpretations of QM, not just general mathematical principles.

Also, there is no need for you to try to invent your own version of "location" in terms of the basic math of QM, since there is already a well known interpretation of QM that has such a concept: the Bohmian interpretation. So you already have a starting point.

Lynch101 said:
With regard to being grounded in the mathematics of QM: the point being made is that the statistical interpretation is not a complete description of the system.
Again, that is not the claim you are trying to make a rigorous argument for in this thread. See above.
 
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  • #13
Lynch101 said:
There are interpretations which are potentially complete descriptions such as Bohmian Mechanics, Everettian Many Worlds, Spontaneous Collapse theories, etc. The reasoning outlined above can be applied to those interpretations so, if it isn't grounded in the mathematics of QM it would imply that those interpretations are also not grounded in the mathematics of QM.
At this point you are venturing into personal speculation. You really, really, really need to spend a lot more time becoming familiar with the basic math of QM and what the different interpretations of QM actually say, and you really, really, really need to ground your discussion in the basic math of QM, or, if you are going to make claims about a particular interpretation of QM, in the literature that discusses that interpretation.

This thread is closed.
 
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Related to Rigorous: Where is the Quantum System Prior to Measurement?

1. What is the meaning of "Rigorous: Where is the Quantum System Prior to Measurement?"

"Rigorous: Where is the Quantum System Prior to Measurement?" is a question that highlights the fundamental uncertainty in quantum mechanics. It refers to the fact that the state of a quantum system cannot be precisely determined until it is measured, and even then, the measurement itself can alter the state of the system.

2. Why is this question important in the field of quantum mechanics?

This question is important because it challenges our understanding of how the physical world works. In classical mechanics, the state of a system can be determined at any given time, but in quantum mechanics, this is not the case. It also has implications for the interpretation of the famous Schrödinger's cat thought experiment.

3. What are some proposed answers to this question?

There are several proposed answers to this question, including the Copenhagen interpretation which states that the quantum system does not have a definite state until it is measured, and the many-worlds interpretation which suggests that all possible outcomes of a measurement exist simultaneously in different parallel universes.

4. Is there a consensus among scientists on the answer to this question?

No, there is currently no consensus among scientists on the answer to this question. Different interpretations of quantum mechanics have been proposed, and the debate continues within the scientific community.

5. How does this question relate to the concept of superposition?

This question is closely related to the concept of superposition, which is the idea that a quantum system can exist in multiple states at the same time. The question of where the system is prior to measurement is essentially asking which of these possible states the system is in before it is observed.

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