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Just a heads up, I’m not a physics major. But if you are, or just in general know more about this subject matter, can you take a second to address my (noob) questions about quantum mechanics being probabilistic:
Stern-Gerlach experiment demonstrates the spin of the electron being either up or down after going through the magnet apparatus, and the spin cannot be predicted. My question is: Is the indeterminacy and randomness of the spin an inherent and fundamental property of quantum mechanics? Or is it possible there's “something” that’s making those electrons spin a certain way that we’re not aware of and not measuring with our instruments? And can that “something” be measured if we had good enough instruments eventually in the future?
I guess to look at it another way, replace the Stern-Gerlach experiment with the Heisenberg Uncertainty Principle, which states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa. Is this uncertainty fundamental to the theory or is it based on experimental results with old instruments (hence the probabilistic description of one variable based on the complete measurement of the other in this position-momentum pair)? (Edit: Answer: The uncertainty is fundamental to this principal because it is the result of combining particle and wave nature) Is it possible we can have better instruments in the future that are good enough to measure both with certainty or is that impossible since instruments are not the issue? (Edit: Answer: Nope, you cannot measure both accurately, instruments are not the issue)
I’m interested because of the Bohr-Einstein debates. From my understanding, Albert Einstein, quoted as saying, “I, at any rate, am convinced that He [God] does not throw dice,” hated the notion that everything on the atomic and subatomic scale is random, and tried to find ways to explain that quantum determinism might in fact be a possibility. But Niels Bohr shot down any of Einstein’s attempts and it is generally accepted that Bohr “won” the debates.
So are we to go with Bohr and accept that everything in the microcosm is a random clusterfuck of chaos in opposite and stark contrast to determinism? Or is it because it seems random because we’ve might have missed something and are limited to what our instruments can currently measure, and is it possible we can overcome those limitations in the future?
I’ve always thought the randomness was inherent and fundamental to quantum mechanics, as Bohr proved there was no missing variable to be accounted for (which was what Einstein tried to go for in these debates). And the Heisenberg Uncertainty Principle says it’s not a question of the limitations of the measuring apparatus, but that having an instrument that can measure everything (which can lead to 100% prediction models) is in fact an impossible idea since the underlying quantum mechanics is fundamentally random.
Am I wrong here? Please help a physics noob out!
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Aotearoa39261 Posts
If you subscribe to the copenhagen interpretation of quantum mechanics, then afaik the spin is a result of the indeterminance of QM. However, if you subscribe to another interpretation which is deterministic like pilot-wave theory (aka de Broglie-Bohm theory) then the spin is determined by something. You'll find this interesting http://en.wikipedia.org/wiki/Pilot_wave or http://plato.stanford.edu/entries/qm-bohm/ for a more in depth explanation (there's a small section on spin as well)
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the copenhagen interpretation does indeed say that qm is fundamentally random, as plexa pointed out.
this bohm stuff is pretty weird imo, but it hasnt been disproved, so its just a matter of taste in some sense.
hidden parameter theories (as einstein tried it) have been mostly debunked and find little support anymore. check out Bell's inequation to learn more.
the best example for true randomness is probably a photon at a beam splitter
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Is it possible we can have better instruments in the future that are good enough to measure both with certainty or is that impossible since instruments are not the issue? This is impossible. The uncertainty is not due to bad methods or instruments, it is a fundamental part of the theory. I can't explain it simply (I don't know enough about it, and have been away from the material for too long), but the heisenberg inequality is a result acquired from the fundamental way quantum theory sees our observables applied to the case of the operators for momentum and position.
Sorry, I really can't say much more than this, because I'm not sure enough.
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Thanks for the responses! I will check out the links by Plexa, looks like good reading material, as well as Bell’s theorem as Paljas suggested.
I thought the Copenhagen interpretation still held the majority consensus in the world of qm? The authors of it were Bohr and Heisenberg after all, almost 90 years ago. Did it fall out of favor recently?
I now know getting both the position and momentum of a particle is impossible due to a fundamental part of qm and not because of measuring instrument limitations, according to Yorbon.
What about this: Let’s say someone said it would be possible in the future to predict the spin of one electron in one instance of the Stern-Gerlach experiment, or where one photon might end up after going through the 50:50 beam splitter, with 100% certainty after we’ve overcome our technological limitations. Is this person wrong because the nature of quantum mechanics renders such confident predictions and the existence of such an instrument impossible? (assuming these experimental results are based on the randomness of qm)
If we’re going by the Copenhagen interpretation, or the majority of physicists for that matter, I’m assuming such predictions/measuring apparatus would be impossible no matter what, since qm randomness is random. But other physicists and some of the less popular interpretations of qm (such as de-Broglie-Bohm theory) might allow the existence of such predictions and the possibility that the randomness we’re seeing might have a deterministic nature behind it?
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Wait, first off, there's a lot of different concepts being mixed together.
On April 15 2015 05:04 riotjune wrote:
Stern-Gerlach experiment demonstrates the spin of the electron being either up or down after going through the magnet apparatus, and the spin cannot be predicted. My question is: Is the indeterminacy and randomness of the spin an inherent and fundamental property of quantum mechanics? Or is it possible there's “something” that’s making those electrons spin a certain way that we’re not aware of and not measuring with our instruments? And can that “something” be measured if we had good enough instruments eventually in the future?
The Stern-Gerlach experiment shows that particles fundamentally possess angular momentum and that angular momentum is quantized. This was called "spin" since that was a familiar and classical situation which seemed to help explain a quantum result.
Let's talk about "randomness". For a general particle, it is true that you cannot predict the direction of "spin". However, once you know the direction, it stays that pointing way unless something happens. So if you measure it with a spin direction of "up", it will stay "up" and not "randomly" turn so it is "down".
Next, electrons are not spinning. They are point particles and there is no central physical location to "spin" around. As mentioned above, "spin" is just the name of this type of angular momentum. This is true of non-point particles too.
Finally, the "something" which gives electrons and particles "spin" is nature itself. These particles pop in and out of existence carrying this angular momentum.
On April 15 2015 05:04 riotjune wrote:
I guess to look at it another way, replace the Stern-Gerlach experiment with the Heisenberg Uncertainty Principle, which states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa. Is this uncertainty fundamental to the theory or is it based on experimental results with old instruments (hence the probabilistic description of one variable based on the complete measurement of the other in this position-momentum pair)? Is it possible we can have better instruments in the future that are good enough to measure both with certainty or is that impossible since instruments are not the issue?
The Heisenberg Uncertainty Principle is actually a fundamental property of waves and thus applies to quantum systems. We can prove that it's fundamental. Suppose we start off by describing a wave mathematically using a position description, ie. with the variable x. Now, if you try to change your description so that you're now using a momentum description, ie. with the variable p, you will find that those two descriptions are linked in a special way. This leads to the Uncertainty Principle that you mention. It's worthwhile to note that energy and time are also linked in this way which is the basis for a lot of really cool quantum phenomena.
On April 15 2015 05:04 riotjune wrote:
I’m interested because of the Bohr-Einstein debates. From my understanding, Albert Einstein, quoted as saying, “I, at any rate, am convinced that He [God] does not throw dice,” hated the notion that everything on the atomic and subatomic scale is random, and tried to find ways to explain that quantum determinism might in fact be a possibility. But Niels Bohr shot down any of Einstein’s attempts and it is generally accepted that Bohr “won” the debates.
So are we to go with Bohr and accept that everything in the microcosm is a random clusterfuck of chaos in opposite and stark contrast to determinism? Or is it because it seems random because we’ve might have missed something and are limited to what our instruments can currently measure, and is it possible we can overcome those limitations in the future?
I’ve always thought the randomness was inherent and fundamental to quantum mechanics, as Bohr proved there was no missing variable to be accounted for (which was what Einstein tried to go for in these debates). And the Heisenberg Uncertainty Principle says it’s not a question of the limitations of the measuring apparatus, but that having an instrument that can measure everything (which can lead to 100% prediction models) is in fact an impossible idea since the underlying quantum mechanics is fundamentally random.
Einstein was wrong in this regard. However, your description of random jumps back and forth. Things are not a "random clusterfuck of chaos in opposite and stark contrast to determinism". If you make a measurement and force a system into a stationary state, it will remain there forever. Repeated measurements will tell you the exact same result. It isn't random. The "randomness" here is that before the first measurement, the system has a probability to be in any number of states.
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On April 15 2015 10:03 riotjune wrote:
Thanks for the responses! I will check out the links by Plexa, looks like good reading material, as well as Bell’s theorem as Paljas suggested.
I thought the Copenhagen interpretation still held the majority consensus in the world of qm? The authors of it were Bohr and Heisenberg after all, almost 90 years ago. Did it fall out of favor recently?
It's not quite correct to view the Copenhagen interpretation as a consensus. It's treated as a default view largely because it was basically the first major interpretation of QM, but the reality is that most of the time a lot of physicists don't bother adhering to a particular interpretation because there is currently no testable difference between them.
Some aspects of the original mechanics of the Copenhagen interpretation would also generally be viewed as rather archaic and absurd, with the modern view being that decoherence is a more logical mechanism for wavefunction collapse.
I now know getting both the position and momentum of a particle is impossible due to a fundamental part of qm and not because of measuring instrument limitations, according to Yorbon.
What about this: Let’s say someone said it would be possible in the future to predict the spin of one electron in one instance of the Stern-Gerlach experiment, or where one photon might end up after going through the 50:50 beam splitter, with 100% certainty after we’ve overcome our technological limitations. Is this person wrong because the nature of quantum mechanics renders such confident predictions and the existence of such an instrument impossible? (assuming these experimental results are based on the randomness of qm)
If we’re going by the Copenhagen interpretation, or the majority of physicists for that matter, I’m assuming such predictions/measuring apparatus would be impossible no matter what, since qm randomness is random. But other physicists and some of the less popular interpretations of qm (such as de-Broglie-Bohm theory) might allow the existence of such predictions and the possibility that the randomness we’re seeing might have a deterministic nature behind it?
You are correct that some interpretations of QM do allow for determinism. Bell's theorem rules out local hidden-variable theories, but nonlocal HV formulations are still entirely possible.
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Nice responses above.
I would also recommend you look for some lectures on entanglement, it might help give you a better viewpoint on thinking about quantum states and their detection.
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On April 15 2015 07:28 Yorbon wrote:Show nested quote +Is it possible we can have better instruments in the future that are good enough to measure both with certainty or is that impossible since instruments are not the issue? This is impossible. The uncertainty is not due to bad methods or instruments, it is a fundamental part of the theory. I can't explain it simply (I don't know enough about it, and have been away from the material for too long), but the heisenberg inequality is a result acquired from the fundamental way quantum theory sees our observables applied to the case of the operators for momentum and position. Sorry, I really can't say much more than this, because I'm not sure enough.
I have to disagree.
Our models of the world are only as good as our experiments.
If our experiments improve, we might find better theories that can replace prior theories.
If we could make better measurements and find say substructures of the electron, we might be able to replace the wave function of the electron with different functions of less probabilistic nature.
It would be the same principle as general relativity replacing classical mechanics. Classical mechanics is still correct, but cannot explain everything. The wave functions and the standard model would still be correct, but would not explain everything.
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On April 16 2015 05:28 fancyClown wrote:
If we could make better measurements and find say substructures of the electron, we might be able to replace the wave function of the electron with different functions of less probabilistic nature.
What is this even supposed to mean?
First off, the electron is a point particle. It has no substructure. In this regard, the theory is well-established and the onus is on the experimental community to disprove them. If an electron had a substructure, we would see it by the way the matter around it behaves. Experimentally, the upper limit of the size of an electron has been reduced to the attometer scale. This is one of the most precise measurements that humans have ever been able to do. For you to just casually throw away half a century's worth of work is jaw-droppingly insulting.
Secondly, if you somehow find "substructures of the electron", you would NOT "be able to replace the wave function of the electron with different functions of less probabilistic nature". If you were working on that scale, you have have MORE probability since you're now dealing with a composite particle. If you are working on any scale that doesn't get close to seeing the fine structure of an electron, ie. anything larger than a tenth of a fermi, you'd have EXACTLY what we have now. There would be no replacement.
What you said makes no sense.
On April 16 2015 05:28 fancyClown wrote:
It would be the same principle as general relativity replacing classical mechanics. Classical mechanics is still correct, but cannot explain everything. The wave functions and the standard model would still be correct, but would not explain everything.
The Standard Model is already "wrong". We've known for about 30 years. You don't need to invent electron substructure for this.
Note: It's actually more accurate to say that the SM is incomplete than to say that it's wrong.
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On April 16 2015 05:28 fancyClown wrote:Show nested quote +On April 15 2015 07:28 Yorbon wrote:Is it possible we can have better instruments in the future that are good enough to measure both with certainty or is that impossible since instruments are not the issue? This is impossible. The uncertainty is not due to bad methods or instruments, it is a fundamental part of the theory. I can't explain it simply (I don't know enough about it, and have been away from the material for too long), but the heisenberg inequality is a result acquired from the fundamental way quantum theory sees our observables applied to the case of the operators for momentum and position. Sorry, I really can't say much more than this, because I'm not sure enough. I have to disagree. Our models of the world are only as good as our experiments. If our experiments improve, we might find better theories that can replace prior theories.
Also, WHAT??? What are you disagreeing with?
ALL waves obey the Uncertainty Principle relating position and momentum. The Heisenberg Uncertainty Principle is simply a restatement that particles at the quantum level can be described by waves.
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You can't even imagine how upset I am about how casual you are about spreading misinformation and disregarding the human experience.
The Uncertainty Principle relating position and momentum is not just some quirk or obstacle that we haven't been able to figure out yet. It's required by mathematics. You can choose to describe a wave in the position basis which most people are comfortable with. If you want to change into the momentum basis, you can easily do it with a Fourier Transform. The problem is that position and momentum are related by a derivative. This leads to the result that you so easily disagree with. IT IS NOT A FAILING OF EXPERIMENTS BUT REQUIRED BY MATHEMATICS.
Note: As I mentioned before, any pair of conjugate variables will lead to its own Uncertainty Principle. The Angular Momentum and the orientation of that Angular Momentum is such a pair. That's the underlying reason for the results of the Stern-Gerlach experiment. Another pair of conjugate variables (which I mentioned above) is energy and time. This is the reason why particles appear for an extremely short period of time and then disappear. It also explains the line widths in nuclear structure.
These phenomena are well-understood. Billions of dollars and countless people-hours have been spent extensively testing this throughout the last 80 years.
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Just to clarify, you’re yelling at fancyClown and not at me right?
I agree with you, kingjames01, and your input is very much appreciated in clarifying some of the concepts I’ve had about quantum mechanics. I know there’s probably already thousands upon thousands of experiments which confirm the accuracy of qm, and I don’t think a complete overhaul of the theory is going to come anytime soon in the face of overwhelming evidence. In fact, the experimental results will probably just keep on building upon the theory making it stronger and more complete from here on.
And by randomness, I probably should’ve clarified I meant that a system can be in any number of states before the first measurement. And going by what I know now, qm pretty much maintains that predicting the exact state of the system in one instance after the first measurement is impossible (hence qm’s indeterminance) because 1) there’s no way to accurately measure both conjugate variables as stated by Heisenberg’s uncertainty principle, and 2) the act of measuring changes the outcome (not too sure about this one). The best we can do is calculate probabilities of which states the system will end up in, and then confirm these probabilities by subjecting the system to the same experiment over and over (hope I’m getting this right).
I’ve had this argument with another person. He said it was very possible in the future to predict the results of the Stern-Gerlach experiment (or any other quantum experiments) after some technological advancements or some revision of the quantum theory. I maintained that such predictions are pretty much an impossibility due to the nature of quantum mechanics, and that any revisions to qm that would make these predictions a reality are almost impossible as well, since at this point in time we have a huge body of experimental evidence that states otherwise.
Obviously, stuff about qm is still up in the air, but I’m not going to casually ignore the research that has been done, and which has, for the most part, confirms the accuracy of quantum mechanics to this day. I guess the person I was arguing with could have been right if he was going by the de Broglie-Bohm/nonlocal hidden variable theory, which I don’t think was popular to begin with.
kingjames01 you mentioned energy and time being a conjugate variable pair, which I thought is very interesting, especially with regards to a particle disappearing.
I know I still have a lot of ground to cover to even attempt a grasp at quantum mechanics, but I find the science extremely interesting nevertheless. Maybe I’ll check out general relativity in the future as well 
Thanks to all who dropped by, and may you discover the ToE~
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On April 16 2015 12:44 riotjune wrote:
Just to clarify, you’re yelling at fancyClown and not at me right?
Right, those last 2 posts were not directed at you.
On April 16 2015 12:44 riotjune wrote:
I agree with you, kingjames01, and your input is very much appreciated in clarifying some of the concepts I’ve had about quantum mechanics. I know there’s probably already thousands upon thousands of experiments which confirm the accuracy of qm, and I don’t think a complete overhaul of the theory is going to come anytime soon in the face of overwhelming evidence. In fact, the experimental results will probably just keep on building upon the theory making it stronger and more complete from here on.
Oh Quantum Mechanics is absolutely correct. Without QM, we would not even be having this conversation. We need QM to explain semi-conductors which is the foundation of modern computing.
We need QM to understand nuclear power, without which we'd still be stuck using chemical power. Just to give you an example of the relative scales of power: take an average-sized coffee cup. Let's fill that cup with Uranium and then try to calculate the amount of energy we can get. Then let's figure out how much coal we would need to burn to get the same amount of energy. It works out that approximately 3 kg of Uranium (the amount in the cup) gives the same amount of energy for 40 metric tonnes of coal (that's a MOUNTAIN of coal).
On April 16 2015 12:44 riotjune wrote:
And by randomness, I probably should’ve clarified I meant that a system can be in any number of states before the first measurement. And going by what I know now, qm pretty much maintains that predicting the exact state of the system in one instance after the first measurement is impossible (hence qm’s indeterminance)
I just want to clarify one thing about this section. Maybe I'm misunderstanding, but let's make this more concrete by offering an example.
Suppose that we have a particle that can exist in stationary states A, B or C, with probability 20%, 30% and 50%, respectively.
Now, before a measurement, we don't actually know which state that particle is in. In fact, it's in all 3 at once. Let's make a measurement. Oh, we have found out that the particle is in state B.
Let's measure it again. It's in state B.
Let's measure it again. It's in state B.
Let's measure it again. It's in state B.
Let's measure it again. It's in state B.
Let's measure it again. It's in state B.
Let's measure it again. It's in state B.
This is a simplified example, but I just want to make clear that subsequent measurements will show that the particle is in the same stationary state. It is not random.
On April 16 2015 12:44 riotjune wrote:
The best we can do is calculate probabilities of which states the system will end up in, and then confirm these probabilities by subjecting the system to the same experiment over and over (hope I’m getting this right).
I think it will be more instructive if we pause here and bring in an extra idea. When we talk about such things, we are envisioning a situation where you can have a very large number of identical experiments. We then make a single measurement on each setup and record the result.
If you have one setup and make the same measurement over and over, you're going to get the same result each time. Again, it is not random. Once I know that you're hiding under your bed, if I look there again, I'm sure to find you. However, before I look, I have no idea if you're actually there.
If you have a lot of setups and make the same measurement once for each of them, then you can work out the probabilities for the system to be in any of the states. Here, we have many identical rooms containing a bed and possibly a riotjune. I will look under each bed to see if there is a riotjune underneath. That will tell me the probability for riotjune to hide under the bed.
On April 16 2015 12:44 riotjune wrote:
I’ve had this argument with another person. He said it was very possible in the future to predict the results of the Stern-Gerlach experiment (or any other quantum experiments) after some technological advancements or some revision of the quantum theory. I maintained that such predictions are pretty much an impossibility due to the nature of quantum mechanics, and that any revisions to qm that would make these predictions a reality are almost impossible as well, since at this point in time we have a huge body of experimental evidence that states otherwise.
It is not possible to predict the result of a single trial in the Stern-Gerlach experiment. The main ideas behind the Stern-Gerlach experiment are:
1) The electron has some built-in angular momentum (remember, angular momentum is a vector).
2) This built-in angular momentum has 3 components (since it is a vector in 3 dimensions).
3) Along any given axis (x, y or z), the angular momentum can only have a value of +1/2 or -1/2. (This is the first bizarre result. Classically, angular momentum could go from a maximum to a minimum and take any value in between.)
4) If we try to separate the electrons using the angular momentum in the z-direction, half of them will have +1/2 and the other half will have -1/2. (This assumes the electrons came from a random source.)
5) If we only use the electrons with z-direction angular momentum of +1/2 and then try to separate them using the angular momentum in the x-direction, half of them will have +1/2 and the other half will have -1/2 (in the x-direction.)
The ABSOLUTELY SHOCKING thing is what happens when separate them again. Let's say, we've taken only +1/2 in the z-direction and pass them separate them along the x-direction. Let's take all of electrons with +1/2 in the x-direction and then see what happens in the z-direction.
Half of them will have +1/2 and the other half will have -1/2 (in the z-direction).
The puzzle was why don't these electrons still only have +1/2 in the z-direction?
The answer is actually really simple in the end. Angular momentum and its orientation form a pair of conjugate variables. You can measure the magnitude and only one component of the orientation simultaneously. You just cannot do any better than that due to the Uncertainty Principle.
Alright, now that we've discussed the experiment and the puzzle, you can see that there is NO WAY, that you can predict the results of a SINGLE Stern-Gerlach trial. Your friend is incorrect. It is not a matter of deficiency in experimental technique, nor is it a failure to find the "true" Quantum model. If you can accept that position and momentum cannot be measured simultaneously, then it's the same argument why you can measure the magnitude of the angular momentum and one component of its orientation, but you cannot measure any other component simultaneously.
On April 16 2015 12:44 riotjune wrote:
Obviously, stuff about qm is still up in the air, but I’m not going to casually ignore the research that has been done, and which has, for the most part, confirms the accuracy of quantum mechanics to this day. I guess the person I was arguing with could have been right if he was going by the de Broglie-Bohm/nonlocal hidden variable theory, which I don’t think was popular to begin with.
My opinion is that it is probably better to understand what the simple parts of QM tell us before you go digging into alternative explanations.
On April 16 2015 12:44 riotjune wrote:
kingjames01 you mentioned energy and time being a conjugate variable pair, which I thought is very interesting, especially with regards to a particle disappearing.
Yeah, this happens all the time. The universe is an amazing place. It's also part of the reason why the LHC needs so much energy in its experiments.
Another phenomena that might be more familiar is tunneling. Classically, let's say that you have a marble rolling around in a valley. If it doesn't have enough total energy to roll over the top of a hill, it will stay confined forever. In Quantum Mechanics, these particles have a non-zero probability to just appear on the other side. The more energy that is required, the longer we'll have to wait for it to happen, but it will happen.
The most important every-day application is in semi-conductors, which again, is what our computers are all built on. Tunneling also explains nuclear fusion which is required for our Sun to create energy.
We know QM is the correct explanation for nature at that small scale. Without it, we wouldn't be able to explain almost anything at that level.
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I feel like once Yokokano arrives, this discussion is going to get even more interesting. And weird.
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Not sure what everyone is so upset about.
I have never said that current quantum mechanics is wrong.
You need to remember that we are using mathematics to model the outcome of experiments.
That means, when we say 'the electron is a point particle', it is a good enough mathematical approximation to experimental results.
The question was: "If our experiments improve, will our models improve?"
You are saying 100% NO.
I am saying, maybe let's see.
Again, current models would not be proven wrong, we already know that they give correct results.
The world also made perfect sense when we thought the atom is indivisible or when we only had Newtonian mechanics.
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But you need to be careful, especially with this topic, to distinguish between ontological infallibility and a model's consistency. Whether there might be some future event (experiment) that contradicts (provides evidence against) QM can never be ruled out, of course. But saying that taking better measurements in the same paradigm might allow us to improve a theory in a way that as been specifically ruled out by mathematical proof (Bell inequality) just shows you don't understand that aspect of the theory. I don't know if this is what you're trying to say actually but that's what it sounds like.
Again, current models would not be proven wrong, we already know that they give correct results.
The world also made perfect sense when we thought the atom is indivisible or when we only had Newtonian mechanics.
But, no, the whole point is that you're leaving the door open to evidence eventuating that would prove the model wrong. e.g. Newtonian mechanics is "wrong" as a description of nature because it doesn't include relativity and quantum effects. That doesn't mean it's not a useful model when the scale you're working in lets you ignore those parts of reality for the purposes of making predictions.
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On April 16 2015 16:45 fancyClown wrote:
The world also made perfect sense when we thought the atom is indivisible or when we only had Newtonian mechanics.
true,
and it was interesting to see how the philosophy of the era was impacted by what leading edge scientists thought was the absolute indisputable truth.
isn't there a new measurement for Planck's constant every few years ...as the technology improves?
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EPT
