Proving the Existence of Particles: An Exploration

In summary, the particle discovery process in physics is not as simple as just looking at data and concluding an entity exists. There are many assumptions and systematic errors that need to be accounted for in order to arrive at a particle.
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Iloveyou
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TL;DR Summary
What is exactly the reasonable context for the existence of particles
I studied physics in University a bit out of interest. Curious on how exactly one proves the existence of particles.

If I look it up, often the most basic example would be the cathode ray experiment. It seems pretty simple to me, but in my eyes it does not prove the existence of particles. What I'm interested is first and foremost how one goes from the physical observable to actually creating a model for it. I don't really see a way to do that without making assumptions.

And if this is done on a basic level of physics and then new theories are limited by the former theory which is based on assumption. If the assumption was incorrect, then would not all qualitative aspects of new theories be compromised.

It is obvious that an infinite different version of qualitative theories can achieve the same quantitative result. For instance if an anomaly within the system of a theory occured, one could just as easily invent any mathematical entity to compensate. I'm interested in what on the fundamental level justifies the jump. Because I did not study physics in depth I am very curious about the fundamental procedure.

With regards to the proof of electrons, I am curious within the context of simple reason hoping the process is not just picking the least absurd theory available. If we had chosen a different model, in my eyes it would still be quantitavely consistent because it would've been modified to account for contrary data, no?
 
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  • #3
Being able to count electrons doesn't convince you?
Then what would?
 
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  • #4
What I'm trying to ask is how exactly is a particle discovered. I'm trying to ask it as simply and directly as possible. To make it more specific let's say an electron. I don't want the answer to be simply referring to the J J Thompson experiment, but how exactly he made the jump from the experimental data to concluding existence of electrons.

I understand how he expected the negative charge, but not how the specific entity being a particle. The nice thing about particles are that they are easy to manipulate mathematically, but I don't see how discovery in physics can invent a particle to posses certain attributes without committing and error because then it would be far more accurate to say that it was the invention of an electron, which would seem absurd.

It is like doing an experiment to answer a question and answering using the data, but inventing an object that does not exist to explain it. For this reason, I was thinking that I must be overlooking something big here. I'm not making any conclusions because so far, I most likely overlooked something, but with that oversight it makes the introduction of particles to have metaphysical rather than physical reasoning, either way I'm interested to know the process and how the jump mentioned is reasonable or justifiable.

Question: How is the existence of electrons proven/justified as particles, (particularly its existence seems to be unjustifiably inferred [according to whatever oversight I'm looking to be addressed]).

We had electricity before we conceived of electrons, so it makes the discovery appear almost speculative. If I went to create a theory that quantitatively does not contradict the data and then do further experiments to define relationships of these entities, it starts to be quantitavely consistent without having anything to do with reality, no?

If I find new inconsistencies, I could invent new particles to compensate and make something simple very complicated and it would be justified? Particularly in physics if one actually discovers things mathematically, what would be a contradiction and death to a former theory can be viewed as a discovery.

[Mentor Note -- post edited for readability]
 
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  • #5
Iloveyou said:
What I'm trying to ask is how exactly is a particle discovered. I'm trying to ask it as simply and directly as possible.
A particle as opposed to a what? A wavelet? A string? A toroid?
 
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  • #6
@Iloveyou, maybe this clip helps a bit?

How to build a cloud chamber (US LHC)
 
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  • #7
berkeman said:
A particle as opposed to a what? A wavelet? A string? A toroid?
I'm not saying as oppossed to anything. To define physical reality I don't get to choose anything apart from what is recorded or observed. If I had a theory of some object being responsible for any physical quality, I could not just go and invent it, it would be making a statement greater than my knowledge would reasonably allow. If I expanded theories off such basis, then the theory would only be a lens complicating new potential understanding.

Suddenly one tiny thing becomes has to be explained by a huge baggage. If an electrons discovery is a mathematical invention, then its role would be far less significant and it would not seem advisable to create new theories on such a cumbersome foundation.

If we studied economy/statistics and you noticed certain patterns, then you could with a little justification at least call using that pattern as a theory, but if the particle aspect is simply a forceful means of inferring existence, that becomes very unscientific.

In science are we not supposed to be studying the real world rather than imagining entities such as particles and then studying the world in terms of it, it starts to seem ridicilous, to become a lens complicating and hiding simple data.

With such a foundation then anything branching off that would have to be quite a fantastic piece of fiction. I'm not saying that it's true or not, I'm assuming it's an oversight on my part because I studied physics only casually in school up to ~quantum mechanics II/ thermo/electro dynamics.

The textbooks always seem to just skip past the actual discovery. When I skip past a discovery, what happens is that I start to understand the world in terms of concepts, which have no counterpart in real world, so we begin to speak of concepts and the implications of new findings on these theories, which in my oversight I don't see at all to be justified.

Understanding concepts with no counterpart to experience is like saying I understand when I don't, or I know when I really do not. I'm not at all saying that electrons don't exist, I'm saying according to my limited understanding of the subject, it seems absurd.

Human beings are adept at advancing understanding and use in new technology, such is the characteristic of humanity, but if I'm honest, I don't understand a single leaf and I hear fantastic theories, but if the assumptions and leaps are not addressed, then one who is educated comes to understand his theory and not the world. With time, I'm interested in learning physics on my own time as a hobby, but it seems to me if I cannot address the assumptions and definitions made in direct relation to my experience, what is observable and also what is inferrable.

We can infer based on what is observed, but if we begin to infer on what is inferred already (especially if it is unjustified as in my oversight it seems to be), then one is very far from a clear picture. Advances in all fields are inevitable, but limited explanations seem to be mental baggage. Knowing and thinking are two exclusive functions, while one is in thinking process he ponders possibilities/direction, but only when outside the thinking process is he not trapped by it to be able to expand his current understanding.

[Mentor Note -- post edited for readability]
 
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  • #8
DennisN said:
@Iloveyou, maybe this clip helps a bit?

How to build a cloud chamber (US LHC)

Thank you. In my eyes it doesn't address my question because it doesn't have to be particles that explain that phenomena, the use of the word is only due to the current context and theory. I'm not saying an alternative way because to me it seems like for the sake of clarity one cannot merely invent explanations, one has to discover them or they are describing an imaginary world.
 
  • #9
Iloveyou said:
Thank you. In my eyes it doesn't address my question because it doesn't have to be particles that explain that phenomena, the use of the word is only due to the current context and theory. I'm not saying an alternative way because to me it seems like for the sake of clarity one cannot merely invent explanations, one has to discover them or they are describing an imaginary world.
If particles were visible, then they would have more complicated geometry as they would have to have volume and dimension/substance to be seen by the eyes. So if we observe that phenomena, it would be the effect of particles that are observed, if that is so, then I could make just as mathematically justifiable and consistent theory of an invisible angel flapping its wings as being the cause because particles themselves according to their mathematical definition would not be visible.
 
  • #10
What - if any - evidence would convince you?
 
  • #11
Iloveyou said:
If I had a theory of some object being responsible for any physical quality, I could not just go and invent it, it would be making a statement greater than my knowledge would reasonably allow. If I expanded theories off such basis, then the theory would only be a lens complicating new potential understanding.
Iloveyou said:
it starts to seem ridicilous
Yes, what you are posting is starting to sound ridiculous, and your thread may be closed soon.

And please stop posting "walls of text" that I have to keep cleaing up. Please pay attention to the quality of the text you are typing here. Thank you.
 
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  • #12
Iloveyou said:
What I'm trying to ask is how exactly is a particle discovered. I'm trying to ask it as simply and directly as possible. To make it more specific let's say an electron. I don't want the answer to be simply referring to the J J Thomson experiment, but how exactly he made the jump from the experimental data to concluding existence of electrons.
You can read many fine biographies of J.J. Thomson or the book his son, also a Nobel winner, wrote titled The Electron. These will give context.

But to really convince yourself, you can shuffle your feet on a carpet on a dry winter's day and touch a metal doorknob and then make a determination if electrons are real particles or not. :smile:
 
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  • #13
berkeman said:
Yes, what you are posting is starting to sound ridiculous, and your thread may be closed soon.

And please stop posting "walls of text" that I have to keep cleaning up. Please pay attention to the quality of the text you are typing here. Thank you.
Perhaps it belongs in the Philosophy section if PF has one.
 
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  • #14
bob012345 said:
Perhaps it belongs in the Philosophy section if PF has one.
We don't have such a section...
 
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  • #15
bob012345 said:
You can read many fine biographies of J.J. Thomson or the book his son, also a Nobel winner, wrote titled The Electron. These will give context.

But to really convince yourself, you can shuffle your feet on a carpet on a dry winter's day and touch a metal doorknob and then make a determination if electrons are real particles or not. :smile:
Found the book, thank you.
 
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  • #17
Iloveyou said:
I don't really see a way to do that without making assumptions
It seems like this is the core concern. The fact is that we do make assumptions in science. We always make assumptions. The data does not speak for itself. There is nothing wrong with making assumptions, you just try to be explicit with your assumptions and not make any hidden assumptions
 
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  • #18
So let's back up. The question from Iloveyou is not about the modern theory of electrons or nor about ways to convince Iloveyou of that theory. I suggest to other participants attempting to convince Iloveyou about electrons that you missed the question. Iloveyou's query is a meta-question about the application of the scientific method, about the concept development path from a sci-method observation of data to an explanatory sci-method hypothesis, then from that to an explanatory sci-method theory. Iloveyou's particular itch is "Curious on how exactly one proves the existence of particles." That is, questioning the conceptual validity of sub-atomic particle theories broadly, ANY theory which uses a particle model, "particle-ness," in part or whole. Iloveyou's first example was electrons-as-particles theory and then everyone charged ;-) off in the direction of JJ Thompson. Consider that Iloveyou's question would not change in an alternate universe where JJ had somehow discovered protons instead.

Take a look at "Understanding concepts with no counterpart to experience is like saying I understand when I don't, or I know when I really do not. I'm not at all saying that electrons don't exist, I'm saying according to my limited understanding of the subject, it seems absurd."

>>> "no counterpart to experience"

I will hazard from this phrase that intuition from human-scale experience is the crux for Iloveyou. Because sub-atomic particles cannot be directly observed in an intuitively macroscopic way, Iloveyou questions why a particle model and not some other.

So then that gets to a dialog of why particle-ness is a better fit to ALL observed data rather than alternatives. Early on, what were the alternatives and why were they eventually discarded? Why did the particle model win in modern physics? Of course, later understanding of wave-particle duality confuses matters, so keep it simple.

I will stop there for the moment least I exceed the moderator's Wall Of Text tolerance. Note: a theory of Walls rather than Membranes.
 
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  • #19
Well, the direct empirical observation of Brownian Motion stuck a pretty bow on the particle nature of matter.
 
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  • #20
Iloveyou said:
With regards to the proof of electrons, I am curious within the context of simple reason hoping the process is not just picking the least absurd theory available. If we had chosen a different model, in my eyes it would still be quantitavely consistent because it would've been modified to account for contrary data, no?
Well, part of the issue is that theories and models that accurately match experiments are, in general, VERY difficult to create. So much so that there usually aren't more than a small number of them at anyone time and the advancement of ever more precise instruments has often quickly reduced that number down to one.

For example, the model of the atom went through several variations over the course of 30-40 years before arriving at basically what we have today. But at anyone time there was basically only a single model that could explain things well enough to fit the observations available.

So quite often it isn't about choosing one out of many competing theories, but about choosing the only one that even makes any sense at all.
 
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  • #21
Iloveyou said:
Summary: What is exactly the reasonable context for the existence of particles

but in my eyes it does not prove the existence of particles

Define "particle"
 
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  • #22
Iloveyou said:
I was thinking that I must be overlooking something big here.
Cuz, a lot of science was /is based on observation and analogy.
We have balls, rocks, dust, pollen, and the smaller one gets they still look and behave for most of the time as a 'particle.'
So why not call an electron a particle just as well from previous experience.
You will not that very few discoveries just come out of thin air.

For example, you might think einstein conceived of the theory of relatively in a deep dark basement all on his own, but he was taking data from previous experiments, building upon that and eureka he had a theory, albeit much different than what was considered the norm at the time.

@malawi_glenn has a good question, though, to think about.
 
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  • #23
I don't think it's possible to prove electrons are particles (even in the historical context of their discovery). What you do is propose that the behaviour of some physical phenomenon ("cathode rays") is well explained if they consist of discrete chunks of matter carrying a negative charge. I believe they were initially thought to be light atoms, in fact, but it turned out that they did not behave like a thing of atom-sized mass, and always behaved the same whatever material you used for the cathode. So it became more plausible that they were some new kind of thing. In that context, it's possibly worth looking up Asimov's essay The Relativity of Wrong, about the fact that generally theories aren't "right" or "wrong", they just make accurate predictions in broader or narrower ranges of circumstances.

I think @Drakkith's point that coming up with new theories is hard is a key one. I think that asking "how did someone come up with the idea" is a bit like the perennial question to writers of where they get their ideas from. Usually not even they know. The difference with science is that there's an agreed process for determining whether an idea is a good one or not - make a prediction and see if experiment bears it out.

And I agree that reflecting on how you would answer @malawi_glenn's question might yield some insight.
 
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  • #24
Iloveyou said:
I understand how he expected the negative charge, but not how the specific entity being a particle. The nice thing about particles are that they are easy to manipulate mathematically, but I don't see how discovery in physics can invent a particle to posses certain attributes without committing and error because then it would be far more accurate to say that it was the invention of an electron, which would seem absurd.
There are several reasons to label an electron a 'particle'. For starters, there is a minimum amount of charge that can be moved around when working with electrons. We label this as the 'elementary charge'. Charge cannot be broken up into smaller pieces than this.

Then we have the fact that there is also a minimum mass. Shoot electrons out of an electron gun into a static magnetic field, one elementary charge at a time, and every electron will take an identical path. This is only possible if the mass accompanying each charge is the same, else the relationship between the electromagnetic field, the elementary charge, mass, and acceleration wouldn't hold.

There's also chemistry to consider. Chemistry is best explained by the transfer and sharing of small, discrete charges between atoms during bonding and chemical reactions. And for any given reaction or bond there is always the same amount of charge involved. No more, no less. You never get half charges or something in between.

Electrons also create a visible path through cloud chambers, much like a bullet moving through a medium (air, ballistic gel, etc). You don't see phenomena like water or sound waves doing this.

So we have an entity or phenomenon that appears to have identical and discrete charge and mass, follows most of the same kinematic laws that macroscopic objects like bullets do, and is always found in multiples of some base charge. What do we call it? I'd call it a particle. Because these are all properties of particles, not of something else.

Iloveyou said:
It is like doing an experiment to answer a question and answering using the data, but inventing an object that does not exist to explain it.
What was invented? We observed electrons, determined their behavior and properties, and then said, "Huh. They mostly behave like tiny little particles. I guess we'll call them particles." It's like observing a new species of creature and seeing that it has four legs, a tail, hair, gives live birth to its young, and produces milk. These are all properties of mammals, so we would be entirely justified in calling it a mammal.
 
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  • #25
According to our current best understanding (Quantum Mechanics, Quantum Field Theory) electrons are not particles, they are rather quantum objects. In 100 years we might have develop a new theory where electrons are something else (neither particles, neither quantum objects).

But according to classical physics understanding they are particles. Being particles (according to classical physics) is not an assumption , it was a hypothesis verified by the experiment.
 
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  • #26
@Delta2 you also have to define "particle" :)

I should qoute my quantum mechanics professor "an electron is an electron, don't tell it what it should be!"

Physics is not about truth, it is about describing our observations and measurments with mathematical models.

Anyways, this is classical physics subforum so I guess the OP want to know what classical physics properties one used to infer that the electron behaves as a "particle".

I think OPs question pretty much boils down to semantics. Is Pluto a planet? Or a Dwarf-planet? Who cares what we call it? What is more interesting is the properties of Pluto and what we can learn from it.
 
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  • #27
malawi_glenn said:
@Delta2 you also have to define "particle" :)
Yes well , hard to do that since it is elementary concept but according to classical physics, I think particle is a localized distribution of matter that can have definite position (and therefore velocity, acceleration and all that).
 
  • #28
DaveC426913 said:
Well, the direct empirical observation of Brownian Motion stuck a pretty bow on the particle nature of matter.
Good -- relates a macroscopically observable mechanical behavior (jerky, random walk, if, heh, through a microscope), observation of this behavior in a variety of experiements by a multitude of experimentors, to ask the question what conceptual, mechanical model (analogy, metaphor, physical human-scale intuitive experience) could be a starting point for formulating rules and math for that model, that makes predictions coming in the neighborhood of observed data. There could be more than one initial model. For @Iloveyou then, what 'practice of theory development' methods then help one discriminate among the initial models to narrow the field?
 
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  • #29
LeakingRoof said:
if, heh, through a microscope
By which you mean it might not count as empirical observation?
What a microscope does is identical to what your cornea and lens do.
 
  • #30
Drakkith said:
Well, part of the issue is that theories and models that accurately match experiments are, in general, VERY difficult to create. So much so that there usually aren't more than a small number of them at anyone time and the advancement of ever more precise instruments has often quickly reduced that number down to one.

For example, the model of the atom went through several variations over the course of 30-40 years before arriving at basically what we have today. But at anyone time there was basically only a single model that could explain things well enough to fit the observations available.

So quite often it isn't about choosing one out of many competing theories, but about choosing the only one that even makes any sense at all.
Good -- gets to why the condition of a multitude of intuitive mechanical analogy models does not persist in the scientific community as experimental data accumulates. However, everyone needs to surface the very first discrminator applied in the sci-method as a requirement -- falsifiability. This gets to the (I take to be tongue in cheek) "invisible angel flapping its wings" counter-argument of theory development. Theories that include mystical or supernatural elements cannot be falsified -- one cannot construct a controlled experiement where such elements are convincingly included and compare the observable data to a control experiment where such elements are convincingly excluded.
 
  • #31
Iloveyou said:
Summary: What is exactly the reasonable context for the existence of particles

I studied physics in University a bit out of interest. Curious on how exactly one proves the existence of particles.

If I look it up, often the most basic example would be the cathode ray experiment. It seems pretty simple to me, but in my eyes it does not prove the existence of particles. What I'm interested is first and foremost how one goes from the physical observable to actually creating a model for it. I don't really see a way to do that without making assumptions.

And if this is done on a basic level of physics and then new theories are limited by the former theory which is based on assumption. If the assumption was incorrect, then would not all qualitative aspects of new theories be compromised.

It is obvious that an infinite different version of qualitative theories can achieve the same quantitative result. For instance if an anomaly within the system of a theory occured, one could just as easily invent any mathematical entity to compensate. I'm interested in what on the fundamental level justifies the jump. Because I did not study physics in depth I am very curious about the fundamental procedure.

With regards to the proof of electrons, I am curious within the context of simple reason hoping the process is not just picking the least absurd theory available. If we had chosen a different model, in my eyes it would still be quantitavely consistent because it would've been modified to account for contrary data, no?
Electric charge comes in discrete, localisable quanta. The electron as a particle models this and is, therefore, a useful concept. Existence beyond that is more philosophy than physics.
 
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  • #32
DaveC426913 said:
By which you mean it might not count as empirical observation?
What a microscope does is identical to what your cornea and lens do.
malawi_glenn said:
Define "particle"
Gold star on where to start the human-scale, observable mechanical analogy intuition dialog. Consider a bin (one of those 6 gallon, bi-fold top mechandise bins) containing copper BBs, another containing water. BBs observably have, and maintain against manipulation (we exclude extreme conditions outside most human experience) a chunkness, an individual separateness even though they are indistinquisable by all the properties we can measure (length dimensions, mass, charge, etc.) of each individually. You can pick up the bin (you've been doing your weight lifting reps, good boy) and slosh the BBs around. They rub and bump against each other as you slosh them, in bulk acting somewhat like a fluid, yet each can be imagined as following an individual trajectory, maintaining chunkness and separateness. If we accept Newtonian physics for macroscopic objects, we can ascribe momentum to each and forces acting on them to change that momentum.

Compare to the human-scale, observable mechanical behavior of the bin of water.

This is a start on particle-ness.
 
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  • #33
Ibix said:
I don't think it's possible to prove electrons are particles (even in the historical context of their discovery). What you do is propose that the behaviour of some physical phenomenon ("cathode rays") is well explained if they consist of discrete chunks of matter carrying a negative charge. I believe they were initially thought to be light atoms, in fact, but it turned out that they did not behave like a thing of atom-sized mass, and always behaved the same whatever material you used for the cathode. So it became more plausible that they were some new kind of thing. In that context, it's possibly worth looking up Asimov's essay The Relativity of Wrong, about the fact that generally theories aren't "right" or "wrong", they just make accurate predictions in broader or narrower ranges of circumstances.

I think @Drakkith's point that coming up with new theories is hard is a key one. I think that asking "how did someone come up with the idea" is a bit like the perennial question to writers of where they get their ideas from. Usually not even they know. The difference with science is that there's an agreed process for determining whether an idea is a good one or not - make a prediction and see if experiment bears it out.

And I agree that reflecting on how you would answer @malawi_glenn's question might yield some insight.
Good -- gets to "chunkness" as a better fit to observation compared to "fluidness."
 
  • #34
DaveC426913 said:
By which you mean it might not count as empirical observation?
What a microscope does is identical to what your cornea and lens do.
Ware literal-mindedness. It was a play on words: macro (scopic) intuition <--> micro (scopic) observation method.
 
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  • #35
Physicists have math that works. People who don't understand that come along and want an explanation in plain English. I have learned to not take these explanations seriously. The English language is inadequate for the task. These explanations are just rough approximations. The best seems to have been Feynman. I'd recommend his book QED.

Physicists often have models that inspire theories. But it is routine for a wrong model to come up with math that works. This confuses the situation, especially since once popular literature latches on to an explanation they hold on to it like a dog with a bone, even after no one in physics is using that model any more.

In particular the word "particle" is short hand for a localized excitation of a field. But then, what is a field? Why does it have any excitations at all? Etc. So when I see "particle" I think "weird thing that sort of has a location and does all sorts of stuff that billiard balls can't do".

The "particle/wave duality" particularly irritates me. I recommend instead to give up trying to relate quantum things to familiar concepts. Wipe your mind clean and see them as completely new.
 
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<h2>1. What is the purpose of proving the existence of particles?</h2><p>The purpose of proving the existence of particles is to gain a better understanding of the fundamental building blocks of the universe. By studying particles, scientists can uncover the laws and principles that govern the behavior of matter and energy, leading to advancements in technology and our understanding of the world around us.</p><h2>2. How do scientists prove the existence of particles?</h2><p>Scientists use a variety of methods to prove the existence of particles, including experiments, observations, and mathematical models. These methods allow scientists to detect and measure the properties of particles, such as mass, charge, and spin, and confirm their existence through repeatable results.</p><h2>3. What types of particles have been proven to exist?</h2><p>There are many types of particles that have been proven to exist, including subatomic particles like electrons, protons, and neutrons, as well as smaller particles like quarks and leptons. Scientists have also confirmed the existence of particles that make up dark matter, which is thought to make up a large portion of the universe's mass.</p><h2>4. Can particles be directly observed?</h2><p>Some particles can be directly observed using specialized instruments, such as particle accelerators, which allow scientists to study the behavior of particles at high energies. However, many particles are too small to be directly observed and can only be detected through their interactions with other particles or through the effects they have on their surroundings.</p><h2>5. How does proving the existence of particles impact our understanding of the universe?</h2><p>Proving the existence of particles has a significant impact on our understanding of the universe. By studying particles, scientists can better understand how the universe was formed and how it continues to evolve. This knowledge can also lead to new technologies and advancements in fields like medicine and energy production.</p>

1. What is the purpose of proving the existence of particles?

The purpose of proving the existence of particles is to gain a better understanding of the fundamental building blocks of the universe. By studying particles, scientists can uncover the laws and principles that govern the behavior of matter and energy, leading to advancements in technology and our understanding of the world around us.

2. How do scientists prove the existence of particles?

Scientists use a variety of methods to prove the existence of particles, including experiments, observations, and mathematical models. These methods allow scientists to detect and measure the properties of particles, such as mass, charge, and spin, and confirm their existence through repeatable results.

3. What types of particles have been proven to exist?

There are many types of particles that have been proven to exist, including subatomic particles like electrons, protons, and neutrons, as well as smaller particles like quarks and leptons. Scientists have also confirmed the existence of particles that make up dark matter, which is thought to make up a large portion of the universe's mass.

4. Can particles be directly observed?

Some particles can be directly observed using specialized instruments, such as particle accelerators, which allow scientists to study the behavior of particles at high energies. However, many particles are too small to be directly observed and can only be detected through their interactions with other particles or through the effects they have on their surroundings.

5. How does proving the existence of particles impact our understanding of the universe?

Proving the existence of particles has a significant impact on our understanding of the universe. By studying particles, scientists can better understand how the universe was formed and how it continues to evolve. This knowledge can also lead to new technologies and advancements in fields like medicine and energy production.

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