the-french-artist/wikipedia_20220301.simple_sentence_embeddings huggingface dataset:
Uncertainty principle 
 Many of the physicists who read his paper said that they could not disagree with his conclusions, but that they could not follow his explanation of how he got to those conclusions.
Uncertainty principle 
 We will come back to the equation above because it is a kind of recipe for writing out and multiplying matrices.
Uncertainty principle 
 The new equations had to be so strange and unusual because Heisenberg was describing a strange world in which some things, such as the orbits of electrons, do not slowly get larger or smaller.
Uncertainty principle 
 Electrons can only jump between certain orbits, and the energy gained or lost in changing between orbits is produced when a photon of the right energy is absorbed or a new photon of the right energy is produced.
Uncertainty principle 
 If electrons in hydrogen atoms most frequently jump down (fall) between two particular orbits, then more photons will be emitted at that energy level, and so the light produced at that level will be the most intense.
Uncertainty principle 
 It was difficult to make equations built for continuous spectra (what you see when you put the sun's light through a prism) fit spectra that just have a few peak frequencies between which there is nothing.
Uncertainty principle 
 Almost everything that had already been learned about light and energy had been done with large things like burning candles or suns, and those large objects all produce continuous spectra.
Uncertainty principle 
 Even though these ordinary-sized things were easy to do experiments with, it had still taken a long time to figure out the law (physics)laws that govern them.
Uncertainty principle 
 Now physicists were dealing with things too small to see, things that did not produce continuous spectra, and were trying to find a way to at least get clues from what they already knew that would help them find the laws of these small and gapped-out light sources.
Uncertainty principle 
 The original equations dealt with a kind of vibrating body that would produce a wave, a little like the way a reed in an organ would produce a sound wave of a characteristic frequency.
Uncertainty principle 
 So there was motion back and forward (like the vibrating of a reed) and there was an emitted wave that could be graphed as a sine wave.
Uncertainty principle 
 Much of what had earlier been figured out about physics on the atomic level had to do with electrons moving around nuclei.
Uncertainty principle 
 When a mass moves in an orbit, when it rotates around some kind of a hub, it has what is called "angular momentum."
Uncertainty principle 
 Angular momentum is the way that something like a merry-go-round will continue to rotate after people have stopped pushing it.
Uncertainty principle 
 On top of that, Heisenberg did not show all of his calculations in his 1925 paper, so even good mathematicians might have trouble filling out what he did not say.
Uncertainty principle 
 Even though many physicists said they could not figure out the various math steps in Heisenberg's breakthrough paper, one recent article that tries to explain how Heisenberg got his result uses twenty math-filled pages.
Uncertainty principle 
 The math started with some really hard stuff and would eventually produce something relatively simply that is shown at the top of this article.
Uncertainty principle 
 Getting the simpler result was not easy, and we are not going to try to show the process of getting from an outdated picture of the universe to the new quantum physics.
Uncertainty principle 
 We need just enough detail to show that almost as soon as Heisenberg made his breakthrough a part of how the universe works that nobody had ever seen before came into view.
Uncertainty principle 
 Heisenberg must have been very excited but also very tired when, late at night, he finally made his breakthrough and started proving to himself that it would work.
Uncertainty principle 
 Almost right away he noticed something strange, something that he thought was an annoying little problem that he could make go away somehow.
Uncertainty principle 
 Heisenberg had been working toward multiplying amplitudes by amplitudes, and now Heisenberg had a good way to express amplitude using his new equation.
Uncertainty principle 
 Naturally he was thinking about multiplication, and about how he would multiply things that were given in terms of complicated equations.
Uncertainty principle 
 Heisenberg realized that besides squaring amplitude he would eventually want to multiply position by momentum, or multiply energy by time, and it looked like it would make a difference if he turned the order around in these new cases.
Uncertainty principle 
 Heisenberg did not think it should matter if one multiplied position by momentum or if one multiplied momentum by position.
Uncertainty principle 
 But they were both complicated equations, and how you got the numbers to plug into the equations turned out to be different depending on which way you got started.
Uncertainty principle 
 In nature you had to measure position and then measure momentum, or else you had to measure momentum and then measure position, and in math the same general situation prevailed. (
Uncertainty principle 
 See the English Wikipedia article Heisenberg's entryway to matrix mechanics if you want to learn the fussy details!)
Uncertainty principle 
 The tiny but pesky differences between results were going to remain, not matter how much Heisenberg wished they would go away.
Uncertainty principle 
 At the time Heisenberg could not get rid of that one little problem, but he was exhausted, so he handed his work in to his immediate supervisor, Max Born, and went on vacation.
Uncertainty principle 
 Max Born was a remarkable mathematician who soon saw that the equation that Heisenberg had given him was a sort of recipe for writing a matrix.
Uncertainty principle 
 Dr. Born was one of the few people at that time who was interested in this odd kind of math that most people figured was not good for very much.
Uncertainty principle 
 He knew that matrices could be multiplied, so doing all the calculations for accounting for one physics problem could be handled by multiplying one matrix by another.
Uncertainty principle 
 Just being able to put a complicated procedure into a standard and acceptable form would make it easier to work with.
Uncertainty principle 
 Born was such a good mathematician that he almost immediately realized that switching the order of multiplying the two matrices would produce a different result, and the results would differ by a small amount.
Uncertainty principle 
 It took a couple of years, but Heisenberg was able to prove the Uncertainty Principle, which says that Δx × Δp = h/2, which is the number that comes out of the original equations but leaves out the π and the i that have to do with phase changes.
Uncertainty principle 
 Heisenberg explained that he derived his uncertainty principle from this earlier result when he wrote a paper in 1927 introducing this theory.
Uncertainty principle 
 The constant written h, called the Planck constant, is a mysterious number that often occurs, so we need to understand what this tiny number is.
Uncertainty principle 
 It was discovered when Planck realized that the energy of a perfect radiator (called a black-body radiator) is emitted in units of definite size called "quanta" (the singular of this word is "quantum").
Uncertainty principle 
 Radiated energy is emitted as photons, and the frequency of a photon is proportional to the "punch" it delivers.
Uncertainty principle 
 At the violet end of the spectrum, each photon has a relatively large amount of energy; at the red end of the spectrum each photon has a relatively small amount of energy.
Uncertainty principle 
 The way to calculate the amount of energy of a photon is given by the equation E = hν (energy equals the Planck constant times "nu" or frequency).
Uncertainty principle 
 The Heisenberg uncertainty principle Δx × Δp ≥ h tells us that whenever we try to pin down certain pairs of numbers we can only get so close, and that if we try to get clearer on one of them, i.e., if we try to make Δx smaller so we have a better idea of the position of something, then we will have to get handed back a bigger number for the other number of the pair, and that the amount the two are off by is closely related to h.
Uncertainty principle 
 Another pair of physical quantities goes according to the uncertainty relationship: ΔE × Δt ≥ h, and that pair indicates, among other things, that if we look in interstellar space, some place where we would not expect to find anything at all, and we reduce Δt closer and closer to 0, then to keep the balance shown in the equation ΔE has to get larger and larger — and suddenly something with momentum can pop into existence just for that brief period of time.
Uncertainty principle 
 It is often said that a new theory that is successful can provide new information about the phenomena under investigation.
Uncertainty principle 
 Heisenberg created a math model that predicted the correct intensities for the bright-line spectrum of hydrogen, but without intending to do so he discovered that certain pairs of physical quantities disclose an unexpected uncertainty.
Uncertainty principle 
 Up until that moment nobody had any idea that measurements could not be forever made more and more precise and accurate.
Uncertainty principle 
 This theoretical position grew out of the discovery of uncertainty, and was not just some personal preference on what to believe.
Uncertainty principle 
 In everyday life it is possible to do something like walking alongside an automobile while marking down the times it crosses points on a grid drawn on the pavement.
Uncertainty principle 
 Perhaps the weight of the automobile itself will depress little levers in the pavement that turn off clocks attached to each of them and record the automobile's weight.
Uncertainty principle 
 In the end we would have a clear record of where the car was at various times, and also could compute its direction of progress and weight.
Uncertainty principle 
 We could then know, at any time on the clock, both its position and its momentum (its velocity multiplied by its mass).
Uncertainty principle 
 We would not even imagine that the force required to move the little levers would have any influence on the progress of the car.
Uncertainty principle 
 We would also not imagine that the automobile had no location or trajectory between the points on the pavement where there are levers, or that the car exists in a kind of three-dimensional blur during those times and only settles down while it is depressing a lever.
Uncertainty principle 
 To locate a ship on the sea during the darkest night we could use a searchlight, and that light would not disturb the position or direction of travel of the ship, but locating an electron with light would require hitting it with one or more photons each having enough momentum to disturb the position and trajectory of the electron.
Uncertainty principle 
 Locating the electron with other means would involve holding it in some kind of physical restraint that would also terminate its forward movement.
Uncertainty principle 
 To locate a photon, the best that can be done without terminating its forward movement is to make it go through a circular hole in a barrier.
Uncertainty principle 
 If one knows the time at which the photon was emitted (by a laser, for instance) and the time that the photon arrives at a detection screen such as a digital camera, then it is possible to compute the time required to travel that distance and the time at which the photon was passing through the hole.
Uncertainty principle 
 However, to permit the photon to pass through it, the circular hole must have a diameter greater than the size of the photon.
Uncertainty principle 
 The smaller the circular hole is made, the closer we come to knowing the exact position of the photon as it goes through it.
Uncertainty principle 
 As the diameter of the hole is decreased, the momentum or the direction of the photon as it leaves the hole is more and more greatly changed.
Uncertainty principle 
 Niels Bohr and his colleagues argued that we get into big trouble if we assume to be true of the things that are too small to be seen even with a microscope anything that we have proof for only on the scale of everyday life.
Uncertainty principle 
 In everyday life, if one observes a factory from the night shift of day one to the day shift of day two and one sees a finished automobile rolled out to the shipping dock it would make no sense to say that it is impossible to tell whether it was delivered during the night shift or during the day shift.
Uncertainty principle 
 But on the atomic scale, we can show instances when we have to count a single photon as having been produced at two times. (
Uncertainty principle 
 If that is not bad enough, we can also show instances where a single photon is produced from two adjacent lasers.)
Uncertainty principle 
 Part of the difficulty with finding out what is happening on the atomic scale is that we would like to know both where something is and what its trajectory is, and to know both things for the same time, but we cannot measure both position and trajectory at the same time.
Uncertainty principle 
 We either measure the momentum of a photon or electron at one time and then without any more delay than necessary measure its position, or we switch things around and measure position first and momentum second.
Uncertainty principle 
 The problem is that in making the first one take on a pretty definite form (by squeezing down on it in some way) we increase the uncertainty involved in the next measurement.
Uncertainty principle 
 If our initial measurements were so crude that lots of error was introduced in each, then we could improve things by using a lighter touch to do each of them, but we could never get beyond a certain limit of accuracy.
Uncertainty principle 
 We know from everyday life that trying to weigh something on a bathroom scale placed on a washing machine in spin cycle will produce inaccurate results because the needle on the scale will jiggle badly.
Uncertainty principle 
 But for very accurate measurements we find that trucks going by in the neighborhood make the needle jiggle, so we can put the scale on a something to insulate it from the outside disturbances.
Uncertainty principle 
 We never consider that the thing on the scale is itself vibrating or that it possesses an indefinite momentum.
Uncertainty principle 
 Arguing backwards from the Uncertainty Principle, it looks as though there is in fact no definite position and no definite momentum for any atomic scale thing, and that experimenters can only force things into definiteness within the limit stated by the Uncertainty Principle.
Uncertainty principle 
 Bohr and his colleagues only argued that we could not know anything without making measurements, and when measurements were made we can push things in the direction of more definite position or more definite momentum, but that we can't get the absolute definiteness or certainty that we would like.
Uncertainty principle 
 But others took the possibility seriously, and argued that if the math is right then there cannot be definiteness or certainty in the world of the ultra small.
Uncertainty principle 
 The nature of science is that the math is only a model of reality, and there is no guarantee that it is a correct model.
Uncertainty principle 
 The math and the practical consequences of the things that the math predicts are so reliable that they are very hard to disagree with, but what the math says about the real world has produced several different ideas.
Uncertainty principle 
 Among the scientists who worked with Niels Bohr in Copenhagen, the uncertainty principle was taken to mean that on an elementary level the physical universe does not exist in a deterministic form.
Uncertainty principle 
 Counter to the story woven around the math by the Copenhagen group, there are other stories such as the "multiple universes interpretation" that says that every time there are multiple possible outcomes according to quantum theory, each outcomes occurs in its own new universe.
Uncertainty principle 
 Einstein argued that there are no multiple possible outcomes, so there is only one universe and it is determinate, or, as he put it, "God does not play dice."
Uncertainty principle 
 Albert Einstein saw that the new quantum mechanics implied a lack of position and momentum in the time prior to measurements being made, and he objected strongly.
Uncertainty principle 
 He firmly believed that things had definite positions and definite momentums before they were measured, and that the fact that measuring one of a pair of things and disturbing the possibility of accurately measuring the other does not argue for there being a lack of either of them beforehand.
Uncertainty principle 
 That paper argues that there must be characteristics that do determine position and momentum, and that if we could see them, or if we can get information about them, then we can mathematically know and predict position and momentum.
Uncertainty principle 
 For a long time people thought that there was no way to prove or disprove what was for Einstein an article of faith.
Uncertainty principle 
 In 1964 John Stewart Bell developed a math method to distinguish between the behavior of two particles that have determinate states that are merely unknown to the two individuals that investigate them, and two particles that have entangled states that are indeterminate or uncertain until they are measured.
Uncertainty principle 
 His method shows that the probabilities for getting certain results are different under the two different assumptions.
Uncertainty principle 
 The initial discussions of Heisenberg's uncertainty principle depended on a model that did not consider that particles of matter such as electrons, protons, etc.
Uncertainty principle 
 If we try to make the wave of a thing like a proton narrower and taller, that would make its position clearer, but then the momentum would get less well defined.
Uncertainty principle 
 If we try to make the momentum part of a wave description clearer, i.e., make it stay within a narrower range of values, then the wave peak spreads out and its position becomes less definite.
Uncertainty principle 
 The wave that is part of the description of a photon is, in quantum mechanics, not the same kind of thing as a wave on the surface of the ocean or the regions of compressed air and rarefied air that make up sound waves.
Uncertainty principle 
 Instead, these wave have peaks or high amplitude regions that have to do with probability of finding something at that point in space and time.
Uncertainty principle 
 More precisely, it is the square of the amplitude that gives the probability of some phenomenon showing up.
Uncertainty principle 
 In that case, the square of the value of every peak would give the probability of observing the photon at that point.
Uncertainty principle 
 Since the amplitudes of the sine waves are everywhere the same, the probability for finding the photon at each of them would be the same.
Uncertainty principle 
 So, practically speaking, knowing the wave for one of these photons would not give a clue about where to look for it.
Uncertainty principle 
 Since in this case we have a pure sine wave, the amplitude of every cycle of the wave is the same and therefore there is only one momentum value associated with this wave.
Uncertainty principle 
 In beams of light that focus on some point on a detection screen, the waves associated with the photons are not pure sine waves.
Uncertainty principle 
 Instead, they are waves with high amplitude at one point and much lower amplitudes on either side of that highest peak.
Uncertainty principle 
 Mathematically it is possible to analyze such a wave into a number of different sine waves of different wavelengths.
Uncertainty principle 
 It is a little easier to visualize the reverse of this process by looking at an initial sine wave of one frequency to which is added a second sine wave of a different wavelength, then a third, then a fourth, and so on.
Uncertainty principle 
 The result will be a complex wave showing one high peak and containing a large number of waves of different wavelengths and therefore of different momentums.
Uncertainty principle 
 In that case, the probability that the photon will appear at a certain point is extremely high, but the momentum it delivers can turn out to be related to the wavelength of any one of the component waves.
Uncertainty principle 
 In other words, the value of p = ħ/λ is no longer a single value because all the lengths of the assembled "waves of different wavelength" have to be taken into account.
Uncertainty principle 
 The simulation shows how to mathematically model the sharpening up of the location of a particle: Superimpose many different wave forms over the original sine wave.
Uncertainty principle 
 The center will form a higher and higher peak, and the rest of the peaks will be increased in number but decreased in height because they will interfere with each other.
Uncertainty principle 
 So in the end there are many different waves in the superposition, each with a different wavelength and (by p = ħ/λ) a different momentum, but only one very high peak, one that grows higher and narrower and gives us something closer and closer to a determinate position.
Uncertainty principle 
 To make momentum more and more definite, we would have to take away more and more of the superimposed sine waves until we had only a simple sine wave left.
Uncertainty principle 
 In so doing we would progressively diminish the height of the central peak and progressively increase the heights of the competing places where one might find the particle.
Uncertainty principle 
 So when we start with a wave picture of subatomic particles we typically will always deal with cases with relatively tall central peaks and relatively many component wavelengths.
Uncertainty principle 
 If the mathematical model is an accurate representation of the real world, then no photon or other subatomic particle has either an exact position or a definite momentum.
Uncertainty principle 
 When we measure such a particle we can choose a method that further squeezes the peak and makes it narrower, or we can choose a method that lowers the peak and evens out the component wavelengths.
Uncertainty principle 
 Depending on what we measure and how we measure it, we can make our location come out more definite or we can make our momentum range narrower.
Uncertainty principle 
 We can take care in designing the experiment to avoid various ways of jiggling the apparatus, but we cannot get rid of the fact that there was nothing completely definite to begin with.
Uncertainty principle 
 Under the theories of classical physics it is possible to argue that the laws of cause and effect are inexorable and that once the universe began in a certain way the interactions of all matter and energy to occur in the future could be calculated from that initial state.
Uncertainty principle 
 Since everything is absolutely the result of what came before it, they argued, every decision a human being makes and every situation into which that human being enters was predetermined since the beginning of time.
Uncertainty principle 
 People who believe in free will argue that the laws of quantum mechanics do not predict what will happen but only what is more and what is less likely to occur.
Uncertainty principle 
 Therefore every action is the result of a series of random "coin tosses" and no decision could be traced back to a set of necessary preconditions.
Uncertainty principle 
 The expressions "quantum leap" and "quantum jump" have become ordinary ways of talking about things.
Uncertainty principle 
 Usually people intend to describe something as involving a huge change that occurs over a short period of time.