Hey everyone! Hope you like the video. I can honestly say that this is one of my favorite results in physics. Also, if you didn't see already, I am considering starting a Twitch channel where I would live-stream informal physics lectures. I want to do Twitch because it would give more opportunities for interaction/questions from the audience in real time. I would also upload the videos to TH-cam after the fact. Let me know your thoughts on this!
Sounds like an amazing idea! I myself was actually looking for educational streamers on twitch, but was not very successful at finding them! Hopefully you'll be the first!
@Ascolano Irl, this is a great idea. I've never really interacted much with TH-cam's livestreaming, so Twitch was my first thought. This is the direction I think I will go!
This may be one of your favorite topics but you got the 100% wrong intuition here. Please take the time to read Mott's 1929 paper about the derivation of alpha ray tracks from wave mechanics. That is how nature creates macroscopic reality, by constant weak measurement. Heisenberg gave a similar example to Mott using Rydberg atoms a year or two earlier in one of his papers. Both are nearly forgotten early results of weak measurement theory. Why they were forgotten and replaced by this mathematical limit nonsense is the real question. Physicists kept observing high energy particle tracks in gas, liquid and solid detector media, but they have never observed the averaging of paths around the classical path in vacuum. Such a thing simply doesn't happen.
As a Physics MSc, I've watched endless videos on the principle of stationary action in order to understand it. This is the first one that actually explains why it makes sense! and it came in a total surprise, thanks man!
Also completely wrong on this one. He got this idea from Feynman, who didn't think it through. Since then everybody is playing monkey-see-monkey-do with path integral averaging.
Wow, this is the absolute best explanation that I have ever seen of the Path Integral! You my friend just earned a cookie and a new subscriber to your channel!
I really appreciate these videos. They are technical enough that I don't feel like I'm being hand-waved at, but not so dense that I can still follow without getting lost in the weeds. Thank you!
I somewhat see a chicken-egg arguments occuring twice here: 1) Explicitly @3:55 with the "backward derivation". 2) Subtly @11:17 when the stated goal was to "derive classical behavior"... But it's just a conformity of mathematical interpretation. The underlying principle, which is that of least action, is the foundational presumpted-result (thus the 'egg-chicken' argument, if you will). So what emerges is a conclusion of something already readily known; which is good... is pretty much a meromorphism. I do reckon this content is quite excellent and admire very much the endeavors of propagating some of the insights that Richard had. Kudos!
You lost me at the connection at 7:40 but this is super interesting! Currently taking an honors physics course, but I'll only get to ap physics next year where they use integrals, I'll return to this video then to see if it's my re understandable
It's that spiral a fresnel spiral, C(t)+iS(t), that only takes form when you integrate e^(ix^2). I'm just curious on how the integral pipes out. I can't quite tell from your discription in the video
This is a fantastic observation! In the case of the free particle, the path integral is a functional integral which is exactly a complex Gaussian. In the very simplest case of a non-relativistic particle, the Lagrangian is just 1/2*m*(dq/dt)^2 where we perform our functional integrover the coordinates q. In more general QFTs, this kinetic term looks different and we can add a separate mass term, but it is always quadratic in the fields, i.e. the Lagrangian (density) looks like A.O.A where A is a field we perform our functional integral over and O is some differential operator.
I used to tell people to read Feynman's book. I don't do that any longer. It's a great piece of science writing, but unfortunately it's technically wrong. Classical mechanics does not emerge from quantum mechanics in this way. Rather than studying the path integral of the quantum mechanical ensemble we need to study weak measurement processes on a single system. The two are not equivalent because weak measurements on a single system introduces conditional probabilities, which do not result from any possible operation of an ensemble of completely independent events.
There is a classical to Quantum transition referred as coherent superposition which is described by non zero off diagonal entries where in quantum to classical transition of a system is described by loss of these off diagonal entries from the matrix,
This was insightful thank you! I liked how you explained the Lagrangian's very well, maybe you could the same for the Hamiltonian one day in the future? :D
Having read this argument many times, I'm still confused about what exactly it is claiming about the relationship between the classical and quantum descriptions. The quantum description gives you the prob amp for the particle starting at source to be measured at some point on the screen. I get that in the path integral description that amp is gotten by doing the path integral and getting a final arrow for that point. However, the classical description gives you the total path which the particle takes (classically) between source and destination. To put it another way, a classical treatment allows us to 'fill in' the history of the particle between source and screen, whereas the quantum treatment does not. It insists that the particle actually does take all possible paths. In Feynman's treatment and this one of yours, I am missing how we go from summing the action over all paths for a given point to 'filling in' the intervening path with a specific classical trajectory.
@Joel Knoll I can definitely see how this can be a sticking point, especially because the argument is actually more of a mathematical one than an intuitive one. It relies on a mathematical concept known as the "stationary phase approximation," which you can read more about here: en.wikipedia.org/wiki/Stationary_phase_approximation. The basic idea is this: when you sum up a bunch of functions which are all oscillatory, if the frequencies of oscillations are very rapid, then all of the contributions end up cancelling except for the one which has minimum frequency. When we look at the path integral, the amplitude from each contributing path is given by exp(i/hbar S) where hbar is Planck's constant and S is the classical action along that path. This is exactly the form of an oscillating function where the "frequency" of oscillation is given by S/hbar. If hbar is large enough to be relevant, then we can't use this approximation and we have to sum up all paths to get our total amplitude. This is the case of quantum mechanics: all paths contribute and we can't say that the system took any one path. However, if we want to "turn off" quantum mechanics, this amounts to taking hbar -> 0. Here, we see that the frequency of oscillation for each individual path, given by S/hbar, gets very, very large. Using our stationary phase approximation, we know that, when we sum all of the paths in this case, all of the contributions from the paths which don't minimize S/hbar will cancel since they are completely out of phase. So, in the limit where hbar->0, we are left with the only contributing path being that which minimizes the classical action, which is, of course, just the statement of Lagrangian mechanics. What I was trying to show in this video is how these paths begin to cancel when we start taking hbar to be smaller and smaller. This is the idea of the neat-looking spiral showing the sum of all of the individual amplitudes. When we take hbar to be smaller and smaller, the swirls which come from paths far from the classical path become tighter (meaning that the individual amplitudes are getting more out of phase) and pulling in more of the arrows from paths closer to the classical path. When we take all of the infinite paths into account and take hbar -> 0, all of the paths which are not the classical paths will perfectly cancel out (i.e. the spirals will become points and contain all of the arrows which do not come from the classical path), leaving only a single non-zero contribution to the total amplitude, which is the classical path. Basically, this is saying that in the hbar -> 0 limit, there is zero probability that the particle took any path other than the one which minimizes the action. Hopefully that helps!
@@zapphysics Alas, it does not help my confusion. It's somewhat subtle and I have a hard time articulating it. It may be that I'm overthinking. I'll try to state it clearly. The problem being solved here, it seems to me, is this: Given the source-point and a screen-point, what is the probability that a particle starting at source-point will be measured to be at screen-point some amount of time later? One way to get the answer, according to Feynman, is the sum-over-paths approach you outlined. And the idea there is that the probability amplitude for the transition in question to happen gets its largest (and in the limit hbar > 0, only non-zero) contribution from the path which corresponds to the one we would call the classical one. I.e. the one that minimizes the action. Where I struggle is in the logical step from "the final prob amp gets its only non-negligible contribution from a certain path" to "that same path is the one we classically observe." Something seems to be missing in the argument there, as far as how quantum gives rise to classical. After all, the whole point of Feynman's model is that it does NOT give us path information, because there IS no path information because the particle "takes all possible paths". How then can this quantum model containing no non-trivial path information give you an exclusive path? I suspect the answer is something like: to get a classical path, we have to do measurements all along the way. Which makes sense, since it is by making such measurements/observations that other paths get excluded, i.e. the wavefunction collapses. So we pack the space between source and screen with infinite other screens, densely packed. Then from source to screen 1, screen 1 to screen 2, screen 2 to screen 3, etc., the amplitude (as calculated from the sum-over-paths procedure) to be measured at any point not on the classical trajectory is zero because it gets canceled. So basically iterate this argument an infinite number of times to get a classical path.
@@joelcurtis562 It turns out that, to get the classical case, all you need to do is take the hbar -> 0 limit. You don't need to make a series of measurements or any of that, and you can still try summing over all paths, it just turns out that the final amplitude in this case is exactly equal to the amplitude from the classical path. So here is the bridge: when we take the hbar -> 0 limit, the result where we sum over all possible paths (quantum) and the result where we only take into account a single path (classical) exactly agree. So in this limit, if we treat the system as if it only evolves along one single path, we don't actually lose any information and we get the exact correct answer (for finite hbar, this would only give an approximation). At the end of the day, it sort of turns out that it is a matter of how we interpret things. For finite hbar, we don't have a choice. To get the correct answer, we have to use the quantum description where we sum over all paths. However, for hbar -> 0, we can instead choose to interpret the system as only following a single, classical path and still get the correct answer. This, of course, is the classical interpretation.
@@zapphysics > You don't need to make a series of measurements or any of that, and you can still try summing over all paths, it just turns out that the final amplitude in this case is exactly equal to the amplitude from the classical path. So here is the bridge: when we take the hbar -> 0 limit, the result where we sum over all possible paths (quantum) and the result where we only take into account a single path (classical) exactly agree. So in this limit, if we treat the system as if it only evolves along one single path, we don't actually lose any information and we get the exact correct answer (for finite hbar, this would only give an approximation). Hm, I'm not sure that answers the question Joel had. I mean, is having an amplitude contribution from only the classical path the same as actually (or at least approximately traveling along side it)? Maybe I just don't know enough, since I'm not sure if the amplitude contribution from each path to a certain point also somehow marks the probability that that specific path was actually traveled along. And then how the amplitude (both of a specific path as well as a sum) relates to amplitudes for other ending points.
What does this say about energy conservation? Of the "Lagrnagian energy" anyway (en.wikipedia.org/wiki/Lagrangian_mechanics#Definition). I mean, deriving that the Lagrangian energy is constant (at least in Landau and Lifshitz' book on mechanics) requires the Euler-Lagrange equation, which presupposes stationary action, which is rejected here. So at best it seems like we could not say whether the Lagrangian energy is constant and at worst it would vary a lot depending on the path. Do you know if there's any work done on this?
Hey man, can I get the code that visualize the arrows and the 100 slits in this video? Is there a place where you upload these so I can mess with them? I'll credit you if I can make something with them (even tho I highly doubt that l I'm capable of doing anything new lol) Thanks! Awesome video btw, I learn a lot from it! :D
@Clyde Nathaniel Glad you liked the video! I would be more than happy to share the code! At the time being, it is in a not-so-user-friendly Mathematica notebook, but if you want to brave that, I can post it to my github. I could also relatively quickly put it into a Mathematica package which would be easier to use. Adapting it to a different language though would be tricky and definitely take some time, but probably could be done. Let me know what you prefer! (You can also feel free to email me and I can send it to you directly: zapphysics@gmail.com)
@@zapphysics thank you! I'm more used to python, but feel free to throw me the mathematica package if you don't feel like wasting time translating it to another language. Just give me some guidance on how do I add more slits, check the phase, etc. would be enough for me. Do you have discord or somewhere I can reach you quick for questions? My email is dokisame@gmail.com, and my discord is doki73#9834.
Yes, that is what we teach in university, it's just not true. This is NOT how classical mechanics emerges from quantum mechanics. Oops... I just noticed that I made the same comment two years ago, already. ;-)
The philosophical interpretation of the Lagrangian and principle of least action is Murphy's Law - anything that can happen will Minimising the difference between kinetic and potential is exactly that, allowing everything that is potential to be utilised given what is currently in motion. This gives a unique path I think using QM to explain this is a step in the wrong direction. It's a much more general, universal principle, of which physics participates in Its more of a law of logic or common sense even
@Neuron Neuron I don't know if I follow this... I'm not sure I see the connection between Murphy's law and the form of the Lagrangian. The claim is that one "allows everything that is potential to be utilized given what is currently in motion." However, the potential energy will be a fixed quantity along the path. When we vary the path, we not only change the potential energy, but also the kinetic energy, so I'm really not sure how one justifies this form of T-U based on this argument. The other thing is that the logic seems a little contradictory to me. Again, the claim is that minimizing the action allows for us to take into account all things that can happen. However, minimizing the action yields a single path, completely disregarding all other paths, so I'm really not sure how Murphy's law is at all connected with this procedure. I think that the quantum mechanical picture is much more conducive to a Murphy's law description because it truly does allow every possible path to contribute, not favoring any one over the other.
> Minimising the difference between kinetic and potential is exactly that, allowing everything that is potential to be utilised given what is currently in motion. But L=T-V, not |T-V|. So it's not necessarily the difference that's being minimized.
“U” Shape Waves This model may be related to the your topic. th-cam.com/video/wrBsqiE0vG4/w-d-xo.htmlsi=waT8lY2iX-wJdjO3 Thanks for your informative and well produced video. You and your viewers might find the quantum-like analog interesting and useful. I have been trying to describe the “U” shape wave that is produced in my amateur science mechanical model in the video link. I hear if you over-lap all the waves together using Fournier Transforms, it may make a “U” shape or square wave. Can this be correct representation Feynman Path Integrals? In the model, “U” shape waves are produced as the loading increases and just before the wave-like function shifts to the next higher energy level. Your viewers might be interested in seeing the load verse deflection graph in white paper found elsewhere on my TH-cam channel. Actually replicating it with a sheet of clear folder plastic and tape. Seeing it first hand is worth the effort.
Yes, that is the usual lore and it is 100% false. That is not the mechanism by which nature creates classical mechanics from quantum mechanics. The real physics is continuous weak measurement. See e.g. Mott's 1929 paper on "The Wave Mechanics of alpha-Ray tracks".
Perhaps another analogy: as Humans we will get breakthroughs in physics proportional to the chaos against which we counteract. Really is 'reward vs punishment.' The parallel universe of evil is not an option for me, anyway. There is a Spiritual Substance that holds everything in STC-perfect order, for Whom nothing is impossible. BTW loved the visuals! At age 5 I understood infinity mirrors yet here at 55 I'm incompetent as ever in math. Taoism says hope is an illusion & I must agree, LOL. 😁
Hey everyone! Hope you like the video. I can honestly say that this is one of my favorite results in physics. Also, if you didn't see already, I am considering starting a Twitch channel where I would live-stream informal physics lectures. I want to do Twitch because it would give more opportunities for interaction/questions from the audience in real time. I would also upload the videos to TH-cam after the fact. Let me know your thoughts on this!
Last point seriously taken. I'm flying solely on intuition. 😒🐒😄
Sounds like an amazing idea! I myself was actually looking for educational streamers on twitch, but was not very successful at finding them! Hopefully you'll be the first!
@Ascolano Irl, this is a great idea. I've never really interacted much with TH-cam's livestreaming, so Twitch was my first thought. This is the direction I think I will go!
到底都是谁需要伪科学，封杀揭发伪科学的客观自然科学？
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（量子力学被推翻了？并没有！量子跃迁需要时间吗？
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（耶鲁科学家验证量子跃迁确属连续过程，并成功开发量子跃迁 ...www.sohu.com › ...
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Jun 15, 2019 - 在研究中，研究人员通过特制的高速监测系统，成功捕捉到了量子跃迁将要发生的起始时间，并以此在量子跃迁进行到一半的时候人为逆转量子态 ...）李辉林注：这是我的论文“动力学基础理论的提出及应用原理示意”发表9年后；“宇宙物质结构动力学”网络公开（国内7年、国外5年）后，看到的第一篇国际论证“物质非能量转化”地足以推翻杨振宁“量子伪科学”为封建专制服务，及抢劫、剽窃“宇宙客观真理”否定欧美文明地权威佐证报道。
李辉林评论：总书记高呼“要发展量子科学”。就将几年前被我驳斥得体无完肤的陈词滥调又都重新放于网上。上Google网上去查看原文，却只有标题没有内容了。看来都是被那些科学巨骗们忽悠的（大陆伪科学的癞蛤蟆们一叫，台湾的伪科学癞蛤蟆们也都跟着一起叫。当然首要的不是为了某个政治人物。而是只有维护文化精神领袖，才能巩固不劳而获靠嘴巴成就生存“真理”的伪科学、文化、知识垃圾精英的荣誉、地位、利益。）！想用“电子光”物理作用，人为地设计一个“量子”伪物理概念替代 [本来想用“电子跃迁存在不连续的突变”来解释存在“物质与能量交换的量子过度现象”。被耶鲁大学的科学实验证明“电子跃迁是连续的”而破碎：薛定谔的“猫”从来就是鲜活的“电子”，是被人类所谓的科学精英们自己始终坚持近百年装死制造出地幻觉“量子”彻底死了。就想干脆用“量子跃迁”混淆、取代占有“电子跃迁”回避掉其“物质变能量地不连续性”。]。设计出“量子”的原始定义是“物质与能量地相互转化”。那么它不论作为物质还是作为能量都应该是可以实际收集储存（存在）的。可它与“光”一样只是在电子的“得”或“失”时地物质结构力地一种反应现象，电子平衡时“光”就没有了踪影（此时的元素没有电子成对地一入一出，而“量子兔成对出现就纠缠无踪影”了？），所以是不可收集储存与直接使用的臆造物质与盗窃物理现象，仅仅只是将“控制储存的基本物质电子运动，引起物质结构能量反应出的光现象及利用其物理特性，进行比特（状态）再分解（光的路径、偏振、角速度还可以做若干地分劈）获取更多的计算机运算、存储的开关单元。但是最终必须还是要还原成电子物质才能进行有效的存放。”。如“有光”（1）与“无光”（0）的计算机存储，只能靠电子的“得电”（1）与“失电”（0）来实现的。其讲解中除了将“量子”当做佛祖的名号供奉于神坛，就没有介绍出任何其具体有地实际作为。所以其“量子”没有任何实际意义，只是盗取、占有了“电子与光的物质结构与能量变化反应”的关系地“伪科学概念”。如果没有电子运动，“光”就不可能存在，“量子”更完全是垃圾科学家们脑袋里多余的“狗屎”。以你们的这种“科学理论思维”：高锟先生的“光纤技术理论”，还有“光碟信息技术理论”就都应该脱离“以电子为基础技术理论”而归依你们臆造的“正统”，作为“量子计算机的辅助材料理论”叫“量子科学技术理论体系”了？如果再对计算机追根寻源，其人类发明的第一台计算机就是使用“电子管”靠其发光运行计算、存储的，不就已经是你们这种欺骗逻辑的“量子计算机”了吗？逻辑理论上就是“挂伪真理的狗头，卖科学的羊肉。”，盗用一些成功的实用科学手段，去“证明伪真理”主观意志。将自然科学主观复杂诡异化，成为被主观封建专治说教控制地典型教义。（国内外都统一地封杀我的个人网页上的科学论文、评论、《证据》，只允许我做些时事评论，不允许做揭露伪科学的评论与论述。预示着“人类社会的专制与“民主”从意识形态上正在接受、效仿伪基础科学欺骗性地思想认识方法，用主观搞乱客观而趋于统一去专制、奴役人类！”。2020年12月7日）
牛顿第二定律，当加速度等于0时，物体要保持原来的运动不变，就必须要克服其物体继续运动的阻力反加速度，平衡这个反加速度的一个相等的加速度存在。爱因斯坦的“能量”也只是抄袭了工程力学缺失了阻力能量、极不准确的：“E（能量）等于m（质量）乘以c（速度）的平方”地物体做功公式，以为这个物体运动公式产生出的不等式，是因为“能量与质量可以互相转化”！所以就假设其中间存在一个转化媒介“量子”使其能够“被平衡”？完整的能量公式应该是“E等于M平方乘以C平方”。质量可以影响速度，速度影响不到质量。能量“消耗”的只是物质地结构（如分子结构、原子结构），没有消耗任何基本物质（电子、质子、中子，“能量”只是在改变这三种基本物质的排列组合时地“结构动态反应”动力现象。）。自然中不存在量子，宇宙不是产生于大爆炸。这不是质疑大师们的数学能力，而是其眼界和思想广度与深度的客观逻辑性地积累没有达到能够包罗宇宙所有事物。
F=ma 0=m0 ? （加速度等于零时，物体运动不需要动力？）
F=mv=-a'm=F' （物体做匀速运动时，加速度等于阻力加速度。）
我在国内（包括网络上）如此讲了十多年，又在国际网络上讲了五年。没有一个专业科技人员肯公开出来讨论，甚至辩论。其一些被我点名的中国国家权威们面对他们的虚伪、欺骗、抢劫，也全都始终选择保持静默？而不是被诽谤？名誉受到攻击？却只能动用权势，在国际网络上对我提出的“国家伪统治体制事实”进行全面封杀！认为我在此种情况下（得罪了东西方文明，遭国内外共同封杀。）不敢回国去。可是我还是于2019年9月至12月回国呆了3个月，有惊无险地回来了。因为我说的是纯粹的自然科学与客观经历、无党无派独往独来，他们不敢轻举妄动并不会是害怕李辉林，而是害怕其潜规则强盗科学与真理在全世界曝光？或者在欧美的态度与其一致的情况下，他们仍然还不坦然地害怕？
【爱因斯坦晚年信精神，怀疑物质！】那是因为爱因斯坦只知道宇宙中的所有物质都是在“相对性”地不断变化，由存在到“消失”的。却不知道他们全都是由“绝对性、客观真理似的”不可被消灭或创造的基本物质：电子、质子、中子地相互关系（只存在万有引力与万有斥力地物质结构能量关系，。）与排列组合（能量只是改变物质分子结构或者原子结构所产生出的，并没有消耗任何基本物质。）而成的。以至于整个宇宙地不断变化却又永存地存在。如果爱因斯坦还健在，他看了我的“论证”，他还能如此为自己提出的“能量与物质相互转化！（并引起人类基础科学界半个多世纪虚假地猜想、臆造、欺骗。）”而迷茫吗？或者也会像整个中国的科学、文化、知识分子们及那几位华人诺贝尔物理学奖获得者那样地不敢面对，却又觊觎将其中拆零分散而获得一个个“重大科学突破”立可见成效的客观事实？
上帝从来不现身于人类，只是为人类创造了一个“三基本物质”组成的客观世界，出了一道“客观宇宙”题，如果人类还不能正确认识、解释这道客观题的结构与运算，怎么有文化、资格谈论与见到上帝？靠什么去描绘、解读上帝？难道上帝也是个希望见到什么都不会表达，只会编造花言巧语吹捧他，表示愿意为他做奴才的那种人类？人类有史以来只是我首先提出了“宇宙客观真理及关系”，就是基本物质电子、质子、中子地性质、结构、运动、变化关系组成了整个宇宙。宇宙中所有的事物，包括人类自身的物质结构与运动变化，及思想、意识、行为都是这种基本物质地运动变化所产生的。在这宇宙整体客观真理面前，人类所有的先知先觉、文化圣人们，所有所谓的科学、文化、知识都只能算是其中主观臆造或者片面客观的认识；所有的自然科学家、劳动生产创造者的知识成就都只是其构成中的沙粒、砖瓦、基石。其区别仅仅只是存在阶段、局部、相对性地优劣。人类所有科技、文化、知识累积到现在，都连面对、探讨这个客观真理的知识能力，甚至态度都还基本不具备。
还可以在下面“评论”栏目中查看另外的评论文章。
This may be one of your favorite topics but you got the 100% wrong intuition here. Please take the time to read Mott's 1929 paper about the derivation of alpha ray tracks from wave mechanics. That is how nature creates macroscopic reality, by constant weak measurement. Heisenberg gave a similar example to Mott using Rydberg atoms a year or two earlier in one of his papers. Both are nearly forgotten early results of weak measurement theory. Why they were forgotten and replaced by this mathematical limit nonsense is the real question. Physicists kept observing high energy particle tracks in gas, liquid and solid detector media, but they have never observed the averaging of paths around the classical path in vacuum. Such a thing simply doesn't happen.
That was a great visualization of how to think about path integrals :D Also, the infinite-frequency part was really well explained!
As a Physics MSc, I've watched endless videos on the principle of stationary action in order to understand it. This is the first one that actually explains why it makes sense! and it came in a total surprise, thanks man!
One of the BEST physics/science creators on TH-cam.
Also completely wrong on this one. He got this idea from Feynman, who didn't think it through. Since then everybody is playing monkey-see-monkey-do with path integral averaging.
@@schmetterling4477 shut up and calculate.
@@schmetterling4477 In what way is it wrong?
Wow, this is the absolute best explanation that I have ever seen of the Path Integral! You my friend just earned a cookie and a new subscriber to your channel!
Come on, friend, he merited more than a mere cookie! I offer a half-dozen chocolate donuts. BTW are you perhaps Pole or Finn? I'm Hungarian.
I second the recommendation of Feynman's book QED. It is entirely readable and explains QED well.
Yes, it does, but it does not explain the quantum to classical transition correctly.
I really appreciate these videos. They are technical enough that I don't feel like I'm being hand-waved at, but not so dense that I can still follow without getting lost in the weeds. Thank you!
I'm SO glad I found this channel! Thanks for the brilliant content
I somewhat see a chicken-egg arguments occuring twice here:
1) Explicitly @3:55 with the "backward derivation".
2) Subtly @11:17 when the stated goal was to "derive classical behavior"... But it's just a conformity of mathematical interpretation.
The underlying principle, which is that of least action, is the foundational presumpted-result (thus the 'egg-chicken' argument, if you will).
So what emerges is a conclusion of something already readily known; which is good... is pretty much a meromorphism.
I do reckon this content is quite excellent and admire very much the endeavors of propagating some of the insights that Richard had. Kudos!
You lost me at the connection at 7:40 but this is super interesting! Currently taking an honors physics course, but I'll only get to ap physics next year where they use integrals, I'll return to this video then to see if it's my re understandable
Found another golden channel. Very nice.
I'm sorry
I can't watch it now as it's 1:30 AM here and my mom is going mad at me for using my phone
But I'll surely watch it in the morning
I love your videos , thanks for spending that huge amount of time creating this videos for us viewers
Very clear explanation. Professors should learn how to teach from this video!
Whaaat those animations :o
Awesome video, this is my favorite of yours.
@Eli Marburger, Thanks! This was probably one of my favorites to make, and hopefully you will see more like it very soon!
What a video! Thanks. You mentioned very deep issues
It's that spiral a fresnel spiral, C(t)+iS(t), that only takes form when you integrate e^(ix^2). I'm just curious on how the integral pipes out. I can't quite tell from your discription in the video
This is a fantastic observation! In the case of the free particle, the path integral is a functional integral which is exactly a complex Gaussian. In the very simplest case of a non-relativistic particle, the Lagrangian is just 1/2*m*(dq/dt)^2 where we perform our functional integrover the coordinates q. In more general QFTs, this kinetic term looks different and we can add a separate mass term, but it is always quadratic in the fields, i.e. the Lagrangian (density) looks like A.O.A where A is a field we perform our functional integral over and O is some differential operator.
This helped me with Feynman’s book. I found it frustrating, and your correspondences helped.
I used to tell people to read Feynman's book. I don't do that any longer. It's a great piece of science writing, but unfortunately it's technically wrong. Classical mechanics does not emerge from quantum mechanics in this way. Rather than studying the path integral of the quantum mechanical ensemble we need to study weak measurement processes on a single system. The two are not equivalent because weak measurements on a single system introduces conditional probabilities, which do not result from any possible operation of an ensemble of completely independent events.
There is a classical to Quantum transition referred as coherent superposition which is described by non zero off diagonal entries where in quantum to classical transition of a system is described by loss of these off diagonal entries from the matrix,
Yes, but you need a dissipative process for that to happen in reality. No such process exists in this "derivation", which is simply false.
"General sadness". Yes. Immediately I knew what that felt like.
I loved the part where action was involved in the Wave equations, made me very eager to play with funny physics sinarios.
This was insightful thank you! I liked how you explained the Lagrangian's very well, maybe you could the same for the Hamiltonian one day in the future? :D
Really great video
Can the Method of Least Action deal with friction?
I've always found the path integral image beautiful.
Having read this argument many times, I'm still confused about what exactly it is claiming about the relationship between the classical and quantum descriptions. The quantum description gives you the prob amp for the particle starting at source to be measured at some point on the screen. I get that in the path integral description that amp is gotten by doing the path integral and getting a final arrow for that point. However, the classical description gives you the total path which the particle takes (classically) between source and destination. To put it another way, a classical treatment allows us to 'fill in' the history of the particle between source and screen, whereas the quantum treatment does not. It insists that the particle actually does take all possible paths. In Feynman's treatment and this one of yours, I am missing how we go from summing the action over all paths for a given point to 'filling in' the intervening path with a specific classical trajectory.
@Joel Knoll I can definitely see how this can be a sticking point, especially because the argument is actually more of a mathematical one than an intuitive one. It relies on a mathematical concept known as the "stationary phase approximation," which you can read more about here: en.wikipedia.org/wiki/Stationary_phase_approximation. The basic idea is this: when you sum up a bunch of functions which are all oscillatory, if the frequencies of oscillations are very rapid, then all of the contributions end up cancelling except for the one which has minimum frequency. When we look at the path integral, the amplitude from each contributing path is given by exp(i/hbar S) where hbar is Planck's constant and S is the classical action along that path. This is exactly the form of an oscillating function where the "frequency" of oscillation is given by S/hbar. If hbar is large enough to be relevant, then we can't use this approximation and we have to sum up all paths to get our total amplitude. This is the case of quantum mechanics: all paths contribute and we can't say that the system took any one path. However, if we want to "turn off" quantum mechanics, this amounts to taking hbar -> 0. Here, we see that the frequency of oscillation for each individual path, given by S/hbar, gets very, very large. Using our stationary phase approximation, we know that, when we sum all of the paths in this case, all of the contributions from the paths which don't minimize S/hbar will cancel since they are completely out of phase. So, in the limit where hbar->0, we are left with the only contributing path being that which minimizes the classical action, which is, of course, just the statement of Lagrangian mechanics.
What I was trying to show in this video is how these paths begin to cancel when we start taking hbar to be smaller and smaller. This is the idea of the neat-looking spiral showing the sum of all of the individual amplitudes. When we take hbar to be smaller and smaller, the swirls which come from paths far from the classical path become tighter (meaning that the individual amplitudes are getting more out of phase) and pulling in more of the arrows from paths closer to the classical path. When we take all of the infinite paths into account and take hbar -> 0, all of the paths which are not the classical paths will perfectly cancel out (i.e. the spirals will become points and contain all of the arrows which do not come from the classical path), leaving only a single non-zero contribution to the total amplitude, which is the classical path. Basically, this is saying that in the hbar -> 0 limit, there is zero probability that the particle took any path other than the one which minimizes the action.
Hopefully that helps!
@@zapphysics Alas, it does not help my confusion. It's somewhat subtle and I have a hard time articulating it. It may be that I'm overthinking. I'll try to state it clearly.
The problem being solved here, it seems to me, is this: Given the source-point and a screen-point, what is the probability that a particle starting at source-point will be measured to be at screen-point some amount of time later? One way to get the answer, according to Feynman, is the sum-over-paths approach you outlined. And the idea there is that the probability amplitude for the transition in question to happen gets its largest (and in the limit hbar > 0, only non-zero) contribution from the path which corresponds to the one we would call the classical one. I.e. the one that minimizes the action.
Where I struggle is in the logical step from "the final prob amp gets its only non-negligible contribution from a certain path" to "that same path is the one we classically observe." Something seems to be missing in the argument there, as far as how quantum gives rise to classical.
After all, the whole point of Feynman's model is that it does NOT give us path information, because there IS no path information because the particle "takes all possible paths". How then can this quantum model containing no non-trivial path information give you an exclusive path?
I suspect the answer is something like: to get a classical path, we have to do measurements all along the way. Which makes sense, since it is by making such measurements/observations that other paths get excluded, i.e. the wavefunction collapses.
So we pack the space between source and screen with infinite other screens, densely packed. Then from source to screen 1, screen 1 to screen 2, screen 2 to screen 3, etc., the amplitude (as calculated from the sum-over-paths procedure) to be measured at any point not on the classical trajectory is zero because it gets canceled. So basically iterate this argument an infinite number of times to get a classical path.
@@joelcurtis562 It turns out that, to get the classical case, all you need to do is take the hbar -> 0 limit. You don't need to make a series of measurements or any of that, and you can still try summing over all paths, it just turns out that the final amplitude in this case is exactly equal to the amplitude from the classical path. So here is the bridge: when we take the hbar -> 0 limit, the result where we sum over all possible paths (quantum) and the result where we only take into account a single path (classical) exactly agree. So in this limit, if we treat the system as if it only evolves along one single path, we don't actually lose any information and we get the exact correct answer (for finite hbar, this would only give an approximation).
At the end of the day, it sort of turns out that it is a matter of how we interpret things. For finite hbar, we don't have a choice. To get the correct answer, we have to use the quantum description where we sum over all paths. However, for hbar -> 0, we can instead choose to interpret the system as only following a single, classical path and still get the correct answer. This, of course, is the classical interpretation.
@@zapphysics wonderful! This has vexed me for some time and that is a clarifying explanation. Thank you!
@@zapphysics > You don't need to make a series of measurements or any of that, and you can still try summing over all paths, it just turns out that the final amplitude in this case is exactly equal to the amplitude from the classical path. So here is the bridge: when we take the hbar -> 0 limit, the result where we sum over all possible paths (quantum) and the result where we only take into account a single path (classical) exactly agree. So in this limit, if we treat the system as if it only evolves along one single path, we don't actually lose any information and we get the exact correct answer (for finite hbar, this would only give an approximation).
Hm, I'm not sure that answers the question Joel had.
I mean, is having an amplitude contribution from only the classical path the same as actually (or at least approximately traveling along side it)?
Maybe I just don't know enough, since I'm not sure if the amplitude contribution from each path to a certain point also somehow marks the probability that that specific path was actually traveled along. And then how the amplitude (both of a specific path as well as a sum) relates to amplitudes for other ending points.
The links to the Feynman's QED lectures don't work anymore 😭
What does this say about energy conservation? Of the "Lagrnagian energy" anyway (en.wikipedia.org/wiki/Lagrangian_mechanics#Definition).
I mean, deriving that the Lagrangian energy is constant (at least in Landau and Lifshitz' book on mechanics) requires the Euler-Lagrange equation, which presupposes stationary action, which is rejected here.
So at best it seems like we could not say whether the Lagrangian energy is constant and at worst it would vary a lot depending on the path.
Do you know if there's any work done on this?
Great video, Keep it up!
Hey man, can I get the code that visualize the arrows and the 100 slits in this video? Is there a place where you upload these so I can mess with them? I'll credit you if I can make something with them (even tho I highly doubt that l I'm capable of doing anything new lol) Thanks! Awesome video btw, I learn a lot from it! :D
@Clyde Nathaniel Glad you liked the video! I would be more than happy to share the code! At the time being, it is in a not-so-user-friendly Mathematica notebook, but if you want to brave that, I can post it to my github. I could also relatively quickly put it into a Mathematica package which would be easier to use. Adapting it to a different language though would be tricky and definitely take some time, but probably could be done. Let me know what you prefer! (You can also feel free to email me and I can send it to you directly: zapphysics@gmail.com)
@@zapphysics thank you! I'm more used to python, but feel free to throw me the mathematica package if you don't feel like wasting time translating it to another language. Just give me some guidance on how do I add more slits, check the phase, etc. would be enough for me. Do you have discord or somewhere I can reach you quick for questions? My email is dokisame@gmail.com, and my discord is doki73#9834.
Great video thank you
That was good even for non-English speaker)
Great !
11:14
hello there, Julia set
Yes, that is what we teach in university, it's just not true. This is NOT how classical mechanics emerges from quantum mechanics. Oops... I just noticed that I made the same comment two years ago, already. ;-)
The philosophical interpretation of the Lagrangian and principle of least action is Murphy's Law - anything that can happen will
Minimising the difference between kinetic and potential is exactly that, allowing everything that is potential to be utilised given what is currently in motion. This gives a unique path
I think using QM to explain this is a step in the wrong direction. It's a much more general, universal principle, of which physics participates in
Its more of a law of logic or common sense even
@Neuron Neuron I don't know if I follow this...
I'm not sure I see the connection between Murphy's law and the form of the Lagrangian. The claim is that one "allows everything that is potential to be utilized given what is currently in motion." However, the potential energy will be a fixed quantity along the path. When we vary the path, we not only change the potential energy, but also the kinetic energy, so I'm really not sure how one justifies this form of T-U based on this argument.
The other thing is that the logic seems a little contradictory to me. Again, the claim is that minimizing the action allows for us to take into account all things that can happen. However, minimizing the action yields a single path, completely disregarding all other paths, so I'm really not sure how Murphy's law is at all connected with this procedure.
I think that the quantum mechanical picture is much more conducive to a Murphy's law description because it truly does allow every possible path to contribute, not favoring any one over the other.
> Minimising the difference between kinetic and potential is exactly that, allowing everything that is potential to be utilised given what is currently in motion.
But L=T-V, not |T-V|. So it's not necessarily the difference that's being minimized.
I honestly find quantum mechanics to be easier than classical mechanics.
Discrete > Continuous
“U” Shape Waves
This model may be related to the your topic.
th-cam.com/video/wrBsqiE0vG4/w-d-xo.htmlsi=waT8lY2iX-wJdjO3
Thanks for your informative and well produced video.
You and your viewers might find the quantum-like analog interesting and useful.
I have been trying to describe the “U” shape wave that is produced in my amateur science mechanical model in the video link.
I hear if you over-lap all the waves together using Fournier Transforms, it may make a “U” shape or square wave. Can this be correct representation Feynman Path Integrals?
In the model, “U” shape waves are produced as the loading increases and just before the wave-like function shifts to the next higher energy level.
Your viewers might be interested in seeing the load verse deflection graph in white paper found elsewhere on my TH-cam channel.
Actually replicating it with a sheet of clear folder plastic and tape.
Seeing it first hand is worth the effort.
Yo
First right?
You lost me at the 37 second mark with formulas.
Not the way to show the layman like me about physics.
Yes, that is the usual lore and it is 100% false. That is not the mechanism by which nature creates classical mechanics from quantum mechanics. The real physics is continuous weak measurement. See e.g. Mott's 1929 paper on "The Wave Mechanics of alpha-Ray tracks".
Perhaps another analogy: as Humans we will get breakthroughs in physics proportional to the chaos against which we counteract. Really is 'reward vs punishment.' The parallel universe of evil is not an option for me, anyway. There is a Spiritual Substance that holds everything in STC-perfect order, for Whom nothing is impossible. BTW loved the visuals! At age 5 I understood infinity mirrors yet here at 55 I'm incompetent as ever in math. Taoism says hope is an illusion & I must agree, LOL. 😁