How Feynman did quantum mechanics (and you should too)
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- เผยแพร่เมื่อ 14 พ.ค. 2024
- Discover Feynman's path integral formulation of quantum mechanics! 📝 Get the notes for free here: courses.physicswithelliot.com...
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⚛️ Additional links:
- Feynman's 1964 lecture on quantum mechanics:
www.feynmanlectures.caltech.e...
- Feynman's PhD dissertation can be found in this book:
www.worldscientific.com/world...
- Feynman's original paper on the path integral:
authors.library.caltech.edu/r...
- Dirac's 1932 paper that inspired Feynman:
www.worldscientific.com/doi/1...
- Feynman's Nobel lecture, including the origin story of the path integral:
www.nobelprize.org/prizes/phy...
- Feynman and Hibbs's textbook "Quantum Mechanics and Path Integrals" from the 1960s:
books.google.com/books/about/...
📖 Video summary:
If you've learned some quantum mechanics before, you've probably seen it described using wavefunctions, and the Schrödinger equation, and so on. That's how quantum mechanics was originally constructed by people like Schrödinger, Born, Heisenberg, and many others in the 1920s.
In the 1940s, though, a 20-something-year-old grad student named Richard Feynman discovered another approach. He found that the motion of a quantum particle can be described by taking a sum over ALL the possible trajectories that the particle could conceivably follow. That sum over all quantum paths is what's nowadays called the Feynman path integral, and it's central to our modern understanding of quantum physics.
One of the most important lessons Feynman's perspective reveals is how the usual laws of classical mechanics emerge from this more fundamental, but seemingly very different, quantum mechanical description of nature. In the video, you'll see why a single, special path emerges from the sea of all possible quantum paths when we consider the motion of a big object like a baseball--called the path of stationary action. And that of course is the path that obeys F = ma!
0:00 Introduction
3:12 Quick overview of the path integral
5:46 Review of the double-slit experiment
8:32 Intuitive idea of Feynman's sum over paths
13:27 Why exp(iS/hbar)?
15:00 How F = ma emerges from quantum mechanics
23:15 Lagrangian mechanics
24:34 Feynman's story
25:22 Next time: how to compute the path integral?
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About me:
I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at www.physicswithelliot.com. - วิทยาศาสตร์และเทคโนโลยี
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For more, read Feynman's book "QED" which is based on his lectures which are also on TH-cam.
17:45
Why hasn't Feynman's interpretation killed off the stupid Copenhagen interpretation yet??? 😂
@@robertwilsoniii2048 Not sure if your question is serious, but Feynman's diagrams were a calculational tool for determining the probabilities of how an event occurs and offers no interpretation of what happens to the wavefunction when an observation is made which is where Copenhagen comes in.
best book on beginner quantum mechanics!
I was so bad at math in high school I thought they called it algebra 2 because you had to take it twice. I’m now 38 and pretty obsessed with understanding at least the basic mathematical language of physics. It’s hard to find content like this, that balances accessibility and detailed explanations of the formulas. Thanks for that.
The more you learn the more stuff you will find on the net it's like a tree of knowledge, as long as you can see between the bullshit and the actual academia
Same age here. Good on you pursuing education, I have nothing but respect for you foe that pursuit 🤭
Same, I sucked at math when I was in high school but for some reason at 32 years old, I’m an undergrad physics major. I’ve been getting straight A’s in my classes. I really hope I can get my PhD in time, before I get too old and people probably wouldn’t wanna hire me.
Same, I am 35 now and become an engineer 😂
Elliot you are the singular best instructor I have ever seen! You have the gift Sir, thanks for sharing it!
Thank you!
The 3blue1brown of physics
A half hour flew by. I clung to every word-excellently constructed argument, very well-explained at each step.
Your ability to explain complex topics in an intuitive way is amazing.
I have rarely seen a clearer explanation!
Well done!
Your explanations are the best, keep up the good work!
Hands down, the most intuitive explanation about Quantum Mechanics. Simple remarkable, Thank you for such a video.
That's a lovely video indeed! It somehow condenses the first month of the analytical mechanics course together with the first chapter of QED book by Feynman. It's been years now so I can't remember the details and so your videos are excellent reminder. Thank!
Fantastic description. I've been through the many paths derivation many times and could never quite figure out how Stationary phase approximation leading to F = ma came about. I understood it was to do with argument of the complex exponential changing a lot but putting it on the Argand diagram made it crystal clear. Thanks so much!
Excellent! This was as brilliantly taught as one could imagine.
I'm finally caught up with all your previous videos, and this is another great one! Your animation and explanation of why the amplitudes near the classical trajectory are the ones which intefere most constructively was particularly nice, as well as talking explicitly about the ratio of S and ħ.
There's one thing I think would be really nice to add to what you said: in the video, you looked at the probability of a particle getting from x1 to x2 between times t1 and t2, and as you explained, the trajectories near the classical one dominate the sum, with only the classical path contributing in the classical limit. However, in order to fully appreciate the difference between the quantum and classical scenarios, I think it's also important to think explicitly about the "other" situations - the ones in which there is no classical trajectory for the particle to get from x1 to x2 between times t1 and t2 given the initial conditions. In those cases, the probability will be zero in the classical limit, because there will be no constructive interference of amplitudes near any of the trajectories, while in the general quantum case, the probability can very well be non-zero.
Awesome point !!!
Information goes so smoothly! I was thinking i'll need to pause and rewind stuff all the time, but you made it so intuitive
Awesome
I just enrolled in the course!
I really love the way you teach and explain physics. I sincerely hope this is the 1st of many. Never stop doing this Elliot! You are honestly amazing at it.
I found your channel while trying to learn quantum mechanics but it looks like I must to learn Lagrangian mechanics first.
Wish me luck!
Thanks Andre! I hope you love the course!
Thank you for your videos! I learn a lot with them. And your voice is extremely soothing!
This is so good, thank you for having made this.
Clearly one of the best physics video I've ever seen! Your work is just amazing
this was great! watching this really helped organize my thoughts about quantum physics
One of the best explanations on this topic
explanation of the concepts and visual was top notch... helped my understanding on this topic.
The phase summation at 18:20 is well done. It also makes me think of Fresnel Zones in point to point telecommunications. You need to keep obstacles outside the first zone, which is shape that contains paths with a phase change of less than 180 degrees (iirc), vs. the classical line of sight (geometric optics)…linking Fermats Principle to Feynmans Path Integral, via Fresnel
Brilliantly illustrated and explained!
Hey Elliot, your channel is a gem. Thanks a lot for existing.
WOW! Dude this video came out right when I needed it the most. I've been struggling with understanding the math behind the path integral for my grad quantum class for the last few weeks. Your fourier transform video was absolutely incredible and left a lasting impression on me. I'm so excited to watch this. My heart leapt with excitement when I saw that this came up when I searched for the path integral explained
Nifty AF ! I'll never forget one of his lectures explaining "simple" mirror reflection, regarding individual photons: it could reflect off of this(rather distant) mirror segment or it could go "the way you want it to go"(the middle section). The sum of the different ways always just turns out to be the classical(intuitive) path. I could still use a refresher on how/why |amplitude|^2 "is" a probability in the 1st place though, now that I think about it. Its all such amazing & interesting stuff.
Spectacular video. Loved every second. Are you planning on going over Yang-Mills sometime in the future. Also I’m really excited for your video on tensor analysis.
Thank you, this was a great journey!
Richard Feynman's contributions to physics needs to be promulgated and celebrated. People will praise Einstein all day long. Yes, general relativity was pretty cool. Riemann curvature, metric tensors, stress-energy tensors blah blah blah. But not that cool. Doesn't work with QM. Along comes Feynman and gives us path integration, diagrams, and QED. The guy was a superlative teacher and science communicator. A genius with math. Able to explain the most complex of subjects so simply that someone who knows nothing about science would be able to understand him. We need more love for this guy. The world is poorer for his loss. In our hearts he lives forever.
That's very incredible. Thanks very much. Helped a lot
Phenomenal, thank you for this!
All your videos are top shelf, but this one is a real treat.
Thanks James!
Amazing video! Beautiful explanation.
Such a great well explicative video!! Thank you very much
It is useful to note that it was Dirac who first thought that Amplitude is proportional to exponential of action (with factor of 'i') divided by Planck's constant.
What prompted him to think so? I’ve never got it but I never really looked too much into it.
Awesome video! Two questions: Does this mean that in the classical limit, the action can never have any extrema other than for the classical path? (The explanation for why the classical path emerges from the sum-over-paths depends only on the fact that dS/dε = 0, and surely any other path where dS/dε = 0 would interfere with that?) Also, this way of reasoning for how all the "nonzero" terms average out is reminiscent of how we find Fourier coefficients; is there any way to relate these two concepts?
Good stuff! I thoroughly enjoyed this. I think it is easier to understand than Feynman's QED book. I also liked the derivation of the classical limit.
Great work. Thanks for sharing
I really liked that transition from two slit, n slit, diffraction grating, Bragg refraction, empty space.
If you look at diffraction: the pattern is the Fourier transform of the aperture function,,…that is a sum over amplitudes with complex phases…it’s the same form as a path integral.
It finally makes sense!! All I have ever heard before was that the extreme paths were cancelled because of some hand-wavey reason about pairing with paths with opposite phases.
It makes complete sense, ~0 first-order change around the stationary path means little change in phase! It all makes sense.
Thanks a ton!
What a beautiful idea, Feynmann was a genius.
Simply brilliant. Thanks.
This is done so perfect, wow
A great video on a great subject! Again
Wow!
Incredible clear!
im starting quantum theory 1 now and the next video could really help!
This is amazing!
Educationally brilliant!
His lectures in physics are the best ever books I read. My favorite theoretical scientist for a reason
Thanks for sharing these videos, they are all really helpful!
One bit of feedback: I find the black writing on purple background to be a bit hard to see, especially on a small device. The graph paper lines also make it a bit harder to see easily.
Excellent video! Though one minor correction: "more often than not the stationary point is a minimum" isn't true. More often than not, it is a saddle point. In fact for all continuous systems (classical fluids, a piece of rope, etc) it is *always* a saddle point. And even when it is a minimum, since the principle is invariant to a scaling of the Lagrangian, we can negate it to make it a maximum (i.e. use U-K instead of K-U and get the exact same dynamics), which further shows that minimality is by no means fundamental.
"Least" action is just a historical misnomer. I think this video is actually great at showing the intuition behind why stationarity is what really matters, and it is a shame that you had to mysteriously and erroneously suggest that "least" is somehow special at the end. But yeah, great video otherwise! I'll definitely be pointing students to this one.
I'm intrigued by your statement that the stationary point is, for continuous systems, always a saddle point. Perhaps you might expand on this statement. I have not heard this before.-ArthurOgawa
@@user-et9ub3dc3jI'd love to give you a link to more info but youtube disallows comments with links to external sites. There are some good Physics Stack Exchange answers on this though, so at the very least I can give you the URL extension for one: */122486/confusion-regarding-the-principle-of-least-action-in-landau-lifshitz-the-clas*
I'm mostly an applied maths (grad) student with not much interest in physics, but this channel is slowly making me fall in love with the subject !
Wuwoo this is all I need , thanks sir yu r amazing
Thank you for this
man i barely understood anything, but that little I understood made me wanna learn about this more. thank you so much
Unbelievably good.
A great day today because I found this channel.
Phenomenal video!
19:41 is where it clicked for me! The parallels with Lagrangian mechanics saying that objects follow minimum action paths (so gradient of S is zero) is beautiful. Thank you!
Edit: Whoa whoa whoa you can't just put all those equations up and not tell us more at 23:31 ! I really want to see how this idea of action generalises to other topics like electromagnetism/relativity. Hope you'll do more like this!
I legitimately laughed out loud in excitement when you said the path of least action was a sort of equilibrium around which the values are stable-it clicked immediately that the lagrangian formulation and thus F=ma would emerge. Absolutely incredible video!
Edit: this whole idea of the complex waveforms representing the kernel reeks of the Fourier transform of something to me: is there any significance to the inverse Fourier transform of the kernel, and does the kernel have any relation to the wave function?
The Fourier transform of the Kernel is related to it's representation in momentum space, just like the wavefunction in the Schrödinger formulation
and now humankind is in its baby steps to recognise that space and time are emergent properties of something much more deeper... mathematical objects that live beyong spacetime
its like trying to observe the inside of your computer using the google search bar... space and time are like pixels on your screen, its like saying "this screen is my fundamental reality" where in reality the computer itself with its motherboard, cpu and gpu are actually the fundamental components in which the arrangement of pixels emerge from.
do you know why we still have problems with gravity in modern physics? its because we treat gravity as an emergent property of spacetime where in reality spacetime ITSELF is an emergent property of something much more bigger
@@ukacip9310 It is quite rich for a nonphysicist to tell physicists that they are wrong about physics.
HI Elliot, Excellent videos, both in terms of production values and pedagogy. I hope you will continue to make videos - they're really of great value to students and all who love to learn more about physics and math. I would like to make a couple requests. (1) I'd love to see the details of the epsilon expansion approach to renormalization-group theory. I'm familiar with Position-Space Renormalization group, but not that much with the epsilon expansion. (2) I'd also like to see the calculations behind the Schwarzschild solution to general relativity, including the Schwarzschild radius and Einstein's initial reaction to it. Many thanks for your top-notch physics videos, Elliot. Jim Walker
I've finally got a "cancellation" part of path integral. Thank you for the clearest explanation on this topic!
Have you studied the lagrangian formulation of classical mechanics, or at least seen Fermats Principle?
Edit: never mind. I watched the video. He killed it
great stuff!!
Thanks a lot sir , I Have a Simple DOUBT , that the quantum particle could choose any path, but how does curve back in space in the middle the path ?
Feynman is the Best. Really Amazing video!
After Encountering 1 min explaination of Quantum Mechanic by Prof. Brian Cox n the fact that I found that detail explaination here is just unbelievable. Thnks alot.
Thanks for this! I get a bit frustrated with so many videos that explain quantum mechanics in an abstract way for simplicity as they raise more questions than they answer... It was always confusing how they talked about wave function like two waves through the slits, and also the probability function, which looks like a wave... I always wondered "um if you're taking about physical locations of slits in a 3d space affecting that function, how are you including a definition of that physical setup in the formulas!? So where you break down the slits and say "imagine there's so many slits they disappear"... Although you're still talking in abstract terms, it actually makes a lot of sense and pulls everything together with the maths! Thanks! I mean I still don't fully "get" everything but I feel I'm on a stronger learning path now and have better questions to ask
Thank you!
This is, by far, the clearest explanation of the Feynman path integral formulation of QM.
I’d rather take 100000011 and times it with 9024 and have some quite time about the speed of light used in some measurements. Not my thing here
IF I have the money, I would definitely get your Lagrangian Formulation course with 1-vs-1 coaching. I'm just sad that I'm just not rich enough to afford your course. You are so great, Dr. Elliot!!!
Very insightful video. I have a question. Maybe this is splitting hairs, but is a certain path CONTRIBUTING to the amplitude K_fi the same thing as the particle ACTUALLY TAKING that path? If the classical particle truly did take all paths at once, wouldn't the result be the same? The paths near the stationary one would still give the greatest contribution to K_fi and the rest would cancel. Do I have that right?
I truly enjoy your videos. You have a knack for tying concepts together and it gives one a sense of how physics evolved. The graphics are very well done and you have an engaging speaking style. I have watched all your content and look forward to more. BTW...an observation...i used to wear button down collar shirts like you wear in the 1960's. We called them Ivy League Shirts!...Just Sayin!!
love it!
yes we should all do it like that
i sure hope computing the trajectory of a free particle won't be comically difficult
A very nice explanation. I realy loved the animation. What program is used for this?.
You helped me figure something out that’s been rattling in my brain. You talk of how it doesn’t show the one way, it shows all the conceivable paths and that’s what I’ve been trying to explain to people and how our brain works by doing the same thing. It leverages the qm to gather differences through our senses and integrate them in our head to show us this. That’s why dark energy and matter were thought of, the canceling out of each other but it’s not cancelling but kinda like opposite reaction like physics so more like it exists as potential until revealed. And i mean potential as in stored energy in the form of neurons and the connections of differences that make up the thought. So not an invisible but a real but not fully formed, like how evolution is connecting us and the pieces. That’s why math can be used in a 1:1 or a reflection kinda like how we reflect the differences through our connections and differences connected. Like sharing words through language and stuff.
23:00 justify using K - U rather than K + U because that results in F = ma rather than F = - ma
But it seems a little circular - using the desired classical result to figure out the more fundamental formulation. Is there not a more fundamental reason for K - U without appealing to the desired classical result?
At however fundamental a level, you have to input some experimental/observational information into your model to determine exactly what form the theory takes.
The insights that led to the path integral were anticipated in a 1929 paper by Mott. He sought to answer the question of why the wave function of an escaping alpha particle was a spherical expanding wave, but what we would see in a cloud chamber would be the straight tracks of an apparently classical particle. His answer was to consider a multi-time, multi-point wave-function. It turned out that only a family of strainght rays would have significant amplitude. I wonder if Dirac was aware of Mott's paper?
Beautiful!!
Hey, could you please make a series on Hamilton-Jacobi theory? That would be very helpful.
When does the next video arrive? Looking forward to it!
Excellent explanation of path integrals and how they can be used to derive Newton's law of motion from the quantum mechanical amplitude! This is a wild but beautiful idea, which seems to involve Hugh Everett's many worlds hypothesis in a very strange way that I still need to get my head around!
To me it just looks like a monte carlo simulation of least action given unknown (or at least unaccounted for) external perturbations. It seems to just be an action-weighted sum of all the potential paths which is roughly equivalent to varying a bunch of potential confounding variables randomly across a large number of runs of least action simulations. And like a monte carlo simulation, you end up with a probability density.
Well, 46 secs in, the graphic rep of all possible paths, amplitudes and Cartesian, is already impressive.
Amazing.
I HAVE DONE THIS MATH SEVERAL TIMES @10 TO 18TH POWER USUALLY LOOKING FOR DIGIT ANOMOLIES AND PRACTICE THANK YOU SIR
Hey Elliot. I'm just wondering: I've heard that in actually working out a path-integral, there's supposed to be a term corresponding to a variational free-energy functional you minimise that you just add on to the end to solve these. What exactly is that, or am I talking nonsense?
Awesome! Thanks Elliot! Is it correct to assume that when talking about one particular path, the particle (no pun intended) is treated as a material point?
Glad you liked it! For each term in the sum, we're writing down the trajectory of a point-like particle, and then taking a kind of weighted average of all those possibilities. Not positive if that answers your question though
@@PhysicswithElliot Yes it does. Thanks!
My only complaint is that you don't post more videos, and your courses are too expensive for the casual learner. But still, one of the best physics channels
You are REALLY good.
Amazing video
Amazing video!
Which program did you use to create this video?
In this video I connected the dots as follows:
1. About minus sign: Once I thought and asked why it is needed for the least trajectory, [HCV class]]
2. About Cancellation: something reminds me Heisenberg matrix mechanics development.
3. About angle's dimesionless: Reminds me them as mere ratios of same physical stuff.
4. exponential limiting case: Reminds a beautiful thought about observable.
5. Revision of recently heard story about Intuitive idea of Feynman's sum over path.
6. What happens if slits count beyond two? an earlier question
Wonderful!
Thanks for the excellent video. One minor comment/criticism: Why do you say (at 26:26) that the sum of two bumps is another bump? Should it be like a bimodal distribution? 🙂
great video
Are there similarities / partial with the eikonal approximation and Wentzel-Kramers-Brillouin approximation ?
In the 2004 book by Hagen Kleinert on path integrals it is mentioned: the Feynman path integral formulation (also) works for the hydrogen atom .
Very well explained. I almost went into a confirmation bias because at so many times (like drawing the unit vector r spherically) which in Recurrent deep learning where normal distribution is a nice statistical way to weight and regularize while introducing non linearity and although the formal expression for a centered random normally distributed vector X and the operation of adding a copy or another random vector Y = Same nice properties random vector
Z (uhh...assuming independence) the function describes the distance from the origin in terms of r (radial distance) - most evident in the case of the spherical Gaussian, where the density at any point depends only on the distance from the mean (or origin in the standardized case). Though the explicit forms are so similar, in it's usage in RNN, it is necessary to ensure that the probability density function integrates to 1 over the entire space, but so is the case in adding all probabilities in Quantum theory also 1? Isn't that where it breaks down as well? Edge cases etc. but you know what, I learned a wee bit of QED but was silly enough to think, "the behavior of particles like electrons as a sum over all possible paths, each weighted by a phase factor determined by the action,....." even though the approach of path integral formulation feels quite abstract and hard to wrap it conceptually, this video explanation was brilliant in it's efforts. Trying to confirmation bias my way into thinking there is anything more than mere metaphorical similarity b/w Gaussian processes/ statistics and machine learning for "modeling distributions over functions" had some literal connection to Path integrals - only shows underlying beauty of mathematics' symbolic similarity even in completely different branches.
in some ways, the Path Integral is like the Lagrangian in classical mechanics and the Wave Function is like the Hamiltonian.
So is Lagrangian better than Hamiltonian approach for QM?
Super teacher!
You are my favorite youtuber.
great video. Here is my thoughts why the Quantum particle changes position. Its because of another variable maybe Magnet field, Electric charge, ... Or a sum of all of those.