How Feynman did quantum mechanics (and you should too)

your experience with things that you have seen before is inadequate is incomplete the behavior of things on a very tiny scale is simply different that was Richard Feynman he won the Nobel Prize the year after that clip was recorded for understanding the quantum physics of light and how it interacts with matter but long before he was a famous Nobel Prize winner as a matter of fact when he was just a 20-something-year-old grad student feynman's first great discovery was an entirely new way of thinking about quantum mechanics which in the 80 years since has proven essential to our modern understanding of quantum physics it's called The Path integral formulation of quantum mechanics and once you understand it feynman's perspective will give you a ton of insight into the counter-intuitive way that things behave in the quantum world and at the same time it will teach you how the laws of classical physics like f equals m a are derived from the more fundamental quantum mechanical description of nature quantum mechanics is all about describing the behavior of really tiny particles like electrons and to give you an idea of just how different it is from classical physics let's start by comparing and contrasting the classical and Quantum versions of a very simple problem so say we've got a particle that starts out at some position x i at an initial time TI In classical mechanics our job would be to figure out where the particle is going to be at any later time we add up all the forces set that equal to the mass of the particle times its acceleration and then solve this equation for the position X as a function of the time T if it's a free particle then the solution to this equation is just a straight line or if it's a baseball that we're throwing up in the air the trajectory would be a parabola either way the point is that in classical mechanics we can predict the final position XF where we'll find the particle at a later time TF quantum mechanics is fundamental different though if we're told that a Quantum particle was found at position x i at the initial time t i then all we can predict for when we measure its position again later is the probability that we'll find it here at position x f if you do the same experiment many times over sometimes you'll find the particle around there and sometimes you'll find it somewhere else this probabilistic nature of quantum mechanics is one of the strangest things about the physics of tiny objects it means that a Quantum particle doesn't follow a single well-defined trajectory anymore in getting from one point to another in fact the incredible thing that Feynman discovered and that you'll understand by the end of this video is that instead of following a single trajectory like in classical mechanics a Quantum particle considers all the conceivable paths and it does a kind of sum over all those possibilities that sum over all trajectories is What's called the Feynman path integral and it's pretty mind-boggling to say the least if you're wondering how that could possibly be consistent with the fact that a baseball most definitely follows a single well-defined trajectory stay tuned because understanding how the classical path in f equals m a emerge from the quantum sum over all paths is in my opinion anyway one of the deepest lessons in all of physics I think what I should do right at the beginning here is just to give you a quick sketch of how the path integral works and what the main formulas are just so you have a rough idea of where we're going don't worry if it doesn't make perfect sense yet we'll spend the rest of the video unpacking where it all comes from and what it means what we're interested in here is the probability for a Quantum particle that started at position x i at time TI to be found at position XF at a later time TF generally speaking to find a probability in quantum mechanics we start by writing down a complex number called an amplitude and then we take the absolute value of the amplitude and square it to obtain the actual probability if you saw my last video you got an idea of how that comes about by looking at a famous Quantum experiment called the double slit experiment I'll put up a link to that if you haven't seen it yet and I'll also review the key takeaways we'll need from that video in just a minute so what we're looking for is the amplitude for the particle to travel from point I to point F and I'll write that as k f i and here's feynman's path integral prescription for computing K again classically the particle would follow a single unique trajectory between these points but in quantum mechanics Feynman discovered that we need to consider every possible trajectory that passes between them each of those possible paths contributes with a particular weight which is written as e to the I times s over h-bar h-bar is Planck's constant which is the fundamental physical constant of quantum mechanics and S is a certain number that's associated to each trajectory called its action I'll explain how that's defined later on but the action is the central object in the more powerful approach to classical mechanics known as the lagrangian formulation which you might have heard about before I've actually created a whole course all about lagrangian mechanics I'll pin a link to that down in the comments along with a very special offer code for the first 100 students who use it to sign up and now to find the total amplitude for the particle to go from point I to point F we add up these contributions from all the possible paths this is feynman's procedure for computing the quantum mechanical amplitude of course the set of all these paths isn't a finite list so this isn't really a discrete sum it's a sort of integral called a path integral and so we more often write it using a notation like this and that's why this is called the path integral formulation of quantum mechanics but anyway now we need to actually understand what the heck all this means and the intuitive idea behind Feynman sum of our paths starts from the double slit experiment again so let's begin by quickly reviewing the key things we learned in the last video here was the setup we took a solid wall and punched two holes or slits in it then we chucked different things at the wall and recorded what made it through to the other side with classical particles like BB pellets or billiard balls or whatever things were very simple the particles that went through the left hole mostly hit the backstop in the region behind the left hole and likewise the ones that went through the right hole wound up on the right the Total distribution was the sum of those two curves because each particle either went through one hole or the other that gave us one broad bump in the center of the backstop next we looked at waves like light waves that was more interesting because the two waves coming out of the holes can interfere with each other and produce what's called an interference pattern on the back screen the bright spots are where the waves add together to make a bigger wave which is called constructive interference and the dark spots are where the interference is destructive and the waves cancel each other out the corresponding intensity curve looks something like this with alternating Peaks and valleys corresponding to the bright and dark spots and we discussed how that comes about by writing down the waves coming out of each hole in complex notation they were of the form e to the I5 where the phase Phi depends on the distance R from each hole to the spot on the screen the total outgoing wave is the sum of those contributions and from that we were able to compute this intensity curve finally We Shrunk The Experiment down and fired Quantum particles at the wall like electrons and the result was something surprising instead of showing one big bump around the center of the backstop like we had for classical particles the quantum particles were distributed according to another interference pattern with lots of particles clustered in some spots separated by gaps with next to none this is nothing like our experience with how things like BB pellets or baseballs would behave it means that an electron does not follow a single well-defined trajectory on its way across the Gap each electron somehow probes both holes at once and interferes with itself last time we saw how to describe what's going on mathematically using Schrodinger's idea of the wave function that's how quantum mechanics was originally constructed by people like Schrodinger and Bourne and many others back in the 1920s in the 40s though Feynman came up with his path integral approach the two are completely equivalent you can derive either formulation from the other but they each give a valuable perspective on the underlying physics so now we'll take findings approach and see how the lessons from this simple experiment lead us to the idea of the path integral the key lesson to take from the double slit experiment is again that a Quantum particle doesn't follow a single trajectory like a classical particle would have we have to consider trajectories that pass through each hole in order to understand the distribution of hits we see on the backstop but now let's push that idea a little further if we drill a third hole in the barrier we'll have to include trajectories that pass through that hole as well and the same goes if we drill a fourth hole or a fifth and sixth and so on while we're at it let's go ahead and add another solid barrier in between and drill a few holes in that now we have to consider all the possible combinations the particle might pass through the first hole of the first barrier and then the first hole of the next barrier or it could go from the second hole to the third and all the other possibilities now take this idea to The Logical extreme we completely fill the region with parallel barriers and through each one we drill many many little holes then we need to account for all the possible Roots the particle could take traveling from any one hole to any other on its way across in fact we can imagine drilling so many holes that the barriers themselves effectively disappear just like when I mentioned hoygen's principle in the last video we drill through all the barriers until we're effectively left with empty space again then what this thought experiment suggests is that to find the total amplitude for the particle to propagate from this initial point to some final point at the detector we need to add up the individual amplitudes from from each and every possible path that the particle might follow in traveling between those endpoints and not just the paths traced out in space but all the possible trajectories as a function of time and that's how what we learned from the double slit experiment leads us to the idea that we need to sum over all trajectories to compute the total quantum mechanical amplitude but what weight are we supposed to add up for each path let's suppose much like in our discussion of waves that each trajectory contributes to the sum with a particular complex phase e to the I Phi where Phi is some number that we assign to each path which determines how it contributes to the total amplitude this is the core idea of the quantum sum over paths and it's pretty incredible compared to our usual experience we're used to finding a single classical trajectory for the position X as a function of the time T that goes from the starting point to the ending point where I'll stick to one dimension X here to keep things simple maybe it's a straight line line or a parabola or whatever but in quantum mechanics Feynman discovered that we need to count every possible trajectory that the particle could conceivably follow between those points for each path we write down the phase e to the I Phi that it contributes and then we add them all up to find the total amplitude strange as it sounds this prescription is at least totally Democratic in the sense that each term in the sum is a complex number with the same magnitude one you can picture e to the I Phi as an arrow in the complex plane in other words we draw a picture with the real Direction along the horizontal axis in the imaginary Direction along the vertical axis then e to the I Phi is an arrow of length one that points at an angle Phi different trajectories will contribute arrows pointing at different angles but they all have the same length of one the question is what angle Phi are we supposed to assign for each possible path well I already mentioned the answer back at the beginning of the video for each trajectory the complex phase it contributes is given by e to the I times s over h-bar h-bar is the quantum mechanical constant called Planck's constant its value in SI units is given approximately by 10 to the minus 34 Joule seconds s meanwhile is the action which is a particular number that we can compute for any given trajectory you might have run into it before because it's something that already plays a central role In classical mechanics but here's how it's defined we take the kinetic energy of the particle at each moment subtract from that the potential energy U and then we integrate that quantity over the time interval from TI to TF the result is a number that we can compute for any given trajectory and that's its action the quantity k minus U that we're integrating here gets its own special name by the way it's called the lagrangian so that the action is defined by integrating the lagrangian overtime and that's the central object in What's called the lagrangian formulation of classical mechanics and yes that really is a minus sign in the middle more on that in a minute now depending on whether you've learned a little bit about lagrangian mechanics before seeing the action and lagrangian appear here might be ringing enormous bells in your head or these formulas might look completely out of left field so let me try to motivate where this weight e to the i s over H bar is coming from well first of all let's just think about the units we have to play with here we certainly expect Planck's constant H bar to appear in our weight Factor since again it's the fundamental constant of quantum mechanics that had units of energy in joules times time in seconds but Phi here is an angle remember it's measured in radians say and doesn't have any Dimensions so we'll have to pair the H bar up with something else with those same units of energy times time in order to cancel them out and the simplest thing we could write is a ratio s over H bar in the action s indeed has those units we're looking for K and U are energies and they get multiplied by time when we integrate over t and the units of s cancel out the units of h-bar and we're left with a dimensionless number for the angle Phi like we needed so the units at least work out correctly otherwise it wouldn't even make sense to write down this quantity e to the i s over H bar you might be wondering though why the heck are we taking the difference between the kinetic and potential energy wouldn't it seem more natural to write the total energy like we're much more accustomed to that's certainly what I would have tried first if I'd been working on this problem 100 or so years ago but that's wrong it's most definitely a minus sign that appears in this formula for the action and we'll see why after we've talked about the second key piece of motivation for where this weight e to the i s over H bar comes from it ensures that the unique classical trajectory emerges when we zoom out from studying tiny quantum mechanical particles to bigger everyday objects it's not at all obvious how that works at first glance at feynman's formula if this is telling us to sum over all paths that the part article could follow each with the same magnitude and just different phases how could that possibly be consistent with what we observe in our daily lives where a baseball most definitely follows a single parabolic trajectory after all quantum mechanics is the more fundamental theory in our everyday laws of classical mechanics must emerge from it in the appropriate limit the answer to this question is one of the deepest insights the path integral reveals about the laws of physics it will show us how f equals m a follows from this more fundamental quantum mechanical description roughly speaking what happens is that for the motion of a classical object like a baseball almost all the terms in the sum over paths cancel each other out and add up to nothing all except one and that's the classical path and here's why let's draw the complex plane and here again on the left is a plot of the position X versus the time T each term in the sum corresponds to an arrow in the complex plane it has length 1 and points at an angle set by S over H bar so we pick any trajectory connecting the initial point to the final point we compute the action s for that path divide by H bar and then we draw the corresponding Arrow at that angle if we pick a different trajectory we'll get some other value for the action and that'll give us another arrow at some other angle and what we need to do is add all these arrows up here's the thing though H bar is really really really tiny again in SI units its value is of order 10 to the minus 34. that's a 1 with 33 zeros to the left of it and then the decimal point by comparison a typical action for a baseball will be something like one joule second maybe give or take a few orders of magnitude in either direction but it's vastly larger than the value of H bar so the angle s divided by h-bar will be an enormous number for a typical path for a baseball on the order of 10 to the 34 radians starting from Phi equals zero it's like we flicked this Arrow so hard that it spins around and around a bajillion times until it lands in some random Direction but now let's pick a slightly different trajectory and consider what that contributes to the song it's a very similar path to the one we started with so its action will only be slightly different from the first one maybe the first path had an action of one joule second and this new one has 1.01 say so that the change in the value of the action between them is 0.01 Joule seconds it doesn't matter what the precise numbers are because again when we divide by the incredibly tiny value of H bar even that small change in the action at the classical scale will produce a massive change in the angle in this case something like 10 to the 32 radians then even though these two trajectories were only slightly different their corresponding arrows point in random different directions in the complex plane and now as we include more and more curves each of them will give us an arrow in some other random Direction too we'll get an incredibly dense array of arrows pointing in all directions around the unit circle according to feynman's formula what we're supposed to do is add up all these arrows for all the different paths just like you'd add vectors together but since they're all pointing in random directions when we add them all up they simply cancel each other out and seemingly give us nothing thus for a classical object where the actions involved are much bigger than h-bar almost all the terms in the sum over paths add up to zero almost there's one crucial exception again the reason a generic path doesn't wind up contributing anything is that it's Neighbors which differ from it only vary slightly in shape have significantly different actions at least on the scale set by h-bar then their corresponding arrows point in random different directions and they tend to cancel out when we sum over many paths but suppose that there's some special trajectory for which the action is approximately constant for it and for any nearby path then the arrows for these trajectories would point in very nearly the same direction and those wouldn't cancel out trajectories that are near this special path would add up coherently and survive whereas everything else in the sum cancels out a special path like this where the action is approximately constant for any nearby trajectory is called a stationary path and those are the only contributions that survive in the limit when h-bar is very small compared to the action what that means is if you start from a stationary path X of T and you make a tiny variation of it by adding some little Wiggles say then the value of the action is the same for the new curve as it was for the old one at least to first order that might sound like something fancy but it's just like finding the stationary points of an ordinary function like a minimum say when you take a tiny step away the value of the function is constant to first order because the slope vanishes at that point finding the state stationary trajectory is totally analogous it's just a little harder since we're looking for a whole path now instead of a single point but at last what we've discovered is that in the classical limit the only trajectory that actually winds up contributing to the sum over paths is the path of stationary action and yes the stationary path is the classical trajectory I've proven that for you in a couple of past videos and I'll also show you how it works in the notes that I wrote to go along with this lesson you can get those for free at the link in the description the notes will go into more detail about a lot of what we've been covering here but the short of it is if you plug the definition of the action into this condition you'll find that a trajectory will be stationary if and only if it satisfies this equation M times the second derivative of x with respect to T equals minus du by DX and that's nothing but f equals m a because Remember the force on the particle and the potential energy are related by force equals minus the slope of the potential this is how the path integral predicts f equals m a it's not that the classical trajectory makes some huge contribution to the sum that dominates over all the other terms every term in the sum has the same magnitude one the classical path wins out because that's where the action is stationary and so the arrows near that trajectory all point at the same angle and they add together instead of getting canceled out but that was for a classical object like a baseball for something like an electron on the other hand the size of the action will be much smaller close to the scale of H bar so the angles s divided by h-bar won't be such huge numbers anymore and that means that the arrows for non-classical paths don't necessarily cancel out then in the quantum regime it's not true that only the single classical trajectory survives there can be a wide range of paths that contribute and f equals m a therefore isn't very relevant when it comes to understanding the behavior of quantum particles oh and like I promised to explain before when we defined the action if we had flipped the sign and used K plus u instead of K minus U like we might have at first guessed the equation for the stationary path would have come out the same except with the sine of U flipped but that would have said that M A equals minus F instead of f equals m a so we indeed need to take the difference K minus u in order to get the correct predictions for classical physics the fact that the trajectory of a classical particle makes the action stationary is called the principle of stationary action actually more often than not the classical trajectory comes out to be a minimum of the action and so it's more common to call this the principle of least action it's one of the most fundamental principles in classical physics much more fundamental than f equals Ma and now we've seen how it emerges from quantum mechanics the principle of least action is the starting point for the lagrangian formulation of classical mechanics which I mentioned earlier and if you want to discover why the lagrangian method is so much more powerful than what you learned in your first physics classes you can enroll in my course fundamentals of lagrangian mechanics the course will guide you step by step starting from the basics of f equals Ma and working all the way up through lagrangians and the principle of least action and all the important lessons this way of thinking about mechanics teaches us lagrangian mechanics is an essential subject for anyone who's serious about learning physics and you'll come away from the course with a much deeper understanding of classical mechanics and the preparation to take on more advanced subjects afterwards like the path integral approach to Quantum Mechanics or field Theory or a dozen other topics in physics that rely on the lagrangian method right now the first 100 students to enroll in the course using the discount code I painted the comments can save a hundred dollars off the regular price so sign up now if you want to take advantage of that and start learning a better way of thinking about classical mechanics feynman's path integral is really the quantum version of classical lagrangian mechanics it's actually a good story how Feynman came up with all this in the first place when he was a 20-something-year-old grad student at Princeton he talks about it in his Nobel Prize speech first of all he had a huge hint thanks to an earlier paper by Paul Dirac from 1932 where Dirac realized that the quantum mechanical amplitude somehow corresponds to this quantity e to the is over h-bar finally tells the story of how he was at a bar in Princeton when he ran into a visiting Professor who told him about this paper of directs in the next day they went to the library together to find the paper and then find and derived the basic idea of the path integral on a Blackboard right in front of the astonished visiting Professor I'll link that story down in the description if you want to read it now I've been pretty vague so far about how we're actually supposed to Define and compute this sum over the space of all possible paths and if you're mathematically minded you've probably been squirming a little in your chair wondering how the heck to make sense of this formula like I mentioned at the beginning the set of all these paths isn't a discrete list and so we're not actually talking about a standard sum here instead it's a kind of integral a path integral and in the next video I'm going to show you how we'd actually go about defining and evaluating this thing in a simple example so make sure you're subscribed if you want to see how to apply the path integral in an actual quantum mechanics problem in the meantime remember that you can get the notes at the link in the description and also check out my course on the grand Gene mechanics that special offer is only available for the first hundred students who sign up so don't wait if you want to enroll as always I want to thank all my supporters on patreon for helping to make this video possible and thank you so much for watching I'll see you back here soon for another physics lesson

Channel: Physics with Elliot

Views: 197,009

Keywords: physics, quantum mechanics, quantum physics, wave particle duality, wavefunction, wave function, feynman, schrodinger, classical waves, quantum particles, planck's constant, double slit experiment, diffraction, interference, physics lesson, path integral, sum over paths, Huygens principle, complex wave, stationary, action, principle of least action, f=ma, stationary phase, classical limit, lagrangian, lagrangian mechanics, Richard feynman

Id: Sp5SvdDh2u8

Length: 26min 29sec (1589 seconds)

Published: Sat Sep 30 2023