How Feynman did quantum mechanics (and you should too)

your experience with things you've seen before is inadequate it's incomplete the behavior of things on a very small scale is just different that was Richard Feynman 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 - in fact, when he was just a graduate student in his twenties - Feynman's first big discovery was a completely new way of thinking about

mechanics

.

quantum

mechanics

, which in the 80 years since has proven to be essential to our modern understanding of quantum physics, is called the integral formulation of quantum mechanics and once you understand it, Feynman's perspective will give you great insight into the counterintuitive way in which how things behave in the quantum world and at the same time, will teach you how the laws of classical physics, such as f equals m a, are derived from the most fundamental description of nature in quantum mechanics.

Quantum mechanics is about describing the behavior of really small particles like electrons and giving you an idea of ​​how different it is from classical physics, let's start by comparing and contrasting the classical and quantum versions of a very simple problem. Let's say we have a particle starting at some position x i at some initial time TI. In classical mechanics our job would be To determine where the particle will be at any later time, we add all the established forces that are equal to the mass of the particle multiplied by its acceleration and then solve this equation for position X as a function of time T if is a free particle so the solution to this equation is simply a straight line or if it is a baseball that we are throwing into 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 will find the particle at a later time.

The quantum mechanics of TF is fundamentally different, although if we are told that a quantum particle was found at position x i at the initial time t i, then all we can predict will be when we measure its position again. later is the probability that we find it here at position x f if you do the same experiment many times, sometimes you will find the particle there and sometimes you will find it somewhere else. This probabilistic nature of quantum mechanics is one of the strangest things about the physics of tiny objects is that a quantum particle no longer follows a single well-defined path to get from one point to another.

In fact, it's the incredible thing that Feynman discovered and that you will understand at the end. of this video is that instead of following a single path as in classical mechanics, a quantum particle considers all the conceivable paths and makes a kind of sum of all those possibilities, that sum of all the paths is what is called the integral on Feynman's way and it's nice. -mind-blowing to say the least if you're wondering how that could be consistent with the fact that a baseball definitely follows a single well-defined trajectory, stay tuned because understanding how the classical path in f equals m a emerges from quantum addition over all paths is, in my opinion, anyway, one of the most profound lessons in all of physics.

I think what I

should

do at the beginning is just give you a brief outline of how the path integral works and what the main formulas are so that you have a rough idea of ​​where we're going, don't worry if it doesn't make much sense yet, we'll move on. the rest of the video analyzing where it all comes from and what it means, what we are interested in here is the probability that a quantum particle that started at position x i at time TI is found at position XF at a later time TF, in terms In general, to find a probability in quantum mechanics we start by writing a complex number called the amplitude and then take the absolute value of the amplitude and square it to get the actual probability.

If you watched my last video, you get an idea of ​​how that happens by looking at a famous quantum experiment called the double slit experiment. I'll put a link to that if you haven't seen it yet and I'll also go over 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 I will write that since k f i and here is Feynman's path integral prescription for calculating K again classically, the particle would follow a single path between these points, but in quantum mechanics, Feynman discovered that we must consider all possible paths passing between. them, each of those possible. paths contributes a particular weight which is written as e to the power of I multiplied by 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 associated with each path called action I'll explain how that is defined later, but action is the central object in the most powerful approach to classical mechanics known as the Lagrangian formulation, which you may have heard of before.

In fact, I've created an entire course on Lagrangian mechanics. pin a link to that in the comments along with a very special offer code for the first 100 students who use it to register and now, to find the total amplitude of the particle to go from point I to point F, we add these contributions. of all possible paths, this is

feynman

's procedure for calculating quantum mechanical amplitude, of course the set of all these paths is not a finite list, so it is not really a discrete sum, it is a kind of integral called a path integral, so we most often write it using a notation like this and that's why it's called the path integral formulation of quantum mechanics, but anyway now we need to really understand what the heck all this means and the Intuitive idea behind the Feynman sum of our paths starts from double. slit experiment again, so let's start by quickly reviewing the key things we learned in the last video.

This was the setup: we took a solid wall and put two holes or indentations in it, then we threw different things at the wall and recorded what made it through. the other side with classical particles like pellets or billiard balls or anything very simple, the particles that passed through the left hole mainly hit the stop in the region behind the left hole and in the same way those that went through the right hole ended up in a On the right, the total distribution was the sum of those two curves because each particle passed through one hole or the other, which gave us a large lump in the center of the back.

Then we saw waves like light waves, which was more interesting because the two waves coming out of the holes can interfere with each other and produce what is called an interference pattern. On the back screen, the bright spots are where the waves meet. add up to form a larger wave which is called constructive interference and the dark points are where the destructive interference is and the waves cancel each other, the corresponding intensity curve looks like this with alternating peaks and valleys corresponding to the bright and dark points and we discuss how that occurs by writing the waves coming out of each hole in complex notation. of the form e to I5 where the phase Phi depends on the distance R from each hole to the point on the screen, the total outgoing wave is the sum of those contributions and from there we were able to calculate this intensity curve finally.

I experimented and shot quantum particles at the wall like electrons and the result was somewhat surprising. Instead of showing a large bulge around the center of the back as we did with classical particles, the quantum particles were distributed according to another interference pattern with many particles clustered together. at some points spaced apart with almost nothing, this is nothing like our experience of how things like birdshot or baseballs would behave. It means that an electron does not follow a single, well-defined path on its path through space that each electron explores in some way. both holes at once and interferes with itself last time we saw how to describe what is happening mathematically using Schrodinger's idea of ​​the wave function, this is how quantum mechanics was originally built by people like Schrodinger and Bourne and many others out there in the 1920s and 1940s.

Feynman came up with his path integral approach. The two are completely equivalent; You can derive either formulation from the other, but each provides valuable insight into the underlying physics, so we will now take the findings approach and see how they lead to the lessons from this simple experiment. Going back to the idea of ​​the path integral, the key lesson we can draw from the double slit experiment is again that a quantum particle does not follow a single path like a classical particle would, we have to consider paths that pass through each hole to power We understand the bump distribution we see on the backrest, but now let's take that idea a little further.

If we drill a third hole in the barrier, we'll need to include trajectories that pass through that hole as well and the same goes if we drill a fourth hole or a fifth and a sixth and so on while we're at it, let's go ahead and add another barrier solid in the middle and drill some holes. Now we have to consider all possible combinations so that the particle can 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 we take this idea to the logical extreme we completely fill the region with parallel barriers and through each of them we drill many small holes, then we need to take into account all the possible roots that the particle could take as it travels from one hole to another on its way; in fact, we can imagine drilling so many holes that the barriers themselves effectively disappear, like when I mentioned Hoygen's principle in the last video, we drill through all the barriers until we are effectively left with empty space again, so what this thought experiment suggests is to find the total amplitude for the particle to propagate from this initial point to some end point in the detector we need to add the individual amplitudes of each and every possible path that the particle could follow when traveling between those end points and not only the paths drawn in space but all the possible trajectories as a function of time and that is how What we learned from the double slit experiment leads us to the idea that we need to add all the trajectories to calculate the total quantum mechanical amplitude, but what? what weight are we supposed to add for each path?

Suppose it is very similar to what we did in our discussion about the waves that each path has. the path contributes to the sum with a particular complex phase and to the I Phi where Phi is a number we assign to each path that determines how it contributes to the total amplitude. This is the core idea of ​​quantum path addition and it's pretty awesome. Compared to our usual experience, we are used to finding a single classical trajectory for position maybe it's a straight line or a parabola or whatever, but in quantum mechanics, Feynman discovered that we need to count all the possible paths that the particle could follow between those points, for each path we write the phase e to the I Phi that it contributes. and then we add them all together to find the total amplitude.

Oddly enough, this prescription is at least fully democratic in the sense that each term of the sum is a complex number with the same magnitude that can be imagined e raised to I Phi as an arrow in the complex plane in other words we do a drawing with the real direction along the horizontal axis in the imaginary direction along the vertical axis then e al I Phi is an arrow of length pointing at an angle Phi different trajectories I contribute arrows pointing at different angles, but They are all the same length. The question is what angle Phi are we supposed to assign for each possible path.

Well, I already mentioned the answer at the beginning of the video for each trajectory, the complex phase. what contributes is given by e to the I times s over h-bar h-bar is the constant of quantum mechanics called Planck's constant its value in SI units is given approximately by 10 at least 34 Joule seconds s meanwhile is the action which is a particular number that we can calculate for any given trajectory, you may have come across before because it is something that already plays a central role in classical mechanics, but this is how it is defined: we take the kinetic energy of the particle at each moment and we subtract 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 calculate for any given trajectory and that is its action. The quantity k minus U that we are integrating here gets its own special name: In this way it is called Lagrangian so that the action is defined by integrating the Lagrangian over time and that is the central object in what is called the Lagrangian formulation of mechanics classical and yes, that really is a minus sign in the middle, more on that in a minute, now depending on If you've learned a little about Lagrangian mechanics before seeing the action and the emergence of the Lagrangian here may be ringing huge bells in your head or these formulas may seem completely off, so let me try to motivate you where is this weight e to the i. about the H bar comes in handy, first of all let's think about the units we have to play with here, we certainly expect the Planck constant H bar to appear in our weight.Factorice, again, is the fundamental constant of quantum mechanics which had units of energy in joules times time in seconds, but Phi, here is an angle, remember it is measured in radians and has no dimensions, so we will have to match the bar H. we invented something else with those same units of energy multiplied by time to cancel them and the simplest thing we could write is a relation s on the bar H in the action s, in fact, it has those units that we are looking for, K and U are energies. and they are multiplied by time when we integrate over t and the units of s cancel the units of h-bar and we are left with a dimensionless number for the angle Phi as we needed for the units to at least work correctly, otherwise it wouldn't even make sense to write this quantity e in bar i over H.

You might be wondering why the heck we are taking the difference between kinetic and potential energy. Wouldn't it seem more natural to write the total energy as? we're much more used to it, that's certainly what I would have tried first if I had been working on this problem about 100 years ago, but that's wrong, it's definitely a minus sign that appears in this formula for the action and we'll do it. see why having talked about the second key piece of motivation where this weight e a i s on the H bar comes from, it ensures that the single classical trajectory emerges when we move away from the study of small particles of quantum mechanics to larger everyday objects.

It's not entirely obvious how that works at first glance in Feynman's formula, if this tells us to add up all the paths the partial article could follow, each with the same magnitude and just different phases, how could that be consistent with what we observe in our daily lives where a baseball definitely follows a unique parabolic trajectory, after all, quantum mechanics is the most fundamental theory in our daily lives. The laws of classical mechanics must arise from it in the appropriate limit. The answer to this question is one of the most profound ideas. The path integral. reveals about the laws of physics, it will show us how f equals m a follows from this most fundamental description of quantum mechanics, generally speaking, what happens is that for the motion of a classical object like a baseball, almost all terms in the path sum cancel each other. take out and add nothing except one and that's the classic way and here's why we're going to draw the complex plane and here again on the left is a graph of position X versus time T each term in the sum corresponds to an arrow in The complex plane has length 1 and points at an angle set by S on the bar H, so we choose any path that connects the start point to the end point, calculate the action s for that path, divide it by the bar H, and then we draw the corresponding arrow at that angle, if we choose a different trajectory we will get some other value for the action and that will give us another arrow at some other angle and what we have to do is add all these arrows together, here's the thing though bar H is really really very small again in SI units its value is of the order of 10 to the power of minus 34. that's a 1 with 33 leading zeros and then the decimal point in comparison, a typical action for a baseball will be something like a joule per second, perhaps.

Give or take a few orders of magnitude in either direction, but it is much larger than the value of the H bar, so the angle s divided by the h bar will be a huge number for a typical baseball trajectory of the order of 10 to the 34th power. The radians starting from Phi are equal to zero, it's like we move this arrow so hard that it spins a billion times until it lands in some random direction, but now let's choose a slightly different trajectory and consider what That contributes to the song. It's very similar. path we started at, so its action will only be slightly different from the first, maybe the first path had a action of a second joule and this new one has 1.01 say, so the change in action value between them is 0.01 joule seconds, it doesn't matter what the precise numbers are because again, when we divide by the incredibly small value of the H bar, even that small change in action on the classical scale will produce a massive change in angle, in this case something like 10 at 32 radians, although these two trajectories were only slightly different, their corresponding arrows point in different random directions in the complex plane and now, as we include more and more curves, each of them will also give us a arrow in some other random direction. we will get an incredibly dense series of arrows pointing in all directions around the unit circle according to Feynman's formula.

What we're supposed to do is add all these arrows for all the different paths just like you would add vectors, but since they all point in random directions when we add them all they just cancel each other out and apparently give us nothing, so for a classic object where the actions involved are much larger than the h bar, almost all the terms in the path sum almost sum to zero there is one crucial exception again, the reason a generic path ends up contributing nothing is that its neighbors that differ from it only vary slightly in shape have significantly different actions at least on the scale established by the bar h so their corresponding arrows point in different random directions and tend to cancel when we add many paths, but suppose there is some special path for which the action is approximately constant for it and for any nearby path, then the arrows for these trajectories would point in almost the same direction and those would not cancel the trajectories that are close to this special path would add up coherently and survive, while everything the rest in the sum cancels a special path like this where the action is approximately constant for any nearby path is called a stationary path and those are the only contributions that survive in the limit when the bar h is very small compared to the action, What that means is that if you start from a stationary path order, which may seem somewhat sophisticated, but it is like finding the stationary points of an ordinary function at the very least, let's say.

When you move away one small step, the value of the function is a first-order constant because the slope disappears at that point. Finding the state of the stationary trajectory is totally analogous, it's just a little more difficult since we are now looking for a complete path instead of a single point, but ultimately what we have discovered is that in the classical limit the only trajectory that really ends up contributing to the sum of paths is the stationary action path and yes, the stationary path is the classic trajectory. I've demonstrated it in a couple of previous videos and I'll also show you how it works in the notes I wrote to accompany this lesson.

You can get them 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 in summary, if you plug the definition of action into this condition, you will find that a trajectory will be stationary if and only if it satisfies this equation M multiplied by the second derivative of this is how the path integral predicts f is equal to m a is it is not that the classical path makes a large contribution to the sum that dominates over all other terms, each term of the sum has the same magnitude and the classical path wins because there is where the action is stationary and that's why all the arrows near that path point at the same angle and add instead of canceling, but that was for a classical object like a baseball, for something like an electron, for another On the other hand, the action size will be much smaller near the H-bar scale, so the angles s divided by h-bar will no longer be such large numbers and that means that arrows for non-classical paths will not necessarily cancel each other, then in the quantum regime it is not true that only the single classical trajectory survives. a wide range of contributing paths and f is equal to m a, therefore it is not very relevant when it comes to understanding the behavior of quantum particles, and as I promised to explain earlier when we defined the action if we had reversed the sign and used K plus u instead of K minus U, as we might have guessed at the beginning, the equation for the stationary path would have turned out the same except with the sine of U reversed, but that would have said that M A equals minus F instead of f equals m a , so indeed it is necessary to take the difference K minus u to obtain 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. In fact, most of the time the classical trajectory turns out to be a minimum of action and that is why it is more common to call this the principle of least action is one of the most fundamental principles in classical physics much more fundamental that f equals Ma and now we have seen how the principle of Least Action emerges from quantum mechanics is the starting point for the Lagrangian formulation of classical mechanics that I mentioned earlier and if you want to discover why the Lagrangian method is much more powerful than what you learned In your first physics classes, you can enroll in my Lagrangian fundamentals course.

Mechanics, the course will guide you step by step, starting from the basics of f equals Ma and moving up to Lagrangians and the principle of least action and all the important lessons that this way of thinking about mechanics teaches us. Lagrangian mechanics is an essential topic. for anyone who is serious about learning physics and will leave the course with a much deeper understanding of classical mechanics and the preparation to take on more advanced topics later, such as the comprehensive approach to quantum mechanics or field theory or a dozen other topics. in physics that are based on the Lagrangian method right now, the first 100 students who sign up for the course using the discount code I posted in the comments can save a hundred dollars off the regular price, so sign up now if you want to take advantage of that and begin. learn a better way to think about classical mechanics the

feynman

path integral is actually the quantum version of classical lagrangian mechanics.

It's actually a good story how Feynman came up with all this when he was a twenty-something grad student. at Princeton he talks about it in his Nobel Prize speech. First of all, he had a big clue 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 bar h. finally tells the story of how he was in a bar in Princeton when he ran into a visiting professor who told him about this Direct article, the next day they went to the library together to look up the article and then find and derive the basic idea. of the path integral on a blackboard right in front of the astonished visiting professor.

I'll link that story in the description if you want to read it now. So far I've been pretty vague about how we're supposed to define and calculate 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. , as I mentioned at the beginning, the set of all these paths. It's not a discrete list, so we're not really talking about a standard sum, but it's kind of an integral, a path integral, and in the next video you'll seeI will show how we would define and evaluate.

This is in a simple example, so make sure you are subscribed if you want to see how to apply the path integral in a real 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 Grand Mechanics of Genetics, that special offer is only available to the first hundred students who sign up, so don't wait if you want to sign up as usual . I want to thank all my followers on Patreon for helping make this video possible and thank you so much for watching.

See you here soon for another physics lesson.