Deriving The Feynman Path Integral Part 1

what's going on smart people bring you another episode no this is not tensor calculus I put up a poll I said what heavy physics video do you guys want to see next you voted and said the Fineman path integrals so that's what we're going to be going over these next couple of videos and they're gonna be just as much for me as they are for you because everyone knows the best way to make sure you understand something is to try to explain it to other people and in going through this derivation I realized towards the end of it that there are a couple things that I don't quite know how I want to explain so just making this into two videos gives me the time to make sure that I can explain it in a good way rather than throwing it all into this video on BB saying something that's not entirely correct and there are plenty of spots in this derivation where you could do that but before we get into the derivation let's talk a little bit about what it is where it comes from so in quantum field theory there's two main approaches there's the canonical quantization or second quantization approach you know in regular quantum mechanics your observables become quantum operators in second quantization the fields also become operators so you have no our operators array with the path integral which is the second approach of quantum field theory it kind of goes in the opposite directions you effectively eliminate operators from the equation altogether by inserting completeness a bunch of times such that all of your operators act on the eigenstates and then you're just left with numbers at the end of the day which is pretty nice it's very formal and very powerful people like Freeman Dyson made sure that it ended up that way but you can still use it to solve things like quantum mechanics and that's actually how we'll be deriving it today but to actually solve a problem in quantum mechanics using the path integral will test every bit of mathematical capabilities that you have it just things that the harmonic oscillator are just a huge challenge using the path integral but you can do it which is important and you can do it to solve the really hard problems as well one of the things that you calculate using the path integral is called the propagator which is how we're going to start off so the propagator it might be a new term to you but it's nothing but the expectation value of the time evolution operator so I'm gonna go ahead write it as okay this is a dead marker so the propagator we're gonna call K as a function of X prime X it's time and it's defined to be the matrix element of the time evolution operator each - I on h-bar which is normally one but whatever times the Hamiltonian and then some time the reason I'm making a capital T is because we're gonna call time a bunch of different things later so we'll just keep it capital for now so this is the propagator gives you an amplitude it's a probability amplitude of a particle propagating from point A to part to point B one of the tools that we're going to use throughout this entire video is the completeness relation the completeness relationship which I hope you all know says that if we take the inner product of our set of orthonormal basis vectors we should get identity and those basis vectors could be discrete or continuous so if we have already little some integral little trick that we or the notation one of my professors used of our basis vector an outer product is equal to identity okay so for position eigen States it would be continuous so we use an integral if we had energy eigenstates that would be discrete we'd use a sum so that's just what that means uh now what is this propagator do so let's take a look at the wave function at some point in space some point in time and how we could define it so sine of X prime T well that's going to be equal to and the X prime basis the inner product time evolved version each the - I h-bar HT of our wave function at T equals zero so we time evolve our wave function and that we represented in the X prime basis that's what this means now what I want to do is I want to insert completeness inserting completeness what that lets you do is it lets you represent something in a different basis you always use it when you want to change a basis so I'm going to go ahead and insert completeness here so that's going to be an integral since I'm doing the coordinate basis and I'm going to call that variable X it's gonna be so good X prime e to the minus ion H bar Antonian time X X sy zero and this is exactly the propagator so this is equal to the integral over DX K of X prime X T sine of X comma K equals zero so what the propagator tells you is that if you know what it is and you have some initial value of the wave function you can find out what the wave function is at some other points space and time so knowing the propagator is equivalent to having a complete solution of the Schrodinger equation it might be a little bit actually more powerful because it's kind of like the greens function to the Schrodinger equation and if you have a greens function you've solved a whole family of problems I've made a couple videos on greens functions if you're curious I'll leave a link in the description okay so now what I want to do is I want to play around with completeness just to make sure everyone's on the same page is everything I'm gonna take a look at the propagator one more time and I'm going to insert energy eigenstates so a complete set of energy eigenstates so we're going to erase this and I'm going to insert completeness here and here so I need two summations so this is equal to let's say sum over n prime and n we're gonna have an X prime ket n prime e to the minus I on H bar H T and X so now I've sandwiched the time evolution operator in between its eigenstates so I can have this guy acting on this eigenstate which is just going to give me the exponential of the energies so this is going to be equal to the sum over n Prime and end of e to the minus PI on H bar P sub n of T and I have an X prime n Prime and then I have an inner product here since they pulled this guy out because this is just a number now of n prime and n these are orthonormal so it's going to be a delta n prime n which is just telling me that I can sum over n prime and that all the end Prime's just become ents that's just n this n prime becomes a 10 and then we also have this guy and X which is equal to sum over n e to the minus PI on H bar a piece of end C this is just going to be the complex conjugate transpose of the wave function of the nth wave function in the coordinate basis so we've got a side dagger stuff and X Prime and this is just going to be the regular wave functions I sub n of X cool so this is sometimes called actually is called the spectral representation of the propagator and in quantum field theory in quantum mechanics you can use the path integral to find the propagator that's one of the tools it's one of the few tools you can actually use the path integral for is to find the propagator and in solving an actual problem once you go through all the disgusting math you can look at the spectral representation and sort of compare terms and say ah so comparing term by term I can see that this must be the energy eigenvalues and things like that next I am going to do nothing and by do nothing I mean I'm going to multiply and divide by n just some arbitrary N and eventually we're going to take n to infinity okay so here whose propagator and the right this is equal to X prime parentheses e to the minus I on H bar H times T over N and I'm going to raise this to the nth power then we have this X so nothing's changed like I said eventually we're gonna have and go to infinity and I'll explain that when the time comes so I'm just multiplying either minus IH bar of the law by itself n times so I'm going to hit go ahead and explicitly write that out and I'm going to insert a complete set of basis vectors n minus 1 times so just just bear with me so this is equal to some integral a bunch of integrals dot d X sub n minus 1 DX 1 and then I'm going to have X prime e to the minus I on H bar H t over N X sub n minus 1 X sub n minus 1 and then the same thing each of the - I on each bar its H 2 u / n X sub n minus 2 at that dot and then we're going to have a an X 1 e to the minus I on H bar H t / n X ok so I inserted at n minus 1 times because we didn't insert completeness here so we just have X Prime no completeness and then all the way over and why the hell am i doing this so this is why I think a lot of people say and I say that the path integral is one of the most intuitive ways of looking at quantum mechanics because what we're doing here we have a propagator that tells us the probability of a particle propagating from point A to point B over some time I think I forgot to mention the time part earlier so we have some point B X Prime we have some point a X and what we're doing when we inserted completeness here is we're dividing up the space between them into a bunch of little slices so we have all these little slices here and - one of them so here's you know X 1 X 2 dot X n - 1 and then we're integrating along that that uh that's slice so why is it this is intuitive is because this is almost like a little generalization of the double slit experiment except for instead of having with one slice with two slits in it and we're adding up the probability amplitude we have well at some point we're gonna have infinitely many slices and infinitely many slits and it's going to say the probability of going from here to here to here to here to here so that's why it's kind of like a generalization of the double slit at a distance it makes things so much more intuitive so in the next video what we're gonna do is we're gonna show how to split up this Hamiltonian this exponentiated Hamiltonian in terms of momentum and the potential or the kinetic term of the potential and then we're going to insert the momentum eigenstates a bunch of times and what that's going to let us do is eliminate operators completely from our path integral formalism we're going to do some more weird stuff we're going to go to imaginary time and then at the end once we massage the theory a little bit we're going to end up with the full-fledged path integral so I hope you guys enjoyed the video let me know in the comments section if you did and I'll see you in part two

Channel: Andrew Dotson

Views: 39,769

Rating: 4.952045 out of 5

Keywords: andrew, dotson, physics, major, degree, nmsu, odu, gradschool, gradstudent, phd, path, integral, quantum, field, theory, qft, feynman, propagator, greens function, derivation, derive, quantization

Id: cMYhdTXpZ4c

Length: 12min 21sec (741 seconds)

Published: Tue Jun 18 2019