How Feynman did quantum mechanics (and you should too)

Summary

The video explains Richard Feynman's path integral formulation of quantum mechanics. It starts by contrasting the deterministic trajectories of classical physics with the probabilistic outcomes of quantum particles, which must consider all possible paths between two points. The path integral sums contributions from each conceivable trajectory, weighted by a phase factor involving the action. Remarkably, this framework predicts both quantum effects at small scales and classical physics at large scales. The video ties this approach to the principle of least action and Lagrangian mechanics, even recounting the origin story of Feynman conceiving the idea in a Princeton bar.

Takeaways

• 😲 Quantum mechanics describes probability of where a particle will be unlike classical mechanics which predicts exact position
• 😯 Particles have wave properties enabling interference unlike classical particles
• 🤯 Feynman discovered particles take all possible paths with phases unlike single classical trajectory
• 😵‍💫 Path integral sums over phases of all paths between endpoints to get quantum amplitude
• 😀 Classical trajectory emerges as special "stationary" path where action doesn't change
• 🧐 Lagrangian mechanics connected to path integral through action depended on kinetic & potential energy
• 👀 Measurement collapses quantum amplitude's squared absolute value to probability
• 🤔 Tiny Planck's constant hbar makes phases vary rapidly except near classical path
• 🤑 Special offer for Lagrangian mechanics course showing classical physics emerges from quantum
• ✨ Inspired by Dirac paper, Feynman formulated path integral talking to visiting professor in a bar

Q & A

What is the path integral formulation of quantum mechanics discovered by Feynman?

The path integral formulation is a way of computing quantum mechanical amplitudes by summing over the contributions from all possible paths a particle could take. Each path contributes a complex phase e^(iS/hbar) where S is the action of that path.

How does the path integral account for the wave-particle duality of quantum particles?

It accounts for wave-particle duality by summing over all possible trajectories, treating the particle as if it is a wave propagating along all paths. This allows it to properly handle phenomena like interference and diffraction.

Why do we need to consider all possible paths in the path integral?

We need to consider all paths because in quantum mechanics, particles do not follow single well-defined trajectories like classical particles. The probability amplitudes depend on contributions from all possible ways the particle could propagate.

What is the action S that appears in the path integral formula?

The action S is defined as the time integral of the difference between a particle's kinetic and potential energies. It plays a central role in classical mechanics and the Lagrangian formulation.

How does classical mechanics emerge from the path integral?

In the classical limit, almost all paths cancel out due to their rapidly varying phases except the path of stationary action. This leads to the classical equations of motion like F=ma.

What is the principle of stationary action?

It is the principle that the classical path makes the action stationary (often a minimum). Small variations in the path do not change the action to first order along this trajectory.

What is the Lagrangian and how is it related to the action?

The Lagrangian L = T - U is the difference between a particle's kinetic and potential energies. Integrating the Lagrangian in time gives the action. It is central to the Lagrangian formulation of mechanics.

How did Feynman originally come up with the idea of the path integral?

The story goes that Feynman learned about Dirac's paper relating quantum mechanics to e^(iS/hbar) from a visiting professor. He then derived a prototype of the path integral formula on a blackboard.

What is the physical meaning behind summing over paths with phases?

It reflects the wave-like propagation of quantum particles. The phases capture the interference between different possible ways for the particle to propagate, like waves traveling along different paths.

How do we actually evaluate the path integral computationally?

We have to find clever ways to approximate the integral, often using numerical techniques. In simple cases there are shortcuts to evaluating it analytically. The next video discusses techniques.

Outlines

00:00

😲 Overview of path integrals and key formulas

This paragraph provides a high-level overview and introduction to the path integral formulation of quantum mechanics. It states the key formulas for computing quantum mechanical amplitudes using path integrals, including the action S and the phase factors involving Planck's constant hbar. The goal is to provide a sketch of the main ideas before diving into the details.

05:02

😃 Double slit review and motivation for sum over paths

This paragraph reviews key takeaways from the double slit experiment with quantum particles. It shows how this leads to the idea that quantum particles don't follow single trajectories, but rather we need to sum over contributions from all possible paths to compute amplitudes. This democratic sum over paths approach motivates Feynman's path integral formulation.

10:04

😮 Defining the path integral sum

This paragraph further develops the idea of summing over all possible paths, taking it to a logical extreme by considering barriers with many small holes. In the limit, this suggests summing amplitudes for all conceivable paths in space and time. A phase factor involving the action S is proposed as the weight for each path.

15:05

❓ Understanding units and emergence of classical physics

This paragraph provides motivation for the phase weight by considering dimensions and units. It also crucially shows how classical physics and the principle of least action emerge from the path integral in the limit as hbar becomes very small compared to action of macroscopic paths.

20:07

😊 Lagrangian mechanics and physics fundamentals

This paragraph discusses relation of path integrals to Lagrangian mechanics and the principle of least action. It advertises a course on Lagrangian mechanics for developing a deeper understanding of physics and classical mechanics.

25:08

😅 Feynman's insight and Dirac's paper

This closing paragraph tells the story of how Feynman came up with the idea for path integrals after reading a paper by Dirac. It mentions that mathematical details of defining the path integral will be covered in the next video.

Keywords

💡Quantum mechanics

Quantum mechanics is the physics of very small things like electrons. It says we can only predict the probability of where we'll find a particle, not its exact position over time. This makes it very different from classical physics.

💡Path integral

The path integral is Richard Feynman's approach to quantum mechanics. It says to compute the probability for a particle getting from one point to another, we need to consider all possible paths it could take.

💡Amplitude

The amplitude is a complex number that gives the 'weight' for each path the particle might take in the path integral. We square its magnitude to get probabilities.

💡Action

The action S is the number we assign to each path in the path integral, given by the time integral of the Lagrangian. It appears in the phase for each path's amplitude.

💡Classical limit

The classical limit refers to objects like baseballs rather than quantum particles. In this limit, particles follow single paths because the action differences between paths become very large.

💡Least action

The least action principle says particles follow paths that make the action stationary. Feynman showed this emerges from the path integral - those paths have arrows pointing the same way so they don't cancel out.

💡Lagrangian

The Lagrangian L = T - U is key to classical and quantum mechanics. Integrating it over time gives the action. In the path integral L appears inside the integral for each path's action S.

💡Interference

Quantum particles passing through double slits can interfere, like waves. The interference pattern led Feynman to realize particles take many paths at once.

💡Phase

The phase factor e^(iS/hbar) for each path contains the action S in the exponent. Varying the action rotates the arrow for that path's amplitude in the complex plane.

💡Superposition

The superposition principle says quantum states combine by simple addition of their amplitudes. This makes interference possible and underlies the path integral formalism.

Highlights

Feynman's first great discovery was an entirely new way of thinking about quantum mechanics called The Path integral formulation

The Path integral tells us a Quantum particle doesn't follow a single trajectory, it considers all conceivable paths and sums over those possibilities

The Path integral formula computes the quantum amplitude by summing contributions from all possible paths, each weighted by e^(iS/ħ) where S is the action

The double slit experiment shows that a quantum particle somehow probes both slits and interferes with itself, so we need to sum over paths

To compute the quantum amplitude, we consider every trajectory the particle could follow and sum their contributions e^(iS/ħ)

For each path, S is the action - the time integral of the Lagrangian (kinetic minus potential energy)

The classical trajectory emerges as the stationary path where S is approximately constant - other paths cancel due to the tiny ħ

The stationary path principle leads to f=ma, showing how classical physics derives from the quantum path integral

For quantum particles, many paths contribute since actions are ~ħ, so classical trajectories aren't relevant

Feynman tells the story of how he derived the path integral in a bar after learning of Dirac's paper relating e^(iS/ħ) to QM

The path integral is the quantum version of classical Lagrangian mechanics, which is more fundamental than f=ma

Understanding Lagrangian mechanics from its basics to advanced applications like path integrals is essential for physics

The course covers Lagrangian mechanics starting from f=ma and working up through actions, path integrals, and more

Supporters on Patreon help make these physics videos possible - thanks to them and to you for watching!

The next video will show how to actually define and evaluate path integrals for real quantum mechanics problems

Transcripts

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