How Feynman did quantum mechanics (and you should too)
Summary
TLDRThe 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
😲 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.
😃 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.
😮 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.
❓ 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.
😊 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.
😅 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.
Mindmap
Keywords
💡Quantum mechanics
💡Path integral
💡Amplitude
💡Action
💡Classical limit
💡Least action
💡Lagrangian
💡Interference
💡Phase
💡Superposition
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
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lagrangian mechanics the course will
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basics of f equals Ma and working all
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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
تصفح المزيد من مقاطع الفيديو ذات الصلة
Brian Cox explains quantum mechanics in 60 seconds - BBC News
Quantum Computers, Explained With Quantum Physics
What is Quantum Tunneling, Exactly?
What can Schrödinger's cat teach us about quantum mechanics? - Josh Samani
Quantum Entanglement: Explained in REALLY SIMPLE Words
Biography of Werner Heisenberg
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