Okay, welcome back everybody. Today we're going to actually use the
Feynman path integral to obtain a few very interesting results and physical
phenomena. And in the first video today, we're going
to actually derive a very unexpected surprising result.
Namely, we're going to see how the Newton equation second Newton law of classical
physics appears in the so-called classical limit of the Feynman path integral.
So more precisely what I'm going to, what I'm going to do, I'm going to start with
the with an expression that we discussed in the previous lecture.
So which tells of that the probability of a quantum particle to go from an initial
point R sub i to find a point R sub f can be written as the sum absolute value of a,
a sum over all classical trajectories. Which sort of symbolically represents what
we call the path integral of this individual quantum mechanical transition
amplitudes. And each of these amplitudes is written as
the exponential of this symmetric constant i, times the classical action, divided by
the Planck Constant. And surprisingly, what we're going to see
is that the familiar equations of motion of the classical physics the, Newton's
second law, appears as a result of taking the classical limit in this expression.
And I'm going to formulate precisely what I mean by the classical limit, but at this
stage let me just tell you that the classical limit will imply suppressing the
essence of quantum mechanics. The interference phenomena, which is,
which are controlled by this, this Planck constant H bar.
Now, just as a side comment here, sort of the historical comment, let me mention
something about this, equation. So, of course, this is I expect.
All of you, or the majority of you to know this equation is something you heard of
sometimes in high school which is the second Newton law that mass times
acceleration of a classical particle is equal to the sum of all the forces acting
on the particle. So, apparently if you read actually the,
the original scientific work by Newton. The first version, the first edition of
this Pincipia was published back in 1687, a long time ago.
So we're not actually see this equation in this modern form.
The closest formulation to what we currently call Newton's second law of
motion. Appeared in his original work in the
following form. The change in motion is proportional to
the motive forced impressed and takes place along the straight line in which
that force is impressed. It probably requires some imagination to
connect this statement toward to well to this equation.
But in any case obviously Newton not only understood, the basic principles of
classical mechanics, he was the one who created it.
So there's no question about that. So this comment, I just wanted to make to
show you that sometimes the original discoveries evolve very strongly from
their original form. And become even though we still give
credit to people who were the fi rst to put them together the final form may be
quite different from what they were envisioning.
And this is a perfect example of that. Now, going back to the main subject of,
this video. So now I'm going to actually, show you how
mathematically this, old and well known Newton, second Newton Law appears from
this modern, more sophisticated quantum theory, seemingly unrelated theory.
In order for me to show you how it happens, I need to present a mathematical
trick or method ih, called Laplace's method or saddle point approximation which
allow us to calculate certain integrals that otherwise cannot be calculated really
quickly. And this method as you all see sort of
propagates to more complicated theories including our ability to calculate certain
path integrals approximately. Now we have seen already the Gaussian
integral, which is an integral of e to the power minus x squared and between minus
infinity and to plus infinity. And we know the result is equal to the
square root of, of pi. So of course what this integral is, Is the
area enclosed by this curve which is nothing but the plot of this Gaussian
function, e minus x squared as a function of x.
Now, of course, this function is special, well, not very, nothing particularly
special about this function but it does appear in many fields of math and physics
and one property of this function is that it, It is very, very fast when the
argument x becomes large, either very negative or very positive.
Now in general, however, if we want to calculate the integral of let's see, from
minus infinity to plus infinity or some other limits of a function e to minus f of
x where f of x is a relatively arbitrary function.
There is no close mathematical expression typically for a Gaussian integral.
But it turns out that if there's a small parameter in the exponential, if this
function actually involves, so not just f of x but f of x divided by some epsilon
and this epsilon is very small Then, 1, in many cases, can simplify things.
And I will explain the logic, behind this simplification on a particular example of
this function of effects. Which is x squared plus, 1 over x squared.
And so I want, I want to integrate the exponential of minus this function, from
my minus infinity to plus infinity. So there's no closed analytical expression
for the integral of this word but it turns out that if indeed we have this small
parameter in the denominator of the exponential things can actually be
simplified. In order to see how it happens, that we
pull up this function, f of x and Soo if we do so we say that there Minimum but of
course due to the minus sign in the exponential.
The smaller the f of x, the larger the exponential itself.
And vice versa. So if f of x on the other hand becomes
very large and positive, e to the power minus f of x becomes negligible.
So if we now plot the function that we actually want to integrate.
It's going to look approximately like this so as my artistic expression of what it's
good look like the point here is that it will have a maximum around where around f
of x is minimal and is going to have the stales which becomes smaller and smaller
as f of x becomes larger. And so what we want of course
geometrically, we want to collect the area enclosed by this curve.
And so some area will come from the original, this maximum, and some area
become, will come from these tails. So, but as epsilon this parameter here,
becomes smaller and smaller, let's say, when it's 0.1 or something like that.
So then it implies that I will exponentiate either by one or 0.1, which
means either power of 10. Of whatever these tails r.
And so these tails will become less and less relevant on the background of this
peak. So the, for instance, for smaller epsilon,
so let's say my f of x were smaller, epsilon is going to look like this.
So, and, and in some sense, it's going to become closer and closer to the Gaussian
form. So the reason for that because if I expand
my function f of x in Taylor series in the vicinity of this point f of x equals 1.
So what I'm going to get so its the first one which is just the value of my function
at h f at f's equal 1 plus f prime of f equals 1 x minus 1 plus f 2 primes,
divided by 2 x minus 1 squared. But since this is a minimum of my
function, so the derivative, the first derivative of my function here of n, I
should say becomes equal to 0. And so the only terms we sh-, sort of in
the leading order that matter i-, are this, this guy and the quadratiture.
So essentially which means the integral that I'm dealing with, Gaussian.
And, as I mentioned, we know how to calculate Gaussian integral.
So without going through the algebra, let me just, sort of present, the results.
So here is, I first present sort of a general result for a, almost arbitrary
function, f or x. And the first factor here accounts
essentially from the just simply calculating the value of this exponential
in the minimum point of f. And the second term is the result of
integrating the Gaussian part much like this Gaussian integral.
So for, for the particular function we consider well we don't, we are not going
to need this result at all. So it is just, it was just example but
nevertheless, here is the, here is the answer.
Let me also mention another fact, which I'm not going to prove it but it's
important for the following, that even if we have a slightly different integral when
we have an exponential of i, the measuring constant i over some.
Small parameter and sum f of x. So this integral often times can too be
simplified using this, very similar saddle-point approximation which we just
discussed. And the reason for that is because e to
power i f of x is, can be written as a bunch of cosines and sines.
And the if epsilon becomes very small they become stronger oscillating functions.
Which on the average give us 0 if epsilon is small, because we have a, a plus, a
positive part of the cosine and negative part of the cosine and then in some sense,
they balance each other out and give on the average, 0.
And there is a similar argument that you can actually write the expression of this
integral in a very similar form and focus only on the minimum of this function.
Now, let me go back to quantum physics and we're going to see how the previous
mathematical discussion is going to become relevant.
Now, remember, so I've, my motivation in the very beginning, I said that I want to
take the classical limit of my quantum mechanical theory.
So what does it mean to take a classical limit?
So let us recall that the most interesting quantum phenomenon, the way how quantum
mechanics was actually discovered Implied interference between different waves, sort
of describing wave functions is describing a particle.
So if we want to kill in some sense the quantum mechanical effects, we want to
suppress the interference phenomenon, or we want to make the wavelength as small as
possible. So let's say if we have a particle with a
momentum p. So the corresponding wavelength is going
to be per, the argument is going to be h bar over 2pi p.
So if you want to set a lambda to 0, that is completely suppress the interference
effect effects of the particlelization wave lengths, we want to set in some sense
h to 0. So the Planck constant, which controls the
quantum effects, which is sort of the essence of quantum mechanics, so it should
be set to zero in order to take the classical limit.
So now, recall the expression for the Feynman path integral.
So we see that in the quasi-classical limit, in the classical limit, we have an
H four always taken to zero. The function that we are actually or
function I'll better say, that we're integrating, becomes in some sense very
similar to the types of functions that we saw in the previous slide.
With the h bar here at applying constant, being the coolant in a sense with the h.
To the epsilon, parameter epsilon that control the applicability of the settle
point approximation. And also in the full analogy with this
previous slide with the settle point approximation we can see that a, as h
becomes smaller and smaller so we can only focus on the trajectories in the vicinity
of the trajectory for which the action is minimal.
So basically the limit of h going to zero therefore reproduces the principle of
least action, which is another sort of cornerstone of classical physics.
Now I should comment here that of course h bar the Planck constant, is a fundamental
physics constant. We cannot take it to zero, we cannot take
the limit physically of h bar going to zero.
What it actually means, when I'm saying that h bar goes to zero.
And this also, you're going to see it in textbooks on quasi classical
approximation. It means that, we have, essentially, two
types of length scales in our problem. 1 type of length scale is, the wavelengths
of our quantum particles. And the other type of length scales is
the, typical length scales in our system. And if with the wave length of our quantum
particles are much more than everything else, this effectively means this limit
and this effectively means the applicability of the quasi classical
approximation. So for those of you who are familiar with
the Lagrangian classical mechanics it should be clear now that a principle of
least action already demands that we essentially are going to reproduce
Newton's equations. As was advertised in the beginning of the
lecture. So you can stop listening to the lecture
right here, and move on to the next part. But for the sake of completeness.
Let me, nevertheless present or remind you how these equations would come about.
So again, how the question we're now asking, okay we have found that while if
we take the Planck constant to zero, or if we consider the wavelengths which are much
more that everything else. We essentially extract we must extract the
minimum of the classical action from the path integral and this minimum would occur
on a particular trajectory that particle would follow.
So how to find this special trajectory? So we can sort of, motivate the standard
what is know as a variational analysis by this again simple analogy using the usual
functions. So, suppose we have an arbitrary function
f of x, which has a minimum or maximum principal too, somewhere, but let's assume
it's a minimum, and we want to determine the point where this minimum actually
occurs. And for this I can again use the Taylor
expansion for the wavefunction, I'm sorry, for this function f in the vicinity of
this point x naught, which I am trying to find.
And so it's going to have the value of the function at x naught, the first
derivative, the second derivitive, etc... And the minimum of this function, see if I
have a functin, so this minumum. Is going to occur where this derivative or
the certain the tangent to this plot f of x as a function of x is going to be
horizontal to the x direction or the first derivative vanishes.
Now, an analogy to this if we have now a function or in particular, our action f of
x. So to determine a particular trajectory,
which we will identify the classical trajectory.
So we have to demand that if we calculate this action in the vicinity of this
classical trajectory in which the action is miminal.
The first sort of variation, the analogy well the anal-, analogous part of this
first derivative is going to vanish. And this symbol delta s equals zero.
So this equation is sort of mathematical expression for this principle of the list
action. And in some sense, all of classical
physics Is contained in this equation which is actually quite remarkable.
Now, if we now go back to, well recall a particular action where study now which is
just the kinetic energy, mv seqared over 2 minus the potential energy for a certain
quantum particle. Similar to find the classical trajectory,
as we know we want to calculate the first variation and we do so by writing this
action in the vicinity of the classical trajectory that we want to find.
So in some sense, what we should think about is that we have this initial point.
And we have this final point and there is some unique trajectory that we're looking
for, but there's also trajectory very close to it.
And this division from this so this black trajectory here is the classical
trajectory and the red ones are, are r classical plus d r.
This d r is a small deviation from the classical path.
So if we now just simply plug this expression into this a expression for the
action. So we're going to have the kinetic energy
true, so of course the velocity is, just d r over d t.
Or for brevity we can write as r dot. So we're gonave have are the two terms and
we're going to have two terms as an argument of the potential energy.
And since dr is very small we're going to keep track only of the linear terms here.
So for example in the kinetic energy part. So here we're going to have.
So M over 2 plus equal velocitiy squared. Plus, the linear term, which, comes from,
2r dot dr. So we're going to have m or plus equal dot
plus dr dot. And this guy, what we're going to do.
We're going to. Expand this function in Taylor series up
to linear order. And again we can do that because this d r
is very small. And, so, we're going to have the classical
sorry our classical minus gv over gr the gradient, dot e r.
And everything integrated over, time. So we have four terms here.
So this term is simple enough. And these two terms actually this guy and
this guy, together they represent a the classical action.
So the one which occurs on the classical trajectory.
Remember what we want is not the classical action itself but in some sense the
derivative of the action, which we, something which is proportional to the
d,r. So the only term which is sort of
non-trivial is this guy. Okay, and it can be further simplified by
using the integration by parts. So just to remind you, I'm sure you have
many of you have seen it before, so if we have an integral.
Let's say from a to b of, of a function f times g prime, dx.
So we can write it as a derivative of everything, of g prime minus f prime g.
And so here I have the full derivative, and I basically calculate, The, the value
of the functions, and the limits and essentially move the derivative from
function g to the function f. Now neuro keys they're all of the function
g is played by this d r dot, and the role of the function f is played by this m r
dot, so we want to move the derivative from here to here.
But the this full derivative theorem. So, if we're going to calculate the,
values of, this function dr in the endpoints at time zero and at time t.
They could respond to the endpoints of, of this trajectory.
So, and our trajectory, remember, is pinned.
So we have to go from the prescribed initial point or prescribed final point.
So there is no freedom for this sort of fluctuation dv from these initial and
final points. They can only it can only give the h from
the classical trajectory in between. So what I'm saying here is that this
theorem, the, the full derivative theorem in this particular case, vanishes, and we
can simply move the derivative from here to here with a minus sign.
So if we put everything together, based on this discussion, so essentially, if we
integrate by parts and replace this in this term with, with classical action.
We're going to have this classical action minus because there isn't going to be a
minus here and there is already minus here in the integral from 0 to r.
And there are two dots second derivative because we moved it plus d v over d r.
And everything is multiplied by this, this thing, delta r.
Now and remember, this is exactly our first derivation, and this is something we
want to set to zero. And this guy might as well be equal to
zero for every possible sort of fluctuation away from the classical
trajectory. And this implies that this Expression in
the curly brackets, so this expression must be identically equal to zero for the
first variation to vanish, for us to be able to find the minimum of the actions.
If we're going to , if we going to derive it we're going to get m r two dots is
equal to minus d v over d r. Well, the second derivative of the
coordinate is known as the acceleration so the left hand side is simply m times a and
the right hand side is the gradient of the potential energy which is the force acting
on the particle. So we see that we indeed have derived the
Newton's second law from the principle of the action being minimal.
And this principle, itself, followed from us taking the formal limit of h bar going
to 0, from setting to 0 the wavelengths of our particle.