Now we are going to proceed with the more technical discussion and derivation of
Feynman's ideas the Feynman path integral. And I should mention here from the outset,
that if you are not interested in this technical details, or if you feel that
your mathematical background is not sufficient to follow them very closely,
you may simply skip this and the following segment because the rest of the course is
not going to be heavily dependent on this particular derivation.
But, otherwise, I would encourage you to actually go through this evaluation and
maybe even work it out on your own after the lecture.
Because understanding it would help you develop intuition not only about the path
integral itself, but about quantum mechanic quantum mechanics more generally.
Now the object that we are going to discuss in this part of our segment is the
so-called propagator, which actually plays an important role in various aspects of
quantum theory. But at this level we can view it simply as
an attempt to make quantum mechanics as close as possible to the concepts that we
understand intuitivelly in classical physics.
And these concepts Involves, for example, the notion of a particle localized in a
point in space and the notion of trajectory that the particle follows.
Now, at the operational level, what, it is, it involves, the following
construction. So, let's assume that, at an initial
moment, moment of time equals zero, our quantum particle was localized in the
vicinity of the original point, R sub i. So by localized I mean that its initial
wave function, psi of 0, T equals 0, is equal to R sub i, this architecture/g,
which represents the Eigenvector of the position operator corresponding to this
point. Or in physical terms it means that this
wave function is a very narrow wave packet localized in the vicinity of this R sub i.
Very similar to the wave packet that we saw in the very first lecture in the last
segment of the very first lecture. And, so this initial, state is going to
evolve under the action of the Schrodinger equation, which is the standard
Schrodinger equation, with some kinetic energy and potential energy.
And as we know from the first lecture, so this evolution, quantum evolution will
evolve spreading out of this wave packet with time.
So instead of the localized sort of particle-like entities, going to become
like a cloud surrounding this R sub i, and this is spreading out [inaudible] fashion,
the presence of potential. Now the question that we're going to ask,
and, which brings us to this, notion of a propagator is what part of this particle
will propagate a certain final point, R sub f.
Or, in other words, what part of this spread out wave packet will be located in,
near this point. And this mathematically implies
calculating the overlap between the final wave function, psi of T which, again, is
governed by the Schrodinger equation. And this R sub f, which describes a wave
packet near this, some other final point. So of course I don't want to give you the
impression that the particle literally propotages from one localized state to
another localized state. It does not, it actually spreads out in
the quantum mechanical language but a part of this wave packet will indeed be located
in, near this point, and this is what we want to calculate.
And this overlap is the propagator we're going to focus on.
Now another question we sort of may ask is how will the particle get there?
How will the particle go from the initial point to this partially final point?
So this question, strictly speaking, doesn't really make sense in the context
of the standard quantum theory based on the Schrdinger equation, because there is
really no notion of the trajectory, which implies the ability to measure momentum
and coordinate at the same time, which is not possible in quantum mechanics.
But, as we will see, in this different formulation, by Feynman, so this question
sort of acquires a meaning, and going a bit ahead of ourselves.
Sort of letting me sort of, reiterate or advertise again that what we're going to
find Is that this particle goes from this initial point to a final point for, like,
all possible trajectories that we can possibly imagine.
So here, if you lose a trajectory that I plot.
Now to, to move, to move on, to actually calculate this overlap, this propagator.
Clearly, what we need to find is well, the final wave function, psi of T.
We know the initial condition. We, we sort of know what, the final state
we want to get. Now we need to, well, actually solve the
Shrodinger equation. Which, in general, is very complicated.
But there is a formal solution we can, write, using, the so called evolution
operator. So we're interested in solving the general
Schrodinger equation here with some initial condition at t equals 0.
So assume that we know the wave function at the initial moment of time.
So, this of course is a completely general formulation which is not specific to the
derivation of any path integrals. But, well in our case, the initial
condition we have chosen to be this localized wave packet near a certain point
R-sub-i. So in any case well to solve the
Schrdinger equation this time-dependent Schrdinger equation basically means to
find the wave function as a function of time, psi of t.
And the evolution operation that I just mentioned, is formally relates the initial
condition to the final wave function, psi of T.
So, it's action is exactly in some sense rotation from psi of 0 to psi of T.
I use the word rotation having in mind this geometric picture that we.
Introduced last week to motivate the direct notations you know, this kept
vector is psi of t which in sums, are meant to represent a sort of vector,
abstract vector in a linear vector space corresponding to the state of the physical
system. Now if we did have a sort of guard variety
a linear vector space as our [inaudible] space which normally is very
multidimensional so we cannot really draw the axis, but if we had, if we mention for
a second that we have such free axis and there is a state psi, oops, psi 0, which
represents our initial state. So, the norm of this vector is so the
absolute value squared of this vector sort of corresponds to the total probability of
finding our quantum state. Our quantum particle in a certain state
and well, from the Borne interpretation we can say that this probability is equal to
one. We will find a particle in which they are
in some state. So the norm of this vector should be
preserved as we perform a quantum evolution.
So in some sense the only thing we can imagine this vector doing is the function
of time under the action of the Schrodinger equation, as its, its rotation
you know, by some angle to a certain new state, psi of T.
And, this rotation, the operator which sort of enforces this rotation is exactly
the solution operator. So, again so here I'm just trying to
represent this in an intuitive way, in general, we're dealing with the generic
quantum mechanic problem. There is no way to draw it, because we
have well, an infinitely dimensional Hubert space, but still, at some level
this picture is preserved. So this evolution operator rotates a
normalized state which describes a particle to a different state, which
describes the same particle. Now it turns out that we can actually,
instead of writing the Schrodinger equation for the wave function, we can as
well just write the same Schrodinger equation for the evolution operator.
The only thing we have to do is just to plug in this equation into the Schrodinger
equation. We're going to see that we have
essentially the identical equation for the evolution operator.
So H U of T. And the initial condition for this
equation Is you observe is equal to 1. Where 1 is just identity operator.
So why? Well it's sort of full as in a very simple
way from it's definition. So if you look at this equation we see
that the evolution operator sort of evolves our initial condition to final
state. But well if physical is there there's
nothing to hold we just stay where we were.
And so therefore psi of 0 is equal to psi of 0, so U of Q is equal to 1, and this is
sort of the initial condition we can enforce.
And one can guess in some sense or just pick a general solution to this matrix or
operator equation, which satisfies both, both the Schrodinger equation itself and
the initial condition by writing u of t, as an exponential of minus i over h bar H
times t. So well, t equals 0, e exponent of 0 is
equal to 1, so it does satisfy the required initial condition.
And if we plug it into this equation, so if we differentiate this sort of ansatz
with respect to time. So well, differentiating the exponential
sort of pulls out this coefficient minus-i over h-bar times h, so minus-i times i is
equal Two one H divided by H is equal to one, so we're just going to have H E to
the power minus I H bar H times T, which indeed is equal to H times U.
Therefore indeed, so we satisfy the required Schrodinger equation.
So an interesting and sort of attractive feature of this evolution operator is that
it solves the Schrodinger equation with all possible initial conditions in one go,
because the evolution operator itself does not actually depend on the initial
condition. So we see that it has a universal initial
condition. And to equal zero, U is equal to one.
So for instance, here in the geometrical interpretation, if we have let's say
different psi of 0, let's say this will be psi of 0.
Well prime. So the evolution of this other initial
condition in this sort of Hilbert space would be a rotation by the same angle
around the, the same axis of the different side of T it would say, side T prime here.
And but the evolution operator that would enforce this rotation would be exactly the
same as before. So it would be, it will be the exactly the
same u of T. So in our case and in particular, so we're
interested in knowing how not this particular initial condition evolves with
time, and therefore the wave function would be equal to e to the power minus i
over e Hamiltonian times time acting on this initial condition.
So this is sort of a exact and universal solution of the Schrodinger equation,
whatever it is. With, with this initial condition of
localized particles in certain region in space.
Now let us go back to the main question that we posed in the beginning of this
segment, namely the question about the overlap between the wave function psi of
t, sort of propagated from the vicinity of the point R sub i, and the wave back at
localized in the vicinity of the point R sub f.
So, what we found in the previous slide is a formal exact expression for this psi of
t. So psi of t is the action of the evolution
operation, the initial condition, R sub i. And therefore, if we put two and two
together, the expression. Or the, propagator is going to be the
fully matrix element is going to be Rf, here we're going to have the solution
operator, ht, r sub i. So this guy is simply the psi of t.
And so we get, the expression that we're actually going to use in the next segment
to derive the Feynman path integral. So a Feynman path integral is another
representation of this metric's element between Rf and Ri of this operator U of T.
So at this stage you may ask me why would we bother to look for any other expression
apart from the one we just derived, which looks pretty compact and simple.
So the answer to this is that this simplicity is deceptive.
Because the main object here, this evolution operator involves exponentiating
an operator, the Camille 2 onion/g. And to actually calculate such an
exponential of an operator, exponential of a matrix.
In a rather complicated exercise and in general even, we define the exponential
open operator, is a tricky business, so as a side comment here, let me remind you
that for an average operator, a matrix x, in our case this act as this minus i over
h bar Ht, so its exponential is defined by the Taylor series, so it's 1 plus x plus x
squared over 2 plus x cubed over 3 factorial, plus etc.
And so to calculate this series with matrix as operator is rather complicated.
And so the Feynman approach, as we will see, essentially circumvents the need to
calculate this complicated exponential and truncates the series, right here, and
basically the end result as we will see will be, an expression that doesn't have
any operators whatsoever. And not only that the final result also
will have a very clear intuitive interpretation that we already, sort of,
advertised in the beginning of this lecture.