In the first lecture I introduced the basics of the Schrodinger formulation of
quantum mechanics, including the key notion of a wave function.
This is the most commonly used formulation of quantum theory, and we're going to use
it throughout the course. We're going to discuss in great detail the
interpretation of the wave function and properties of the Schrodinger equation.
But today, I would like to give you an idea about an alternative formulation of
quantum theory using so-called Feynman path integral.
Which in my opinion, is a very beautiful formulation of quantum mechanics.
If you define I should mention that this material is almost never taught, at least
at the undergraduate level, and when it is taught it's usually said towards the end
of the course. But today I will experiment a little bit
with this material, and will introduce it right away.
And one of the goals here is for me to tell you that what you read in regular
text books and actually here in this course is just one way to think about
quantum physics, the most commonly accepted way to describe quantum mechanics
but there are many other ways, actually. Some of them don't even include the notion
of the Boolean function. And you should be aware of their
existence, at least, and keep an open mind here and actually in general whenever you
study science. The derivation that I'm going to present
later in this segment follows very closely this paper by Richard Feynman well a part
of the paper, the paper is much more detailed written back in 1948 and this
paper is a typical Feyman. So if you read the abstract the first
sentence of the abstract says non-relativistic quantum mechanics is
formulated here in a different way. So he suggest to the completely new a way
to and to think about things here. Now I should mention that well, notice the
date, 1948. So this time was actually very difficult
time for Feynman, so there is this book which I would recommend for you to take a
look if you're interested in Feynman as a person Perfectly Reasonable, it's called
Perfectly Reasonable Derivations From the Beaten Track.
So, this book is really, there is no shortage of books about Feynman, but this
book is pretty much a collection of letters that Feynman wrote through,
throughout his life and when you read, when you read this book this literature
going back to this period tho the 40s, you see that it was a very, very painful time
for Feynman because his wife Arleen died, the school, the high school sweetheart
whom he married died in 1945, in June of 1945 of tuberculosis and he was, it was
very different and it was not easy for him, had to deal with it so the ladders
would sort of bared this feeling very closely.
And despite this difficult time and maybe suffering some sort of inspiration, so he
came up with the number of very influential very, very unusual ideas and
one of those ideas is this path integral relation.
Naniki is going back to the actual physics, so I'm going to present the
derivation but and this I mentioned the derivation is rarely introduced in the
very beginning of quantum mechanics. And one of the reasons here, of course, is
that technically this issue is quite involved.
And furthermore the end result of this calculation is actually a new mathematical
object that hadn't even existed before Feynman wrote it.
And after he did, mathematicians have been arguing to this day about its precise
meaning. So, there is some technical issues, there
are some technical issues which exist, but for those of you who are not really
interested in these technicalities, again I just would like to briefly tell you the
main ideas without going into, into math. So let us consider a quantum particle
localized to the initial moment of time equals zero in the vicinity of a certain
point, initial point, R sub i. And let's ask the question of what is the
probability for this particle to reach a final point, R sub f in a time t.
So if it were a classical particle it would have followed a unique well defined
classical trajectory. Let's say if it were a free particle it
would have been just a straight line connecting two points, but in general it
would have been a solution of the, the second Newton equation or the Lagrange
equations, but it would have been a unique classical trajectory.
So the main result of Feynman in this paper is that in quantum mechanics, the
particle goes over all possible trajectories at the same time, and In some
sense. So it falls into classical trajectory, you
know, this trajectory, any trajectory you can imagine and there is a weight complex
weight associated with each trajectory which is the exponential of i times
classical action divided by the Planck constant.
So classical action here we're going to discuss it in more detail later, but just
to remind you that classical action is an integral from 0 to t of the Lagrange which
is the kinetic energy, basically mb squared over 2 minus the potential energy
times et. So this in classical physics the minimum
of this action gives rise. To Newton equations and to classical
equations of motion. In quantum mechanics there's no principal
of least action as Feynman showed but all actions are allowed and all of them give
rise to some terms in quantum theory. Now going back to the probability of going
from the initial point to the final is some sense this probability can be
represented as a sum over all well, the absolute value squared of the sum, of all
possible classical actions. Overall fads that I labeled here by an
index L and this sum itself is essentially symbolically represents what we're going
to see as a path integral which will come out of the theory naturally.
And it's really a very remarkable result. Now it turns out that again so the
mathematical part of it is quite subtle but one can actually solve some problems
just by using this is as sort of a cartoon picture of what a path integral is.
And we're going to I'm going to give you an example such a solution So, even if you
don't follow, very closely and carefully, the derivation of the, mathematical
formalism. You can still follow the, main results,
The main qualitative results later on.