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In the first lecture I introduced the
basics of the Schrodinger formulation of

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quantum mechanics, including the key
notion of a wave function.

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This is the most commonly used formulation
of quantum theory, and we're going to use

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it throughout the course.
We're going to discuss in great detail the

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interpretation of the wave function and
properties of the Schrodinger equation.

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But today, I would like to give you an
idea about an alternative formulation of

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quantum theory using so-called Feynman
path integral.

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Which in my opinion, is a very beautiful
formulation of quantum mechanics.

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If you define I should mention that this
material is almost never taught, at least

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at the undergraduate level, and when it is
taught it's usually said towards the end

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of the course.
But today I will experiment a little bit

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with this material, and will introduce it
right away.

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And one of the goals here is for me to
tell you that what you read in regular

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text books and actually here in this
course is just one way to think about

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quantum physics, the most commonly
accepted way to describe quantum mechanics

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but there are many other ways, actually.
Some of them don't even include the notion

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of the Boolean function.
And you should be aware of their

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existence, at least, and keep an open mind
here and actually in general whenever you

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study science.
The derivation that I'm going to present

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later in this segment follows very closely
this paper by Richard Feynman well a part

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of the paper, the paper is much more
detailed written back in 1948 and this

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paper is a typical Feyman.
So if you read the abstract the first

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sentence of the abstract says
non-relativistic quantum mechanics is

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formulated here in a different way.
So he suggest to the completely new a way

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to and to think about things here.
Now I should mention that well, notice the

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date, 1948.
So this time was actually very difficult

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time for Feynman, so there is this book
which I would recommend for you to take a

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look if you're interested in Feynman as a
person Perfectly Reasonable, it's called

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Perfectly Reasonable Derivations From the
Beaten Track.

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So, this book is really, there is no
shortage of books about Feynman, but this

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book is pretty much a collection of
letters that Feynman wrote through,

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throughout his life and when you read,
when you read this book this literature

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going back to this period tho the 40s, you
see that it was a very, very painful time

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for Feynman because his wife Arleen died,
the school, the high school sweetheart

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whom he married died in 1945, in June of
1945 of tuberculosis and he was, it was

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very different and it was not easy for
him, had to deal with it so the ladders

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would sort of bared this feeling very
closely.

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And despite this difficult time and maybe
suffering some sort of inspiration, so he

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came up with the number of very
influential very, very unusual ideas and

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one of those ideas is this path integral
relation.

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Naniki is going back to the actual
physics, so I'm going to present the

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derivation but and this I mentioned the
derivation is rarely introduced in the

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very beginning of quantum mechanics.
And one of the reasons here, of course, is

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that technically this issue is quite
involved.

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And furthermore the end result of this
calculation is actually a new mathematical

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object that hadn't even existed before
Feynman wrote it.

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And after he did, mathematicians have been
arguing to this day about its precise

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meaning.
So, there is some technical issues, there

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are some technical issues which exist, but
for those of you who are not really

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interested in these technicalities, again
I just would like to briefly tell you the

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main ideas without going into, into math.
So let us consider a quantum particle

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localized to the initial moment of time
equals zero in the vicinity of a certain

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point, initial point, R sub i.
And let's ask the question of what is the

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probability for this particle to reach a
final point, R sub f in a time t.

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So if it were a classical particle it
would have followed a unique well defined

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classical trajectory.
Let's say if it were a free particle it

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would have been just a straight line
connecting two points, but in general it

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would have been a solution of the, the
second Newton equation or the Lagrange

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equations, but it would have been a unique
classical trajectory.

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So the main result of Feynman in this
paper is that in quantum mechanics, the

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particle goes over all possible
trajectories at the same time, and In some

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sense.
So it falls into classical trajectory, you

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know, this trajectory, any trajectory you
can imagine and there is a weight complex

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weight associated with each trajectory
which is the exponential of i times

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classical action divided by the Planck
constant.

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So classical action here we're going to
discuss it in more detail later, but just

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to remind you that classical action is an
integral from 0 to t of the Lagrange which

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is the kinetic energy, basically mb
squared over 2 minus the potential energy

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times et.
So this in classical physics the minimum

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of this action gives rise.
To Newton equations and to classical

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equations of motion.
In quantum mechanics there's no principal

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of least action as Feynman showed but all
actions are allowed and all of them give

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rise to some terms in quantum theory.
Now going back to the probability of going

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from the initial point to the final is
some sense this probability can be

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represented as a sum over all well, the
absolute value squared of the sum, of all

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possible classical actions.
Overall fads that I labeled here by an

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index L and this sum itself is essentially
symbolically represents what we're going

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to see as a path integral which will come
out of the theory naturally.

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And it's really a very remarkable result.
Now it turns out that again so the

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mathematical part of it is quite subtle
but one can actually solve some problems

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just by using this is as sort of a cartoon
picture of what a path integral is.

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And we're going to I'm going to give you
an example such a solution So, even if you

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don't follow, very closely and carefully,
the derivation of the, mathematical

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formalism.
You can still follow the, main results,

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The main qualitative results later on.