In the first lecture, I introduced the basics of Schrodinger formulation of quantum mechanics, including the key notion of the 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 proper use of 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 and used to Feynman. I should mention that this material is almost never taught at least at the undergraduate level. And when it is taught, it's usually near the end of the course. But today, I will experiment a little bit with this material and will introduce a try to way. And one of the goals here is for me to tell you that what you read in regular textbooks 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 a wave function and you should be aware their existence at least. And keep an open mind here and actually in general whenever you started signs. 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 Feynaman, 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 suggested the completely new way 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. It's called Perfectly Reasonable Deviations from the Beaten Track. So this book is real, there's no shortage of book about Feynman. But this book is pretty much a collection of plagiarism that Feynman throughout his life. And when you read this book, when you read these last years going back to this period, to the 40s. You see that it was a very, very painful time for Feynman because his wife Arlene died. The high school sweetheart whom he married died in 1945, in June of 1945 of tuberculosis. It was not easy for him to deal with this, so the last resort is to bury this feeling very closely. And despite this difficult time and maybe the suffering and sort of inspiration, he came up with a number of very influential, very original, very unusual ideas. And one of those ideas is this going back to the actual physics. So I'm going to present an iteration but, as I mentioned, this derivation is rarely introduced in the very beginning of quantum mechanics course. And one of the reasons here, of course, is a technical involved. And furthermore, the end result of this calculation is actually a new mathematical object that hadn't even existed before Feyman wrote it. And did, mathematicians have been arguing to this day about its precise meaning, so there is some technical issues which exists. But, for those of you who are not real interested in these technicalities, again, I just would like to briefly tell you are the main ideas without going into math. So, let us consider a quantum particle localized at the initial moment of time to equal zero in the vicinity of a certain 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 we're a free particle, it would have been just a straight line connecting two points. But in general, it would have been a solution or the second mutant equation, or the 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 in some sense. So at full is at the classical trajectory, this trajectory, any trajectory you can imagine and there is a weight, a complex weight associated with the each trajectory which is the exponential of i times classic collection divided by the flying. So a classical action here, we're going to discuss it in more detail later. But just to remind you, the classical action is an integral from zero to t or the which is the kinetic energy, basically nb squared over two, minus the potential energy times dt. So this, in classical physics, the minimum of this action give rise to the mutant equations and to classical equations of motion. In quantum mechanics, there is no principle 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 point, in some sense this probability can be represented as the sum over the absolute values squared of the sum, of all possible classical actions over all paths 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 theorem 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 sort of cartoon picture of what a path integral is. And I'm going to give you an example such as solution, so even if you don't follow very closely and carefully the derivation of the mathematical formulas, you can still follow the main qualitative results later on.