Greetings everyone. Welcome to the fourth week of the course, quantum confusion. Less formula's, more understanding brought to you by Saint Petersburg State University, the faculty of mathematics and mechanics. On this week, we are going to learn a bit of mathematical language of quantum mechanics. Now I can feel a bit of disappointment with some of you since I promised you less formulas in this course, and here apparently will have some formulas. Well, fear or not. It is quite possible that you will understand less than you want in this week. It is especially true if you were previously unfamiliar with complex numbers, vector spaces, matrices and operators, linear functionals, etc. But let me reveal big secret to you. Most of the students don't understand new mathematical material from the first learn. But experienced students know that they can understand it if they want. The only thing you need for this is to review the material several times. It is very much the same as watching video in the new language that you are trying to learn. You understand almost nothing from the first few but as you've watched it again and again, you hear more and more familiar vaults and constructions, until you are able to retrieve every vault. Now, some of you may even decide that you don't want to learn this mathematical language of quantum mechanics since you are not going to go deeper into things, and you're just curious about quantum confusion. That is okay. But still I would advise you to watch the videos of this week at least for one time. It can be helpful even if you don't understand much. But for those who want to dig deeper and to take other courses on the subject, I strongly suggest to struggle until you understand as much as you can. As always in our lectures, we are going to touch only the tip of the iceberg, and we usually go fast enough. Students often ask if they can read something on the subject in order to learn more or to be able to reflect on the material with their own comfortable base. The lectures of this week are very much connected to the second chapter of the book you can see on this slide. It is a very famous text book by Claude Cohen-Tannoudji, Bernard Diu, and Frank Laloe. Originally it was written in French, but it's now translated into many languages, including English. I was happy to find this book in Russia, in our university bookstore where it costed noticeably less than any copy sites online. Before trying to buy this book online, check your local university to stores and libraries. You'll be able to save a lot of money. Now, even these book though you can see it is very thick, assumes that you have some background in mathematics before you reach it. I won't suggest any particular groups online, linear algebra or complex numbers or mathematical analysis, since there are plenty of them available for free online. I just can't choose. I encourage you to look at for ourselves and to choose whatever courses or books you like the most. The syllabus for this week is, first we are going to learn what is an inner or scalar product in the vector space. We would be able to define at last, what does it mean for two vectors to be orthogonal to each other, and how the vector length can be calculated. This would allow us to distinguish the special basis and our vector spaces which are the orthonormal basis. Then we are going to learn the Dirac's notation for the vectors and how the inner product looks like in this new notation. To do this, we would first need to introduce the conjugate space of the original vector space, which is the space of linear functionals. After that, we will learn linear operators and how they act on the vectors of both the original and the conjugate space. Then, you'll be able to approach the procedure of Hermitian conjugation, and how it acts on every entity in our consideration. Complex numbers, vectors, functionals, and operators. After that, we'll learn some special types of linear operators which have a physical meaning for us. They are connected to measurement of quantum system or with this evolution. At last, we'll discover the eigenvalue equation and it's physical meaning. We'll consider some basic and very useful examples of operators which are often used in quantum confusion. There's so much to learn. Let's start.