I'm an aspiring physicist who wants to self study some Quantum Physics. My thirst for knowledge is unquenchable and I can not wait 2 more years until I get my first quantum physics class in university, so I want to start with a self study. I am enrolled in a grammar school and the most gifted among the gifted (not my description, mind you, I hate coming off as cocky, sorry) are enrolled in a special 'project'. We are allowed to take 3 school hours a week off in order to work on a project, which can be about anything you want, from music to mathematics. On the 4th of April we have to present our projects. Last year an acquaintance of mine did it about university level mathematics, so I thought, why not do it about university level physics? It is now the 3rd of October so I have half a year. My question is, where can I conduct a self study of quantum physics? Starting from scratch? And is it possible for me to be able to use and understand the Schrödinger equation by April? What are good books, sites, etc. that can help me? My goal is to have a working knowledge of BASIC quantum physics and I would like to understand and be able to use the Schrödinger equation. Is this possible? What is needed for these goals?
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Just pick up Dirac's book "The Principles of Quantum Mechanics" and read it in conjunction with "The Feynman Lectures on Physics Vol III". Don't waste time with linear algebra, the entire content of the undergraduate courses can be learned in half a day. Don't worry about the infinite dimensional nature of the thing, just reduce all the spaces to finite dimensions.
Also, be aware that "gifted" is a political label that has nothing to do with you, it's just a way for schools to segregate students by their future social class. It's not the analog of special needs, because the students in gifted classes are no different from the students in usual classes, except that they are given a slightly better education. Don't be fooled by a label into thinking you are somehow special, everyone is ordinary, including Einstein and Dirac. One has to do good work despite this, and those folks show it is possible by assiduous effort.
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Without having understood matrices and their interpretation as linear mappings (operators) it is very difficult to get a reasonable understanding of quantum mechanics. So you should spend some time on elementary linear algebra. Wikipedia is not bad on this, so you could pick up most from there. (To start with. For basic math, Wikipedia is almost completely reliable, which is not the case for more specialized topics. In case of doubt, cross check with other sources.)
Today, the shortest road to quantum mechanics is probably quantum information theory. For online introductory lecture notes see, e.g.,
In quantum information theory, all Hilbert spaces are finite-dimensional, wave functions are just complex vectors, and the Schroedinger equation is just a linear differential equation with constant coefficients. So you also need to learn a little bit about ordinary differential equations and how linear systems behave. Again, this can be picked up from Wikipedia.
In more traditional quantum mechanics, the Schroedinger equation is a partial differential equations, and wave functions are complex function depending on one or more position coordiates. On this level, you need to understand what partial derivatives are and have some knowledge about Fourier transforms.
Again, this can be picked up from Wikipedia. Then you might start with
You may also wish to try my online book http://lanl.arxiv.org/abs/0810.1019
If you want to understand quantum physics, you have to understand Fourier series and Fourier transforms. The best introductory text ever, is the book Who Is Fourier?. Do not be fooled by its cartoonish appearance, this is a serious book as can be demonstrated by the fact that the name on the top of the list of advisers is Yoichiro Nambu who is the 2008 Nobel prize co-winner:
Then I would work to gain an understanding of the Heat Equation. The Schrodinger equation can be described as the quantum version of the heat equation (except, what is diffusing is probability).
Fourier developed the Fourier series in order to solve the question of how heat diffuses in a material. If you understand these things, you can understand quantum mechanics within a few months.