High School Student Preparing for College

[Time_]

Hello all! I am a 14 year old student attending an elite private school in North Texas and I aspire to someday become a theoretical physicist. I currently take mostly AP classes and am doing better than a good portion of my peers. I have knowledge of high school math (taught it all to myself when I was 10/11 in one to two weeks), Calculus I, II, III, D.E., topology, statistics, calculus based physics, and have a working knowledge of theoretical/quantum physics, cosmology, and astrophysics. If you look up Jacob Barnett, I am fairly analogous to him. My school is not a huge fan of student's excelling too quickly, sigh, but I do what I can, and do it as well I can. Furthermore, I solve equations on my whiteboard in my bedroom, windows, mirrors, really anything that will not absorb Expo markers. I have a theory I am currently developing, and I would say it is going quite well (considering professors (some very esteemed) find validity in it).​

My question to you all is if I hope to attend a college such as MIT what can I do to help my chances? I have contemplated building a particle accelerator, which sounds quite interesting and would prove useful to my research. In addition, I have contemplated attending a program such as RSI (if I am even so fortunate enough) later on my high school career or even entering the Google Science Fair. With a fair amount of you being in college/graduated (I would presume), does anyone have any pointers to help my dream come true? Thanks for the advice!​

 

[cytochrome]

I wish I started at your age. The best thing you can do is realize how much time you have, and to NOT take it for granted!

Building a particle accelerator would definitely help your chances getting into MIT, as well as exhausting your school's math/physics course and even auditing for college courses.

Although it may not be the most exciting thing in the world (I thought this at the time, but my mind has since then changed), learn as much calculus and linear algebra as possible. As. Much. As. Possible. Understand the theorems of calculus and linear algebra and learn how to understanding mathematical writing and proof. Theoretical physicists "proove" things much like mathematicians do. Understanding mathematical proof and analysis is therefore vital.

Here's my input and order in which I would make sure I completely understand, 100%, every topic (each one is somewhat of a prerequisite)

1) Discrete math - logic, set theory
2) Calculus - derivation, integration, fundamental theorems - seek to view these from a "proof" point of view!
3) More calculus - multivariable, methods of integration, series
4) Linear algebra - vectors, matrices, basis, subspace, linear spaces
5) Vector calculus - vector derivation and integration, line integrals, surface integrals, theorems of Green and Gauss
6) Differential geometryI think this order of things will set you up for everything you need to know before college, plus more.

In the mean time, STUDY PHYSICS. Work problem sets. Don't get stuck on the "quantum physics" hype and be blind to everything else. Physics started as applied to classical systems, so learn your classical mechanics. Learn the basis of the Lagrangian and Hamiltonian formalisms as applied to classical systems and learn the gradual connection with quantum mechanics. Master the idea of the classical oscillator as well, and learn to apply Lagrangian and Hamiltonian methods to find the equations of motion for oscillating systems. As a young theoretical physicist, you should be comfortable mathematically describing any oscillatory system in a HUGE variety of ways.

You will find that oscillations and the Hamiltonian are a vital part of quantum mechanics and virtually every other subject in modern physics. Wouldn't you want to learn it from a "real" world point of view, first? It will make the study of modern physics easier and visual, trust me. Differential equations, of course, will be important as well. But you can solve D.E.'s on any math software (Mathematica, MATLAB, etc.), which would also be a good skill to have and put on a resume.

 

[Time_]

Thanks so much for the advice! It means a lot, honestly. I will be sure to get back to work going back and review the basics. Aside from studying is there anything else you would recommend I do to develop my theory in the mean time? Furthermore, what textbooks would you recommend I use? I already have Multivariable Calculus from Larson and "Mathematical Methods for Physicists" from Arfken, both which are great books. Also, it isn't so much of a textbook, but compilation, the other more math intensive book I own in The Dreams That Stuff is Made Of from Stephen Hawking. It is essentially a compilation of all the major physics papers ever written, and is pretty thorough at that.

 

[cytochrome]

Not only should you review the basics, but you should strive to understand them from more advanced formalisms (which of course requires more math, particularly the calculus of variations for Lagrangian mechanics). Mastering every "basic" principle (pendulums, springs, sliding blocks, projectiles, etc.) in terms of Lagrangian and Hamiltonian mechanics is crucial.

As for your theory, I nor no one else knows what it is. If it is entirely a mathematical physical theory then it should be something that you can easily work on at your own discretion.

 

[micromass]

Thanks so much for the advice! It means a lot, honestly. I will be sure to get back to work going back and review the basics. Aside from studying is there anything else you would recommend I do to develop my theory in the mean time? Furthermore, what textbooks would you recommend I use? I already have Multivariable Calculus from Larson and "Mathematical Methods for Physicists" from Arfken, both which are great books. Also, it isn't so much of a textbook, but compilation, the other more math intensive book I own in The Dreams That Stuff is Made Of from Stephen Hawking. It is essentially a compilation of all the major physics papers ever written, and is pretty thorough at that.

The most important math-thing you need to master right now is probably linear algebra. That is really essential in physics and especially QM.
A good introduction to linear algebra is this book by Lang: https://www.amazon.com/dp/3540780602/?tag=pfamazon01-20
It is a math book, so it's not focused on physics. But I still think it's a very good first introduction to linear algebra. After this book, you might want to get a second book on linear algebra as well.

 

[Time_]

A good introduction to linear algebra is this book by Lang: https://www.amazon.com/dp/3540780602/?tag=pfamazon01-20
It is a math book, so it's not focused on physics. But I still think it's a very good first introduction to linear algebra. After this book, you might want to get a second book on linear algebra as well.

Good to know, thanks for the advice. As soon as I am done with exams I'll be sure to go out and buy that.​