Does anyone here know any resources I can use to expand my math skills?

Question

I've just finished with algebra in the 10th grade and I'm on my way to working with calculus. Yet, I'm increasingly interested in quantum mechanics and I can't quite get a grip on the mathematics they use. Does anyone know any resources that could help me understand higher level math? Or should I just work on calculus for now and let it go from there?

Answer 1

It isn't really recommend you learn high level math skipping the stuff in the middle. I'm not sure it's even possible. You could learn some of the terms, but you won't really understand them properly as in mathematics (like in most things) advanced study relies on previous knowlege.

Calculus is a good place to start as it's the basic nesasry mathematical tool for any branch of physics. Quantum mechanics is very intersting, but the mathematics they use is incredibly complex. It's used pretty much only in quantum mechanics, or other, even more hellishly complicated, branches of physics.

Answer 2

Okay, first of all, I applaud both your early interest in quantum mechanics and your immediate recognition that you'll need to learn specific and advanced math to handle it.



Work on the calculus. Really work on understanding it VERY well. Especially, concentrate on the basic integrals and derivatives. Remember learning to multiply in, perhaps, third grade? Remember doing worksheets that had 50 problems and learning to do them accurately and fast, as drills? That's about what I mean. Integrals and derivatives, anything up to and including x^5 and all of the trigonometry fun, should become like the multiplication tables to you.



Simultaneously, get a book or two on Q.M. that are used as text or supplementary reading for an intro-level college course. I STRONGLY reccomend the book, Introduction to Quantum Mechanics by Griffiths. It's very well-written and accessible to a beginner, but doesn't scrimp on the important stuff. The math behind the problems will probably be a bit over your head, but press on! Read it for the concepts. When you get to a mathematical bit you don't follow, just chalk it up to "well, they stick *this* equation into a big black box and get *that* equation out". You'll learn how to be that big black box later. Getting a feel for the subject right away, from a quality source, is crucial.



Once you've gotten the hang of integrals and derivatives, and given the first half of the Griffith's book a good honest try, find yourself a college-level general physics textbook (I used University Physics [Young] during my undergrad and it was okay; I hear good things about the Giancoli, Physics: Principles and Applications), and try the problems in the sections on quantum and modern physics. Those books have worked-out examples. You should be able to teach yourself how to do the easier sort in the homework problems of those books without the help of a teacher or any calculus, really.



If that gets easy and you're still hankering for more, get the Schaumm's Guide to vector and tensor calculus (it's a reference, more than a stand-alone book, but it has everything you need, without a single extra word), and with in nearby, try the problems in the Griffith's book.



Also, if you like quantum mechanics, look into quantum optics. It's thick, but fascinating stuff.

Answer 3

Start with Calculus, beginning with differentiation and get comfortable with that, ensure you have a thorough understanding of the theory of limits. Then proceed to integration (or the area under curves, as it is sometimes called). First year calculus is usually broken down into these two blocks of instruction, each a semester in length. As for quantum mechanics, try "The Road To Reality" by Roger Penrose.This is a broad background historical survey of math and physics and how they model the universe. There is a large section on quantum mechanics, with all the "bells and whistles" (meaning Roger gives you the perspective, gives you the symbology, although without much explanation, except by working through the examples included in the text footnotes). The tensor and spinor symbology is very difficult to understand, until you stare at it awhile, then it does begin to make sense. As for understanding the formulas and gaining a sense of the meaning of quantum mechanics, simply take in as much as you can, then keep coming back to it periodically as time passes, and you'll discover that either you get it, or it's over your head. Either way, it will become clear. Don't let anyone tell you because of your age and the grade you're in that you're in over your head, not necessarily true. I spent my sophmore summer before my junior year studying advanced math in the fields of topology (abstract geometry), advanced algebra (groups, rings, and fields), and number theory (theory of continuity, cardinality). Can't believe how much I was able to absorb, and retain to this day, but now I hold an MA in Mathematics specializing in the field of topology and also an MSc in Astrophysics, concentrating in theories of Cosmology (String Theory, Symmetry, Multiple Universes). That took a lot of effort over a lot of years however, even though I may have been a quick study on much of the material. I got where I am today because of that summer and that special studies program in school and what it taught me about myself and my self-confidence in my intellectual abilities. And you can do it too, just believe in yourself and "never give up, never give in".

Answer 4

It sounds like someone has given you some excellent advice on books to select, though I don't know enough to be of much help in evaluating the recommendations.



I know that you probably won't need the How to Ace calculus book. I've never read it , but I've seen presentations by Colin Adams, and they were entertaining (and besides it IS summer).



Speaking from personal experience, take your time and don't get impatient. At your age and beyond, I would read articles in Scientific American, and not understand them, and eventually just gave up on math and biology altogether, deciding that I could NEVER catch up. (Somehow, the fact that I was discouraged from these fields, to have engineering shoved down my craw, did not make me feel less of a failure.) My best friend was learning biology and I wasn't and that's what mattered to me.



Although calculus is important to most of physics, and essential, and very powerful, I found my early exposure to it frustrating. There are problems without exact answers, although you can get as approximate the answer as closely as you want. This was very counter-intuitive to me, it almost turned me away from math altogether. The math started to get messy. Not that it was necessarily hard, but it violated my sense of aesthetics. I don't know if this is common to people, but I wanted to warn you. This may happen, but you can get over it. In fact I now look at those same ideas, and see how powerful and beautiful they are.



Try to find a mentor. You can start with your instructors, but be forewarned, there are a lot of high school instructors who are where they are because they were thwarted at some point in their education. Don't let their disappointment discourage you. With the on line community, there are lots of forums where you can lurk and see how you would fit in.

While it is true that there are people who don't go on to graduate school, and there is competition, you need to try. If you give up, you have ZERO chance of becoming a Quantum Physicist.



Also, you are at a point where universities are starting to look for new recruits. See if any local schools have summer programs for high-schoolers. For that matter, there may be non-local programs.



I encourage you to pursue your goals. Don't cave into high-school counselors trying to live vicariously through you. If you fail at your original goal of quantum physics, the engineering schools will still be around later.



I would add this, that if you would like a change from calculus, group theory is very different, and would not require calculus. I think it is also used in quantum mechanics. I don't see any reason that you should need to put this off until you finish calculus.



However, Calculus is regarded as a sort of a "threshold" into higher math, and your abilities will be judged by it.



Most of all have fun and don't get impatient with yourself, as I did.



But DO try to gather a supportive community around you, to help you through any rough spots.

Answer 5

The full theoretical description of quantum mechanics uses advanced calculus. In the meantime, I would suggest sites discussing the conceptual side of quantum mechanics - this can be done using a minimum of mathematics and is a thorny issue all in itself.



Here are some links:



http://newton.ex.ac.uk/research/qsystems/people/jenkins/mbody/mbody2.html



http://www.chemistry.ohio-state.edu/~grandinetti/teaching/Chem121/lectures/Quantum%20Theory/waveparticlelight.html



http://www.newscientist.com/channel/fundamentals/quantum-world/

Answer 6

Everyone so far is right, don't skip steps. It leads to bad things. It's like going to bike on your own (without a parent's help), but they haven't taught you the importance of the break yet. You need it all to do it properly.



That being said, practice, practice, practice. Do your homework. You'll thank yourself in the long run.

Answer 7

Try algebra 2 and pre-cal. Those two classes are very similar. Don't skip from one step to the next because all its going to do is confuse you

Answer 8

start with 5th grade math and work your way up. every step in math builds on the previous steps. you can't skip to the top.



if all else fails, peyote.

Answer 9

Here is a great place where you can also practice and then look at the answers to problems:



http://www.math.temple.edu/~cow/

Answer 10

You should ask your friends which have calculas that if they can help you and teach you the basis.

Answer 11

Dude its summer.