What math classes are necessary for a deep understanding of quantum physics?

[Fizix]

Wazzapnin'.

I'm 14 and I'm going to be a 10th grader this year. My school doesn't offer post-BC math, so I have to take all of my math classes online. Currently, I am studying Real and Complex Analysis, Linear Algebra, Abstract Algebra, and Topology [W00T for Dover. ] on my own. Specifically, I am using Shilov's Linear Algebra as well as Shilov's Real and Complex Analysis. I'm using Hocking and Young for Topology, and Deskins' Abstract Algebra. Are these good books? What other books would you recommend that are relatively cheap [I'm talkin' Dover price ranges. ]

JHU-CTY offers Multivariable Calculus and Linear Algebra. I'm fairly confident that I can finish both of these within 2 months [as far as the classes go]. Stanford's EPGY offer Ordinary Differential Equations, Partial Differential Equations, Real Analysis, Complex Analysis, Modern Algebra, and Point-Set Topology. Apart from ODE and PODE, I think that I will have taught myself the rest of the subjects well enough such that the classes would be review for me. Do you think that it would be worth my money to retake these courses online? If I don't take these classes, what should I do next? Are there any online distance education programs that offers classes beyond these? The local state university offers classes, but that's not an option due to transportation difficulties. What other options are there?

Thank you.

~ Fizix.

 

[mathwonk]

it is unbelievable to think you can master the ones you already mention. just work on those, or a few of those.

 

[Fizix]

Indeed. I will continue to work on the subjects far after the class finished and I will explore the subjects in much more depth, but I simply cannot say that to my school. They will need some sort of transcript or certificate or something from a university to verify it. Should I ask for an independent study?

Sorry if I came off as pretentious. By no means do I think I will have mastered these subjects within the duration of an online course.

 

[Fizix]

Any suggestions?

 

[end3r7]

Not that it's any of my business, but why are you trying so hard to "get ahead"?
You seem to be following a regular mathematics curriculum at a college, so I figured something like Discrete Mathematics and a more rigorous take on Statistics wouldn't hurt, but to study pure mathematics without the joy of applying it to something you like can be a bit... well... boring, especially for somebody your age (unless you want to be a pure mathematician, but even then you will be encountering all those things again in a couple of years).
Do you have some more specific interests? Such as astrophysics, etc.
My first recommendation would be to slow down a bit. Reading those all at the same time can't all be that helpful.

You can buy old editions of used textbooks at amazon.com or half.com. Those usually can come fairly cheap.

Good luck with your efforts.

 

[Fizix]

Thank you for the response.

I am reading these subjects with the goal of being able to understand Quantum Mechanics, one of my main interests, with all of it's mathematical rigor. My other main interests include Cosmology, Astrophysics, Quantum Cosmology, and Gravitational Physics.

Here are some of the books I have: Shankar, Greiner, and the two volumes of Tannoujdi for QM, Quantum Physics by Gasiorowicz, a few books on General Relativity, over 20 books on Cosmology and Particle Physics, and a few books on black holes. I also have all three volumes of the Feynman Lectures. I have Serway and Jewett for general physics.

I want to learn this math so I will be able to understand all of the mathematics in these books. That's the goal I have in mind.

I will be taking AP Statistics this year in school. Are there any statistics books you would recommend? I am also considering Discrete Mathematics and Graph Theory.

 

[ice109]

when you say read books on these topics do you mean scholarly books or pop sci books?

 

[Fizix]

Scholarly? Not sure. By popular science, I'm assuming you mean books like A Brief History of Time, The Universe in a Nutshell, and The Emperor's New Mind.

 

[ice109]

Scholarly? Not sure. By popular science, I'm assuming you mean books like A Brief History of Time, The Universe in a Nutshell, and The Emperor's New Mind.

yes those are pop sci books, scholarly meaning textbooks or journal articles

 

[Fizix]

Ah, I see.

 

[end3r7]

The Road to Reality by Roger Penrose is very good with plenty of math rigor. But it's tough.
There is nothing wrong with reading those books. In fact, it makes sense for you to gradually go up a gradient level of difficulty. And make no mistake--most adults can't even understand those "pop sci" books (Universe in a Nutshell is particularly deceivingly tough).

Of course, if bad comes to worse, there is always Wikipedia. =D

 

[end3r7]

AP stats is a joke btw. Watered down to the extreme.

 

[Fizix]

^ So I've heard, Ender. :rofl: Perhaps there are some good "real" Statistics books out there?

I actually am reading The Road to Reality at the moment. I'm also reading Quantum by Jim Al-Khalili. I also came upon another interesting book, called The Hole in the Universe by K.C. Cole, but I haven't gotten around to reading it. Previously, I've read The Universe in a Nutshell, A Brief History of Time, and Simply Einstein. My interest in these subjects actually started with popular science books.

And of course, the almighty Wikipedia.

 

[end3r7]

Quantum is a gorgeous book, if I remember correctly.

About Stats... if you stick with Dover, you can't go wrong with https://www.amazon.com/dp/0486637603/?tag=pfamazon01-20

 

[^_^physicist]

Consistering that you made the comment that you want to touch quantum on a highly rigorous level, I would suggest going to a techinical bookstore online (such as Powells.com), or stopping by a university library and stopping by the QC140-200 sections. Just flip through the texts and take note as to what types of notation and formalisms they are using. Also read the prelimanary sections and try to work through any of the given problems their, as they are normally introductions to all of the mathematics necessary to get started on that author's take on the subject.

Granted, it is unlikely, even with your level of mathematics, to move through these texts with ease; it will give you a list of topics to investagate.

As for whether you should take university math classes or not, well that is to be decided by you. If you want to start paying early for education go for it. Take the math classes you list. If, on the otherhand you feel that you don't want to waste the money just yet- talk to your school administrators and develop a circulum and syallibus for an independent study of the mathematics necessary to approach your topic of interest. If you need help developing this syallibus/circulum write or email professors at universities you are interested in, and ask them for suggestions for developing a course.

If it were me, I would take the time to develop an independent study course that corresponds to a university course, and maintain contact with school administrators and university facility so that when you go to college you can make the argument that you deserve either to have prerequsits waived or credit granted for free.

 

[Pseudo Statistic]

Specifically, I am using Shilov's Linear Algebra as well as Shilov's Real and Complex Analysis. I'm using Hocking and Young for Topology, and Deskins' Abstract Algebra. Are these good books? What other books would you recommend that are relatively cheap [I'm talkin' Dover price ranges. ]

Finally, another Dover fan. :D
Shilov's Real and Complex Analysis is a little dry, I think-- atleast, I couldn't follow it too well around the metric spaces chapter 'till after I took a real analysis course and read a bit of another book. (Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger, another Dover book with some really nice chapters on inner product and normed linear spaces with applications to Fourier Series-- definitely useful for what you want to understand. It's also got an introduction to sigma-algebras and Lebesgue integration-- especially useful if you plan on doing some probability theory in future.)
However, I guess you should keep in mind that you probably won't find analysis books too easy to read unless you've had some classroom experience of watching stuff being proven; atleast, it's been like that in my experience.
Deskins' Abstract Algebra book is pretty interesting I think-- I'm still reading through it myself, but I like what I see so far. (Except for the fact that there're no solutions at the back of the book-- a big drawback for some Dover books I think)
Apart from those, some other Dover books I really like are "The Variational Principles of Mechanics" by Cornelius Lanczos and "Principles of Electrodynamics" by Melvin Schwartz, though you probably need to learn multivariable calculus and some ODEs before you move onto those. :)
With respect to the math you want to get through, though, I think you'd be at an advantage if you take classes at your local university or something; it makes it a lot easier to absorb the math over time if you intend to remember it all later on.
Also, you might want to learn Multivariable Calculus and ODEs out of a computational rather than rigorous book first to get a feel for applications, rather than learn how to prove existence and uniqueness first. :)
That's what I think, anyway.
Good luck.

 

[mathwonk]

lighten up. life is long. youll be fine.

 

[math_owen]

lighten up. life is long. youll be fine.

Fizix, Mathwonk has just generously given you some of the most profound advice I have ever heard given to another person. Indeed, if it was a theorem, it would produce 1,000 more theorems.

Please let this post and its length serve to you as an abstract template of how mathematics/ physics is truly done. Mathwonk dropped the nugget. Here comes the depth behind it. The same way math is really done.

Lets get some things straight. You're very young and it appears you're very bright and beyond your years. A sincere congratulations. And I definitely applaud your efforts and achievements. But. (Here comes the but)

Life is infinite dimensional young friend, and not discretely either. With that, you may have stumbled upon the interconnectedness of everything. I am not being pretentious, I mean what I say. It is all interrelated.

Within that interelation of all things is your youth. Yeah yeah, sounds like parental rhetoric, I know. It's not.

My friend, you may have a great superficial understanding of all these subjects and indeed that is a very worth while and meaningful understanding. With it you can go deeper into everything. And you'll find many an academic in college who is sorely lacking that understanding - which often causes all kinds of problems for everyone.

But, with all due respect, it is not likely that you could pass without books any serious college level test administered by Mathwonk or I or any other older adult.

I absolutely don't mean to dampen your spirits, just a caution to the wind and a possible challenge that you may not being going to the depths truly required in understanding these texts.

My advice young friend is to honestly slow down and peer into the depths of what you're learning. I was once told if you spend less than an hour on a page of mathematics you've gone too fast. And with my experience, I'm going to tell you that is absolutely true.

There is a difference between reading a theorem knowing vaguely what it means - vice truly deeply understanding a theorem; what it does, where it goes, came from, and is going to be.

Moreover knowing a theorem implies I could ask you anytime to go the blackboard and proof it - while being able to explain in painful detail the necessity of every line in the proof.

They are different understandings my friend. And my advice is to make sure the later is the priority not the former.

Mathwonk said in that dense nugget these things. Stop and think about it friend. Truly, stop, and walk into the unknown without the aid of a book. That is a real scholar. A real scholar does not consume book after book after book without spending inifinitely more time in between books and pages and theorems looking at the depths of what was just been said.

So here's my 2 advices in short for you.

1) For every theorem you read, spend at a minimal 1 hour trying to do something with that theorem - without the aid of a book. Just a pencil and a blank piece of paper. Do something with it and then it yours for life. But don't just do trivial things or simple things, really truly take it to the depths.

2) Get in touch with the philosophy of math and physics. This is where you should spend your next dollars on dover books. Here you will find true power in mathematics and physics. Not knowing and not being able to communicate and think about the philosophy of these subjects relegates you to a mere foot soldier of those who can. Not the place for a mind like yours.

Answers to your questions.

My advice, take no classes. That's right, none. You don't need them until you're in college, and you are truly squabbling your intelligence if you do take them. Many many many many students get caught in this "prestige" game. You've been snared in at a young age.

It's been awhile since I was in 10th grade, but I thought I remember getting my driving permit. I speculate that you've skipped a grade or two. Which in of itself is a great ticket.

Here's another reason not to take these classes. No one is impressed. You as a "young'n" are impressed, we who are elder are not. If you wish to impress us, do what I said earlier and take one of these subjects to its depths on your own time and learn to enjoy your youth. Moreover, it would be more impressive if you studied something else other than hard science and mathematics.

But at the days end, you're a fool to try to impress other fools. Indeed you should seriously only do any of this for yourself. Why not then truly take these subject to their depths?

Question: How many languages do you know natively?

That is the point of this whole post. It's great you can say "what's up" in 20 languages and can order food in 10. But you still only speak then 1 language. The point is to actually take the time and truly learn, to the native level a language or a single subject of mathematics. Then and only then will you have impressed someone. Otherwise you only fool yourself my friend.

Dover books.

Dover books are great, but they only go so far. They're usually (always) outdated works. Not to mention they are often written in ancient tongues. But they're a great place to start.

Fundamental Concepts of Algebra by Bruce E. Meserve (Not what you think)

Challenging Problems in Algebra by Alfred S. Posamentier and Charles T. Salkind

Problems in Group Theory by John D. Dixon

The next two are highly recommended for you young friend.

Introduction to Mathematical Philosophy by Bertrand Russell (5 stars on a 4 star scale)

Physics and Philosophy by Sir James Jeans

The only one of these I have honestly broached but not finished is Russell's. Some I have tried problems out of, but they are very difficult. But that IS the point. Challenge yourself and forget trying to impress people with what college courses you've taken online early.

Do yourself a huge favor now while it is still easy, and ignore this game of prestige and who has done what - and truly learn something on your own. I believe you can do it. You sound like you believe you can do it. Why aren't you doing it?

My concluding remark is this, and please listen up:

If you, Fizix, are destined to be a great mind (like Einstein), then the courses you suggest and the books you mentioned are worthless. They are absolute garbage and with bare minimal resources you will find a way to surmount the heights of great mountains on your own.

I say, you sound like you might just be one of those minds, so then, get started with doing this. Take a single subject to its depths and take the long haul up a mountain to your throne.

And a final note. It took Mathwonk one minute to type that dense nugget. It took me over an hour to compose this post. So too is the way of all theorems in mathematics.

 

[^_^physicist]

I hate to take math_owen down the knees; however, I feel that he misses one point: You haven't tasted enough of each subject superfically enough to even begin to understand where to start for an indepth, mature view of the work.

For instance, I am sure you took a euclidean geometry class; however, I very much doubt you understand exactly where a large number of those properties come from; nor, do you understand where to start in non-euclidean (I.E. non-flat space) geometries.

Before you can take the more advanced lesson, I believe it is necessary to establish some basic ground work, by gaining a superfical understanding of the subjects.

I would seriously consister consulting a group of instuctors and administrators and creating a survey course of the material ncessary for a undergraduate/beginning graduate level course on quantum physics, if that is your interest.