Optimizing Math Study for Quantum Mechanics and Beyond

[bobsmith76]

These are the contents of the books I'm going to be reading to prepare myself for Quantum Mechanics. I was wondering if there any chapters that are not really necessary to learn. For instance, in studying Calculus it seems unlikely to me that infinite series will every be useful, though I might be wrong. I would like to eventually try Particle Physics, Nuclear Physics and Quantum Field Theory and a few basic of String Theory, so if there is some math in these books that is not needed for QM but is needed for the other physics then I should study that too. All this is for my own personal enjoyment and I'm studying it not because I want to be a physicist but because I want to understand the math behind the Big Bang, Cosmology, the Multiverse and the Fine-Tuning Argument.

Also I realize that more math is needed for physics beyond QM. I'm not worried about that now.

 

[homeomorphic]

These are the contents of the books I'm going to be reading to prepare myself for Quantum Mechanics. I was wondering if there any chapters that are not really necessary to learn.

There are too many chapters there that are unnecessary to learn to even list them all. Actually, to some extent, it's a matter of taste or what you want to do with it. I'm not sure the books you are looking at are going to be a very effective way to learn it. Maybe try Susskind's lectures.

For instance, in studying Calculus it seems unlikely to me that infinite series will every be useful, though I might be wrong.

You couldn't be more wrong. Infinite series are so basic, it's not even funny. For example, quantum field theory is all based on perturbation theory, which, in turn, is based on Taylor series.

All this is for my own personal enjoyment and I'm studying it not because I want to be a physicist but because I want to understand the math behind the Big Bang, Cosmology, the Multiverse and the Fine-Tuning Argument.

I suggest Penrose's book, the Road to Reality. It's a good study plan for that sort of thing. It does talk about a lot of that stuff, but you'll need to read other things to understand it. But, you should realize what you are talking about is basically almost becoming a physicist. If you're entertained by it, by all means, go for it.

Also I realize that more math is needed for physics beyond QM. I'm not worried about that now.

Maybe you should. More math can help to understand things.

 

[intwo]

If you're mostly interested in the physics, I recommend reading through a mathematical methods book like Mathematical Methods in the Physical Sciences by Boas. You'll experience a good chunk of the mathematic used in upper level undergraduate physics courses.

 

[bobsmith76]

I'm still looking for a good answer to this question. I dont' really want to get a book on math for physicists because I can only read math books if there is a solution manual. Maybe I could get one of those math books for the physical sciences and just look at what subjects are covered then try to find those subjects in the math books.

 

[homeomorphic]

I dont' really want to get a book on math for physicists because I can only read math books if there is a solution manual.

That's a very serious limitation. I'm not sure you are going to get very far if you insist on that. I never even pay attention to solutions or whether there are any, so I can't help with that.

I guess I can say chapters 1-5 of that linear algebra book are what you really need to know.

4, 5, 9-11 in the second book. I wouldn't do more than that. I'm not sure if I would do the 3rd book at all, but maybe chapters 1-5, there.

 

[bobsmith76]

That's a very serious limitation. I'm not sure you are going to get very far if you insist on that. I never even pay attention to solutions or whether there are any, so I can't help with that.

I'm just starting out. Once I get used to reading math books, hopefully the problem will go away. It's the same with any other foreign language. You start out reading the original text and then the translation, eventually you get better and better and you can read it on your own.

 

[jgens]

I'm just starting out. Once I get used to reading math books, hopefully the problem will go away. It's the same with any other foreign language. You start out reading the original text and then the translation, eventually you get better and better and you can read it on your own.

I know people have varying opinions on solutions manuals, but I would argue that a solutions manual can only hinder your mathematical development. To be frank, if you absolutely need a solutions manual, that is a sign you are not fully understanding the material you're covering; and hence, you're likely not prepared to move forward with your mathematical studies.

 

[Jorriss]

For instance, in studying Calculus it seems unlikely to me that infinite series will every be useful, though I might be wrong.

Anything covered in a mathematical methods book or in a lower division calculus or diff eqs class is useful or else why teach it (basically)? There are things that can be skipped and won't kill you, like that section on the fast Fourier transform, but topics in those classes are chosen by their utility.

Power series are, outside of the notion of differentiation and integration, very likely the most important things you learn in calculus. At this point, many functions such as the exponential, are equivalent to their power series in how I think about them because the power series is so incredibly useful.

 

[Vanadium 50]

Learning math by relying on a solution manual is like bodybuilding by watching someone else lift weights.

 

[twofish-quant]

All this is for my own personal enjoyment and I'm studying it not because I want to be a physicist but because I want to understand the math behind the Big Bang, Cosmology, the Multiverse and the Fine-Tuning Argument.

There's surprisingly *little* quantum mechanics around the big bang and cosmology. Except for the very, very early stages, thermodynamics is more important then QM in cosmology, and what QM there is can be "black boxed".

 

[bobsmith76]

Well, there's so much hooplah around QM that you can't really ignore. besides, it all also has real fundamental implications on the ultimate stuff of reality. i also want to find out how nature really ticks, i realize that's a cliche but I'm not in an inventive mood right now.

 

[Number Nine]

Learning math by relying on a solution manual is like bodybuilding by watching someone else lift weights.

In what sense? It's extremely useful, particularly early on, to be able to verify your solution to a problem. It's more like bodybuilding, and following each session by reviewing a video tape of your workout routine to verify that you have the correct form.

 

[jgens]

In what sense? It's extremely useful, particularly early on, to be able to verify your solution to a problem. It's more like bodybuilding, and following each session by reviewing a video tape of your workout routine to verify that you have the correct form.

I might agree if most students used solutions manuals this way, but most do not in my experience. Having a solutions manual often discourages students from spending hours and hours on the difficult problems (which is an important part of learning mathematics). And even if they do still struggle on the problems, the solutions manual discourages them from working on verifying independently that they in fact have a valid solution (which is another crucial part of mathematics).

Students in general aside, the OP mentioned that he cannot make it through math books without a solutions manual. This suggests a lack of understanding of the material. So in the OP's case, it would seem that he is using solutions manuals in a detrimental way.

 

[jgens]

besides, it all also has real fundamental implications on the ultimate stuff of reality.

Although this is really beside the point, I think that it is worth mentioning:

It's important to realize that quantum mechanics--as well as any other theory put forth in physics--is just a model for 'reality'; it is empirically a very good model for phenomenon on the quantum scale, but it is still a model nonetheless. And honestly, I don't think that any sane physicist would tell you that quantum mechanics is a perfect description of the world (even on the quantum scale).

 

[espen180]

You can never learn too much linear algebra!

http://math.stackexchange.com/questions/43974/importance-of-linear-algebra

 

[bobsmith76]

the OP mentioned that he cannot make it through math books without a solutions manual. This suggests a lack of understanding of the material. So in the OP's case, it would seem that he is using solutions manuals in a detrimental way.

Learning math without a teacher or without a SM is like learning french without a dictionary and a grammar. You can't rely on the text because that's written in French too since mathematicians too often assume that the reader has a high level of knowledge. There is no difference between listening to a teacher or reading a SM. You learn the stuff first, then you gradually get to the point where you can do it on your own. It's the same with French, you read the English translations first, then you gradually get to the point where you don't need the translations. I find that I can learn the material three times faster if I have a SM than if I don't.

 

[clope023]

Learning math by relying on a solution manual is like bodybuilding by watching someone else lift weights.

You'd be surprised how much of a weightlifters training is watching and analyzing the technique of other lifters on video.

 

[jgens]

Learning math without a teacher or without a SM is like learning french without a dictionary and a grammar.

I disagree wholeheartedly here. The way I learned math (starting at the level of calculus) was by sitting down with a book and by working through all of the theorems and exercises without solutions manuals. So it certainly is possible to learn math this way.

In any case, I know my experience with learning math this way isn't unique either; all of the best math students I know started learning this way from a young age.

You can't rely on the text because that's written in French too since mathematicians too often assume that the reader has a high level of knowledge.

This is a sign that you lack the mathematical maturity for the particular text.

There is no difference between listening to a teacher or reading a SM.

Again, I disagree wholeheartedly. Listening to a teacher is akin to reading the proofs of whatever theorems are presented in the text. Reading a solutions manual is like asking the smart kid in your class for the answers to your homework problems.

I find that I can learn the material three times faster if I have a SM than if I don't.

Your goal should be to really learn the material, rather than to cover the greatest amount of material you can in a short period of time. Really learning and understanding mathematics comes from practice and many hours spent taking wrong turns. That is how you develop an intuition and how you learn to attack the more challenging problems.

 

[micromass]

Learning math without a teacher or without a SM is like learning french without a dictionary and a grammar. You can't rely on the text because that's written in French too since mathematicians too often assume that the reader has a high level of knowledge. There is no difference between listening to a teacher or reading a SM. You learn the stuff first, then you gradually get to the point where you can do it on your own. It's the same with French, you read the English translations first, then you gradually get to the point where you don't need the translations. I find that I can learn the material three times faster if I have a SM than if I don't.

Certainly, you can learn it faster. But is your learning effective??
The best way to learn is by trying things yourself without help. Only checking the final answer should be allowed. And perhaps letting somebody else (on this forum for example) check out the form once in a while.
But one should be able to do things without solution manual. It only hinders your learning.

 

[AlephZero]

You need to decide what your objective is. Do you want to get a good grade, assuming the test questions will be pretty much the same as the questions in the book, or do you want to learn the subject?

I'm not moralizing about this - it's your choice. The solution manual is a very good tool for the first objective, but a very bad one for the second IMO.

 

[twofish-quant]

Well, there's so much hooplah around QM that you can't really ignore. besides, it all also has real fundamental implications on the ultimate stuff of reality. i also want to find out how nature really ticks, i realize that's a cliche but I'm not in an inventive mood right now.

It turns out that most of the weird stuff about QM doesn't require much math to understand, and the things that do require a ton of math to get right involves the "non-weird" stuff. Typically most physics courses that involve QM first teach the math by applying it to "non-weird" stuff. Waves and springs.

 

[twofish-quant]

Students in general aside, the OP mentioned that he cannot make it through math books without a solutions manual. This suggests a lack of understanding of the material. So in the OP's case, it would seem that he is using solutions manuals in a detrimental way.

I don't think it is. The way that I learned most math is to go through worked solutions. Once I understand the worked solutions, then I apply it to other problems.

 

[jgens]

I don't think it is. The way that I learned most math is to go through worked solutions. Once I understand the worked solutions, then I apply it to other problems.

I would argue that this is the purpose of the proofs of theorems in texts and the few worked examples texts usually include.

 

[nucl34rgg]

You need calculus, differential equations, statistics and linear algebra to understand an average undergraduate level quantum mechanics course.

 

[clope023]

I would argue that this is the purpose of the proofs of theorems in texts and the few worked examples texts usually include.

I agree with twofish here.

Those minimal theorem proofs and textbook examples aren't always sufficient to the task.

 

[clope023]

You need calculus, differential equations, statistics and linear algebra to understand an average undergraduate level quantum mechanics course.

I would say partial differential equations and Fourier analysis as well.

 

[homeomorphic]

A solutions manual won't hurt you if you use it the right way, as in only as a last resort once you have already checked your work for yourself. Even if you use it to work through example problems along with it, I don't think it will hurt, provided that you practice plenty of problems completely on your own, and practice checking and deciding for yourself that your solution is correct as well.

I always considered solutions manuals beneath my dignity, personally. If I am really hopelessly stuck on a problem, then I might want a hint, but that's it.

 

[zahero_2007]

You should train yourself in linear algebra ,fourier transforms and partial differential equations well .they are very important