What Is the Math Method in Quantum Mechanics?

[romsofia]

I recently emailed a professor at the local university on what math would be necessary to understand quantum mechanics (other than calculus, differential equations and linear algebra) and he replied with "It is good to be familiar with the materials in the Math Method, but it is not necessary." Anyone know what that is? I don't want to seem stupid and reply asking what it is, but after searching google I found nothing :x

Thanks for the help guys!

 

[eri]

Math methods is a typically 2-semester class physics majors take that covers many areas of math without having to take a full class in it. I remember it included calc III, some linear algebra, differential equations. This is the most commonly used book for the class, Mathematical Methods by Boas.

https://www.amazon.com/dp/0471044091/?tag=pfamazon01-20

 

[fluidistic]

Another book commonly used is the one of Arfken. See http://books.google.com.ar/books?id...ge&q=mathematical methods for physics&f=false and specially the first pages to see the contents of the book.
At my university the course is a 1 semester one. It covers linear algebra (up to Jordan normal form), tensors, ODE, PDE, special functions and probably more but I didn't take the course yet.

 

[romsofia]

Thanks guys!

 

[Fredrik]

You can get very far on a solid understanding of a small range of topics in linear algebra. (linear independence, bases, the relationship between linear operators and matrices, eigenvectors, inner products, orthonormal bases). You need very little from calculus. For example, you need to understand what an integral is, but you don't have to know how to integrate weird combinations of elementary functions. I would say that you don't need anything from differential equations. The QM book will tell you what you need to know.

There was a "The mathematical methods of physics" class given at the physics department of my university, but it was useless for someone who just wants to understand QM (and even more useless for someone who wants to understand general relativity). I'm not sure who it was supposed to be useful for. A book like Arfken covers a lot of stuff that you really don't need, and isn't a very good place to learn the things you do need. You're better off studying a good linear algebra book. (My favorite is Axler).

I should also say that there's an enormous difference between "the mathematics you need to understand QM" and "the mathematics of QM". I would say that you need the latter to really understand QM, but most physicists know very little about it. To learn the mathematics of QM, you need at least one, probably two, courses on functional analysis. (One of them should include operator algebras). Unfortunately you can't even begin to read most books on functional analysis if you don't know general topology (a.k.a. point set topology), and you would find a topology book really hard if you haven't taken an advanced course in real analysis first. (Something like Rudin's "Principles of mathematical analysis").

I've been told that the functional analyis book by Kreyszig makes things much easier for its readers, so maybe it doesn't have to be as hard as it was for me to get started with these things.

So you would have to take about 4 additional math courses that physics students don't normally take to understand the mathematics of QM. To understand the mathematics of quantum field theories, you also need one, probably two, courses on differential geometry (one of them should include fiber bundles), and a course on representation theory. So that's 3 more. Of course, you don't need any of those courses to be able to take a course in quantum field theory. There's always a big difference between a branch of physics and the mathematics associated with it.

 

[lisab]

I recently emailed a professor at the local university on what math would be necessary to understand quantum mechanics (other than calculus, differential equations and linear algebra) and he replied with "It is good to be familiar with the materials in the Math Method, but it is not necessary." Anyone know what that is? I don't want to seem stupid and reply asking what it is, but after searching google I found nothing :x

Thanks for the help guys!

I think other posters are correct, he probably meant Mathematical Methods.

But the best thing to do would be write him back with something like, "Thanks, I appreciate your help. And just to be sure: by Math Method, do you mean the material covered in Phys xxx?" with the appropriate course number.

 

[romsofia]

You can get very far on a solid understanding of a small range of topics in linear algebra. (linear independence, bases, the relationship between linear operators and matrices, eigenvectors, inner products, orthonormal bases). You need very little from calculus. For example, you need to understand what an integral is, but you don't have to know how to integrate weird combinations of elementary functions. I would say that you don't need anything from differential equations. The QM book will tell you what you need to know.

There was a "The mathematical methods of physics" class given at the physics department of my university, but it was useless for someone who just wants to understand QM (and even more useless for someone who wants to understand general relativity). I'm not sure who it was supposed to be useful for. A book like Arfken covers a lot of stuff that you really don't need, and isn't a very good place to learn the things you do need. You're better off studying a good linear algebra book. (My favorite is Axler).

I should also say that there's an enormous difference between "the mathematics you need to understand QM" and "the mathematics of QM". I would say that you need the latter to really understand QM, but most physicists know very little about it. To learn the mathematics of QM, you need at least one, probably two, courses on functional analysis. (One of them should include operator algebras). Unfortunately you can't even begin to read most books on functional analysis if you don't know general topology (a.k.a. point set topology), and you would find a topology book really hard if you haven't taken an advanced course in real analysis first. (Something like Rudin's "Principles of mathematical analysis").

I've been told that the functional analyis book by Kreyszig makes things much easier for its readers, so maybe it doesn't have to be as hard as it was for me to get started with these things.

So you would have to take about 4 additional math courses that physics students don't normally take to understand the mathematics of QM. To understand the mathematics of quantum field theories, you also need one, probably two, courses on differential geometry (one of them should include fiber bundles), and a course on representation theory. So that's 3 more. Of course, you don't need any of those courses to be able to take a course in quantum field theory. There's always a big difference between a branch of physics and the mathematics associated with it.

Thanks for the reply, I really appreciate the help!

The course description is: This is a course in which both wave mechanics and matrix mechanics are developed and applied to selected problems in atomic physics. Particular topics include solutions to the time-independent Schršdinger equation for the hydrogen atom, tunneling, the harmonic oscillator, electron spin and approximation methods.
And the book is: Introduction to Quantum Mechanics by Griffiths

What math would you suggest would be needed?

I think other posters are correct, he probably meant Mathematical Methods.

But the best thing to do would be write him back with something like, "Thanks, I appreciate your help. And just to be sure: by Math Method, do you mean the material covered in Phys xxx?" with the appropriate course number.

Thanks, I'll go search for the course number! Also, quick question about referring to a doctor... He has a hyphenated last name, but in his reply he used only one of them, should I use both parts of his last names when replying or just the one he used? I know this has to do with English, but I don't want to mess up :x

 

[Fredrik]

The course description is: This is a course in which both wave mechanics and matrix mechanics are developed and applied to selected problems in atomic physics. Particular topics include solutions to the time-independent Schršdinger equation for the hydrogen atom, tunneling, the harmonic oscillator, electron spin and approximation methods.
And the book is: Introduction to Quantum Mechanics by Griffiths

What math would you suggest would be needed?

That's the type of course I had in mind when I wrote the first paragraph in my previous post. I guess I should have mentioned that you need to understand complex numbers as well.

 

[ZapperZ]

The suggestion on Mary Boas's "Mathematical Methods in the Physical Science" is highly seconded. In fact, if you do a search on it, you'll find at least a couple of threads recommending that text. If you are an undergraduate, and just finished your 2nd year of college, this is a very readable book and something you can pick up on for independent study. It doesn't require as much mathematical sophistication as, say, Afken text.

As with most areas of physics, one can't say what mathematics one would need. You just never know what tools is necessary for a particular area of physics. Therefore, you need to learn a wide range of mathematics. That's why a comprehensive survey of various mathematical methods such as those covered in Boas' text is the way to go.

Zz.