What maths in preparation for quantum mechanics

[NeilB]

I would like to know what mathematics should be studied in preparation for understanding quantum mechanics. I would preferably like to know the appropriate sequence of topics I should study and the level to which they should be studied in order to get a grasp of this discipline. It might be that a list of physics topics may need to be included in the sequence as well.
I am a chemistry teacher of A level students of 20 years standing and I feel the need to undestand the theoretical basis of the chemistry I teach.

 

[haael]

Linear algebra, vector analysis, group theory - obvious.

Tensor algebra - to understand relativity, which is connected to QM to some extent.

I had been underestimating topology for very long, until I realized that it greatly helps in understanding mathematical apparatus used in QM, mainly the group theory.

For me it was very important to understand the idea of noncommutativity. It appears in many theories: noncommutative algebra, noncommutative geometry and so on. The transition from commutative to noncommutative theory is the core of quantization, so it was the single most important topic in my learning of QM. The moment I understood this, I could say that I knew QM.

 

[The_Duck]

Linear algebra and Fourier analysis (on top multivariable calculus + differential equations) are what underlie undergrad-level quantum mechanics, I think. Group theory if you want a really deep understanding.

I think relativity is not at all connected unless you get into QFT?

 

[Fredrik]

This is a part of what I said in another thread:

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.

Get yourself a linear algebra book (Axler has to be the best choice for this) and make sure that you at least understand the topics I mentioned. Also, get Isham's QM book. It's more about how to understand the theory and less about how to calculate stuff than other textbooks. (You should probably get a standard introductory text as well, e.g. Griffiths).

This answer is based on the assumption that you want to understand quantum mechanics, and is OK with not really understanding infinite-dimensional vector spaces. (Linear algebra is about finite-dimensional vector spaces). If you really want to understand the mathematics of QM, you're going to have to spend years studying topology, functional analysis, differential geometry and representation theory.

I see that Fourier analysis was mentioned above. You only need to understand the concept "Fourier transform", and I think most intro books cover that, at least non-rigorously.

 

[haael]

I think relativity is not at all connected unless you get into QFT?

Even the most basic QM books I have read mention relativistic versions of Schrödinger equation.

 

[Daverz]

I would like to know what mathematics should be studied in preparation for understanding quantum mechanics. I would preferably like to know the appropriate sequence of topics I should study and the level to which they should be studied in order to get a grasp of this discipline. It might be that a list of physics topics may need to be included in the sequence as well.
I am a chemistry teacher of A level students of 20 years standing and I feel the need to undestand the theoretical basis of the chemistry I teach.

I think some people here are confusing "how much math do I need to start" with "how much math do I need for a sophisticated understanding".

Calculus through vector calculus, introductory linear algebra (with matrices), and introductory ordinary differential equations (with Fourier series) should do. Most intro QM books cover the rest of the math needed.

I hope you kept your calculus-based physics text for reviewing things like angular momentum, the harmonic oscillator, and motion under an inverse-square force law.

The theory of radiation from atoms will require some familiarity with Maxwell's Equations in differential operator form and the vector potential, but this is usually a late chapter in most intro QM books.

You can probably also get by without much relativity. If a relativistic formula is used, you can probably take it "on faith" without serious damage to the "plot". It rarely goes beyond the energy-momentum relation. However, you might want to look for a text on "modern physics" that includes relativity and basic QM.

 

[Brown Arrow]

i just took a course its basically an into in QM... i had to do 1&2 year calc and linear algebra.. and differential equation...

 

[Fredrik]

Why does everyone say differential equations? There's just one differential equation in the theory, and the QM book will tell you how to solve it for each set of boundary conditions the author considers relevant. (Note: The book by Isham that I recommended doesn't do these things. That's why I added that the OP might want to get a standard introductory text as well).

You need stuff from calculus, but only a small part of it. It's not like you need to be able to integrate weird combinations of elementary functions, or to determine if a given series is convergent. You just need to understand the definitions.

When you get to the concept of "spin", you need to understand the relationship between linear operators and matrices (i.e. the first quote in this post). It takes minutes to learn, but most people skip it anyway. I have no idea why. Edit: No wait, I do know. It's because bad textbooks on linear algebra define the term "linear" on page 300 instead of on page 3.

 

[Daverz]

It's true that you probably don't need much beyond simple ordinary differential equations.

 

[G01]

For a standard undergraduate preparation, The standard calculus sequence + linear algebra + some Differential equations will get you through the course.

For an intro grad course at the level of Sakurai:

I would add some basic knowledge about group theory to the list. A working knowledge of special functions, specifically, the spherical harmonics will help.

Sakurai covers Tensor operators in his book, but it may help to have an external reference for stuff like spherical tensors, Wigner-Eckhart Theorem, etc. At least I found Sakurai's coverage of this material to be lacking.

 

[Cruikshank]

"Why does everyone say differential equations? There's just one differential equation in the theory,"
No, there is one *template* for differential equations. Change the potential and you're not just changing "boundary conditions." The solutions for the square well are different from those of a harmonic oscillator are different from those for an inverse square law because they are solutions to *different differential equations.*
I pity any poor soul who tries to learn the needed differential equations from a physics book. I keep having to go to math books to undo damage done me by physics texts.